000 final 3264

Transcript

1 3264 & All That Intersection Theory in Algebraic Geometry c David Eisenbud and Joe Harris

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3 Contents Preface xv Introduction 1 Chapter 0 13 Chapter 1 Introducing the Chow ring The goal of intersection theory . . . . . . . . . . . . . 14 1.1 . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Chow ring 15 Cycles . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1 1.2.2 Rational equivalence and the Chow group . . . . . . 16 Transversality and the Chow ring . . . . . . . . . . 1.2.3 17 1.2.4 19 The moving lemma . . . . . . . . . . . . . . . . . . Some techniques for computing the Chow ring 22 1.3 . . . 1.3.1 The fundamental class . . . . . . . . . . . . . . . . 22 Rational equivalence via divisors . . . . . . . . . . 1.3.2 22 1.3.3 Affine space . . . . . . . . . . . . . . . . . . . . . 24 1.3.4 Mayer–Vietoris and excision . . . . . . . . . . . . . 24 1.3.5 Affine stratifications . . . . . . . . . . . . . . . . . 26 1.3.6 Functoriality . . . . . . . . . . . . . . . . . . . . . 28 Dimensional transversality and multiplicities . . . . 1.3.7 31 1.3.8 The multiplicity of a scheme at a point . . . . . . . 33 The first Chern class of a line bundle . . . . . . . . . 37 1.4 The canonical class . . . . . . . . . . . . . . . . . . 39 1.4.1 1.4.2 The adjunction formula . . . . . . . . . . . . . . . . 40 1.4.3 Canonical classes of hypersurfaces and complete in- tersections . . . . . . . . . . . . . . . . . . . . . . 41 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Exercises Chapter 2 First examples 43 n The Chow rings of P 2.1 and some related varieties . . 44 ́ 2.1.1 B ezout’s theorem . . . . . . . . . . . . . . . . . . . 46

4 vi Contents Degrees of Veronese varieties . . . . . . . . . . . . 48 2.1.2 49 Degree of the dual of a hypersurface . . . . . . . . . 2.1.3 Products of projective spaces . . . . . . . . . . . . . 51 2.1.4 2.1.5 Degrees of Segre varieties . . . . . . . . . . . . . . 52 The class of the diagonal . . . . . . . . . . . . . . . 53 2.1.6 54 2.1.7 The class of a graph . . . . . . . . . . . . . . . . . 1 P . . . . . . . . . . . . 55 Nested pairs of divisors on 2.1.8 n The blow-up of at a point . . . . . . . . . . . . . 56 2.1.9 P 61 2.1.10 Intersection multiplicities via blow-ups . . . . . . . Loci of singular plane cubics 2.2 62 . . . . . . . . . . . . . . Reducible cubics . . . . . . . . . . . . . . . . . . . 2.2.1 64 2.2.2 Triangles . . . . . . . . . . . . . . . . . . . . . . . 65 2.2.3 Asterisks . . . . . . . . . . . . . . . . . . . . . . . 65 The circles of Apollonius . . . . . . . . . . . . . . . . . 66 2.3 2.3.1 What is a circle? . . . . . . . . . . . . . . . . . . . 66 Circles tangent to a given circle . . . . . . . . . . . 67 2.3.2 2.3.3 68 Conclusion of the argument . . . . . . . . . . . . . 2.4 Curves on surfaces . . . . . . . . . . . . . . . . . . . . 68 2.4.1 The genus formula . . . . . . . . . . . . . . . . . . 69 2.4.2 The self-intersection of a curve on a surface . . . . . 70 3 Linked curves in . . . . . . . . . . . . . . . . . 70 2.4.3 P The blow-up of a surface . . . . . . . . . . . . . . . 2.4.4 72 2.4.5 Canonical class of a blow-up . . . . . . . . . . . . . 73 The genus formula with singularities . . . . . . . . 74 2.4.6 Intersections on singular varieties 2.5 75 . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.6 Introduction to Grassmannians and Chapter 3 3 lines in P 85 3.1 Enumerative formulas . . . . . . . . . . . . . . . . . . 85 3.1.1 What are enumerative problems, and how do we solve 86 them? . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 The content of an enumerative formula . . . . . . . 87 Introduction to Grassmannians . . . . . . . . . . . . . 89 3.2 ̈ The Pl ucker embedding . . . . . . . . . . . . . . . 3.2.1 90 3.2.2 92 Covering by affine spaces; local coordinates . . . . . 3.2.3 Universal sub and quotient bundles . . . . . . . . . 95 3.2.4 The tangent bundle of the Grassmannian . . . . . . 96 3.2.5 The differential of a morphism to the Grassmannian 99

5 Contents vii Tangent spaces via the universal property . . . . . . 100 3.2.6 G The Chow ring of 102 .1;3/ 3.3 . . . . . . . . . . . . . . . . .1;3/ . . . . . . . . . . . . . . G Schubert cycles in 3.3.1 102 3.3.2 Ring structure . . . . . . . . . . . . . . . . . . . . . 105 3 3.4 . . . . . . . . . . . . . . . . . . 110 P Lines and curves in How many lines meet four general lines? . . . . . . 110 3.4.1 3.4.2 Lines meeting a curve of degree d . . . . . . . . . . 111 Chords to a space curve . . . . . . . . . . . . . . . 3.4.3 113 3.5 . . . . . . . . . . . . . . . . . . . . . . . . 115 Specialization Schubert calculus by static specialization . . . . . . 115 3.5.1 117 3.5.2 Dynamic projection . . . . . . . . . . . . . . . . . 3.5.3 Lines meeting a curve by specialization . . . . . . . 120 3.5.4 Chords via specialization: multiplicity problems . . 121 Common chords to twisted cubics via specialization 3.5.5 122 3 P 3.6 . . . . . . . . . . . . . . . . . 122 Lines and surfaces in Lines lying on a quadric . . . . . . . . . . . . . . . 122 3.6.1 3.6.2 Tangent lines to a surface . . . . . . . . . . . . . . . 123 3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Chapter 4 Grassmannians in general 131 Schubert cells and Schubert cycles . . . . . . . . . . . 4.1 132 134 Schubert classes and Chern classes . . . . . . . . . 4.1.1 4.1.2 135 The affine stratification by Schubert cells . . . . . . 4.1.3 Equations of the Schubert cycles . . . . . . . . . . . 138 Intersection products 4.2 139 . . . . . . . . . . . . . . . . . . . 4.2.1 139 Transverse flags . . . . . . . . . . . . . . . . . . . Intersections in complementary dimension . . . . . 141 4.2.2 Varieties swept out by linear spaces . . . . . . . . . 144 4.2.3 4.2.4 Pieri’s formula . . . . . . . . . . . . . . . . . . . . 145 4.3 Grassmannians of lines . . . . . . . . . . . . . . . . . . 147 4.4 Dynamic specialization 150 . . . . . . . . . . . . . . . . . . Young diagrams 152 4.5 . . . . . . . . . . . . . . . . . . . . . . Pieri’s formula for the other special Schubert classes 4.5.1 154 4.6 Linear spaces on quadrics . . . . . . . . . . . . . . . . 155 4.7 Giambelli’s formula . . . . . . . . . . . . . . . . . . . . 157 4.8 . . . . . . . . . . . . . . . . . . . . . . 159 Generalizations 4.8.1 Flag manifolds . . . . . . . . . . . . . . . . . . . . 159 4.8.2 Lagrangian Grassmannians and beyond . . . . . . . 160 4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6 viii Contents Chern classes 165 Chapter 5 Introduction: Chern classes and the lines on a cubic 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . surface 165 167 . . . . . . . . . . . . . . Characterizing Chern classes 5.2 Constructing Chern classes . . . . . . . . . . . . . . . 170 5.3 5.4 172 The splitting principle . . . . . . . . . . . . . . . . . . Using Whitney’s formula with the splitting principle 173 5.5 5.5.1 Tensor products with line bundles . . . . . . . . . . 174 5.5.2 Tensor product of two bundles . . . . . . . . . . . . 176 Tautological bundles 5.6 177 . . . . . . . . . . . . . . . . . . . 5.6.1 177 Projective spaces . . . . . . . . . . . . . . . . . . . Grassmannians . . . . . . . . . . . . . . . . . . . . 5.6.2 178 5.7 Chern classes of varieties . . . . . . . . . . . . . . . . . 179 5.7.1 Tangent bundles of projective spaces . . . . . . . . . 179 Tangent bundles to hypersurfaces . . . . . . . . . . 179 5.7.2 The topological Euler characteristic . . . . . . . . . 5.7.3 180 First Chern class of the Grassmannian . . . . . . . . 183 5.7.4 Generators and relations for A.G.k;n// . . . . . . . 183 5.8 5.9 Steps in the proofs of Theorem 5.3 . . . . . . . . . . . 187 5.9.1 Whitney’s formula for globally generated bundles . . 187 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 191 193 Lines on hypersurfaces Chapter 6 What to expect . . . . . . . . . . . . . . . . . . . . . . . 194 6.1 Definition of the Fano scheme . . . . . . . . . . . . 196 6.1.1 6.2 Fano schemes and Chern classes 198 . . . . . . . . . . . . Counting lines on cubics . . . . . . . . . . . . . . . 199 6.2.1 6.3 . . . . . 201 Definition and existence of Hilbert schemes 6.3.1 A universal property of the Grassmannian . . . . . . 201 6.3.2 A universal property of the Fano scheme . . . . . . 203 6.3.3 The Hilbert scheme and its universal property . . . . 203 Sketch of the construction of the Hilbert scheme . . 205 6.3.4 6.4 . . . . . 208 Tangent spaces to Fano and Hilbert schemes Normal bundles and the smoothness of the Fano scheme 208 6.4.1 6.4.2 First-order deformations as tangents to the Hilbert scheme . . . . . . . . . . . . . . . . . . . . . . . . 212 6.4.3 Normal bundles of k -planes on hypersurfaces . . . . 219 6.4.4 The case of lines . . . . . . . . . . . . . . . . . . . 223 6.5 Lines on quintic threefolds and beyond . . . . . . . . 227

7 Contents ix 6.6 The universal Fano scheme and the geometry of fam- . . . . . . . . . . . . . . . . . . . . . . . . . ilies of lines 229 233 Lines on the quartic surfaces in a pencil . . . . . . . 6.6.1 6.7 . . . . . . . . . . 234 Lines on a cubic with a double point . . . . . . . . . . . 236 6.8 The Debarre–de Jong Conjecture 238 6.8.1 Further open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 6.9 Exercises Chapter 7 244 Singular elements of linear series 7.1 Singular hypersurfaces and the universal singularity 245 Bundles of principal parts . . . . . . . . . . . . . . . . 247 7.2 Singular elements of a pencil 7.3 . . . . . . . . . . . . . . 251 251 From pencils to degeneracy loci . . . . . . . . . . . 7.3.1 The Chern class of a bundle of principal parts . . . . 252 7.3.2 7.3.3 Triple points of plane curves . . . . . . . . . . . . . 256 Cones . . . . . . . . . . . . . . . . . . . . . . . . . 257 7.3.4 Singular elements of linear series in general . . . . . 258 7.4 Number of singular elements of a pencil . . . . . . . 259 7.4.1 7.4.2 Pencils of curves on a surface . . . . . . . . . . . . 260 7.4.3 The second fundamental form . . . . . . . . . . . . 262 r Inflection points of curves in P 7.5 . . . . . . . . . . . . . 265 7.5.1 Vanishing sequences and osculating planes . . . . . 266 ̈ Total inflection: the Pl 268 7.5.2 ucker formula . . . . . . . . . 272 The situation in higher dimension . . . . . . . . . . 7.5.3 Nets of plane curves . . . . . . . . . . . . . . . . . . . . 273 7.6 7.6.1 273 Class of the universal singular point . . . . . . . . . 7.6.2 274 The discriminant of a net of plane curves . . . . . . The topological Hurwitz formula 7.7 277 . . . . . . . . . . . . 7.7.1 Pencils of curves on a surface, revisited . . . . . . . 279 7.7.2 Multiplicities of the discriminant hypersurface . . . 280 7.7.3 282 Tangent cones of the discriminant hypersurface . . . 7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Chapter 8 Compactifying parameter spaces 289 8.1 Approaches to the five conic problem . . . . . . . . . 290 8.2 . . . . . . . . . . . . . . . . . . . . . . 293 Complete conics 8.2.1 Informal description . . . . . . . . . . . . . . . . . 293 8.2.2 Rigorous description . . . . . . . . . . . . . . . . . 296 8.2.3 Solution to the five conic problem . . . . . . . . . . 302

8 x Contents Chow ring of the space of complete conics . . . . . 306 8.2.4 . . . . . . . . . . . . . . . . . . . . Complete quadrics 309 8.3 310 8.4 . . . . . . . . . . . . . . . Parameter spaces of curves Hilbert schemes . . . . . . . . . . . . . . . . . . . 310 8.4.1 311 8.4.2 Report card for the Hilbert scheme . . . . . . . . . . 312 8.4.3 The Kontsevich space . . . . . . . . . . . . . . . . 316 8.4.4 Report card for the Kontsevich space . . . . . . . . 8.5 How the Kontsevich space is used: rational plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 321 8.6 323 Chapter 9 Projective bundles and their Chow rings Projective bundles and the tautological divisor class 323 9.1 324 9.1.1 Example: rational normal scrolls . . . . . . . . . . . . . . . . . . . . . . . . . . 327 9.2 Maps to a projective bundle . . . . . . . . . . . . 331 Chow ring of a projective bundle 9.3 9.3.1 k -plane over G .k;n/ . . . . . . . . . 335 The universal n 9.3.2 The blow-up of P along a linear space . . . . . . . 337 1 Nested pairs of divisors on P 339 revisited . . . . . . . 9.3.3 9.4 Projectivization of a subbundle . . . . . . . . . . . . . 340 341 Ruled surfaces . . . . . . . . . . . . . . . . . . . . 9.4.1 Self-intersection of the zero section . . . . . . . . . 343 9.4.2 . . . . . . . . . . . . . . . . . . 9.5 Brauer–Severi varieties 344 Chow ring of a Grassmannian bundle . . . . . . . . . 346 9.6 3 P 9.7 meeting eight lines . . . . . . . . . . . . 347 Conics in The parameter space as projective bundle . . . . . . 348 9.7.1 9.7.2 The class ı of the cycle of conics meeting a line . . . 349 8 The degree of ı 9.7.3 . . . . . . . . . . . . . . . . . . . 350 9.7.4 The parameter space as Hilbert scheme . . . . . . . 350 Tangent spaces to incidence cycles . . . . . . . . . . 9.7.5 352 Proof of transversality . . . . . . . . . . . . . . . . 354 9.7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 355 9.8 Chapter 10 Segre classes and varieties of linear spaces 362 10.1 . . . . . . . . . . . . . . . . . . . . . . . . 362 Segre classes 10.2 Varieties swept out by linear spaces . . . . . . . . . . 366 10.3 Secant varieties . . . . . . . . . . . . . . . . . . . . . . 367 10.3.1 Symmetric powers . . . . . . . . . . . . . . . . . . 367

9 Contents xi 369 10.3.2 Secant varieties in general . . . . . . . . . . . . . . . . . . . . 373 10.4 Secant varieties of rational normal curves 373 10.4.1 Secants to rational normal curves . . . . . . . . . . 10.4.2 Degrees of the secant varieties . . . . . . . . . . . . 375 10.4.3 Expression of a form as a sum of powers . . . . . . 376 . . . . . . . . . . . . . . . . . . . 377 10.5 Special secant planes 378 10.5.1 The class of the locus of secant planes . . . . . . . . 380 10.5.2 Secants to curves of positive genus . . . . . . . . . 10.6 Dual varieties and conormal varieties . . . . . . . . . 380 The universal hyperplane as projectivized cotangent 10.6.1 bundle . . . . . . . . . . . . . . . . . . . . . . . . . 382 10.7 . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Exercises 389 Chapter 11 Contact problems Lines meeting a surface to high order . . . . . . . . . 390 11.1 11.1.1 Bundles of relative principal parts . . . . . . . . . . 391 393 11.1.2 Relative tangent bundles of projective bundles . . . 11.1.3 Chern classes of contact bundles . . . . . . . . . . . 394 The case of negative expected dimension . . . . . . . 396 11.2 3 P . . . . . . . . . . . 397 11.2.1 Lines on smooth surfaces in 11.2.2 The flecnodal locus . . . . . . . . . . . . . . . . . . 400 11.3 Flexes via defining equations 401 . . . . . . . . . . . . . . 403 11.3.1 Hyperflexes . . . . . . . . . . . . . . . . . . . . . . 405 11.3.2 Flexes on families of curves . . . . . . . . . . . . . 11.3.3 Geometry of the curve of flex lines . . . . . . . . . 408 11.4 Cusps of plane curves . . . . . . . . . . . . . . . . . . . 409 410 11.4.1 Plane curve singularities . . . . . . . . . . . . . . . 413 11.4.2 Characterizing cusps . . . . . . . . . . . . . . . . . 11.4.3 Solution to the enumerative problem . . . . . . . . . 414 11.4.4 Another approach to the cusp problem . . . . . . . . 417 11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 420 Chapter 12 Porteous’ formula 426 12.1 Degeneracy loci 426 . . . . . . . . . . . . . . . . . . . . . . Porteous’ formula for 12.2 .'/ . . . . . . . . . . . . . . 429 M 0 12.3 Proof of Porteous’ formula in general . . . . . . . . . 430 12.3.1 Reduction to a generic case . . . . . . . . . . . . . 430 12.3.2 Relation to the case k D 0 . . . . . . . . . . . . . . 431 12.3.3 Pushforward from the Grassmannian bundle . . . . . 432

10 xii Contents Geometric applications . . . . . . . . . . . . . . . . . . 12.4 433 433 12.4.1 Degrees of determinantal varieties . . . . . . . . . . 12.4.2 Pinch points of surfaces . . . . . . . . . . . . . . . 436 12.4.3 Pinch points and the tangential variety of S . . . . . 439 12.4.4 Quadrisecants to rational curves . . . . . . . . . . . 440 . . . . . . . . . . . . . . . . . . . . . . . . . . 442 Exercises 12.5 Chapter 13 Excess intersections and the Chow ring of a blow-up 445 First examples . . . . . . . . . . . . . . . . . . . . . . . 447 13.1 447 13.1.1 The intersection of a divisor and a subvariety . . . . 3 P containing a curve . . . . . . . 449 13.1.2 Three surfaces in 13.2 Segre classes of subvarieties . . . . . . . . . . . . . . . 453 The excess intersection formula . . . . . . . . . . . . . 13.3 454 13.3.1 Heuristic argument for the excess intersection formula 456 13.3.2 Connected components versus irreducible components 458 4 13.3.3 Two surfaces in P containing a curve . . . . . . . . 459 13.3.4 Quadrics containing a linear space . . . . . . . . . . 460 13.3.5 The five conic problem . . . . . . . . . . . . . . . . 462 Intersections of hypersurfaces in general: Vogel’s ap- 13.3.6 464 proach . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Intersections in a subvariety . . . . . . . . . . . . . . . 465 13.4.1 Specialization to the normal cone . . . . . . . . . . 467 468 13.4.2 Proof of the key formula . . . . . . . . . . . . . . . Pullbacks to a subvariety . . . . . . . . . . . . . . . . . 469 13.5 13.5.1 The degree of a generically finite morphism . . . . . 470 13.6 The Chow ring of a blow-up . . . . . . . . . . . . . . . 471 13.6.1 The normal bundle of the exceptional divisor . . . . 472 13.6.2 Generators of the Chow ring . . . . . . . . . . . . . 473 3 13.6.3 Example: the blow-up of P along a curve . . . . . 473 13.6.4 Relations on the Chow ring of a blow-up . . . . . . 476 13.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Chapter 14 The Grothendieck Riemann–Roch theorem 481 14.1 The Riemann–Roch formula for curves and surfaces 481 14.1.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . 481 14.1.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 483

11 Contents xiii Arbitrary dimension . . . . . . . . . . . . . . . . . . . 14.2 484 484 14.2.1 The Chern character . . . . . . . . . . . . . . . . . 14.2.2 The Todd class . . . . . . . . . . . . . . . . . . . . 487 14.2.3 Hirzebruch Riemann–Roch . . . . . . . . . . . . . 488 14.3 Families of bundles . . . . . . . . . . . . . . . . . . . . 489 14.3.1 Grothendieck Riemann–Roch . . . . . . . . . . . . 489 3  S on G .1;3/ . . . 490 14.3.2 Example: Chern classes of Sym Application: jumping lines . . . . . . . . . . . . . . . . 493 14.4 1 P with given splitting type . . . 494 14.4.1 Loci of bundles on 2 P . . . . . . 497 14.4.2 Jumping lines of bundles of rank 2 on 14.4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . 501 Application: invariants of families of curves . . . . . 502 14.5 14.5.1 Example: pencils of quartics in the plane . . . . . . 504 506 14.5.2 Proof of the Mumford relation . . . . . . . . . . . . 14.6 Exercises 507 . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A The moving lemma 511 A.1 Generic transversality to a cycle . . . . . . . . . . . . 512 A.2 Generic transversality to a morphism 518 . . . . . . . . . Direct images, cohomology and base Appendix B 520 change B.1 Can you define a bundle by its fibers? . . . . . . . . . 520 Direct images . . . . . . . . . . . . . . . . . . . . . . . . 523 B.2 Higher direct images . . . . . . . . . . . . . . . . . . . 528 B.3 B.4 The direct image complex . . . . . . . . . . . . . . . . 533 B.5 Proofs of the theorems on cohomology and base change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 B.6 Exercises 540 . . . . . . . . . . . . . . . . . . . . . . . . . . Topology of algebraic varieties Appendix C 543 C.1 GAGA theorems . . . . . . . . . . . . . . . . . . . . . . 543 C.2 Fundamental classes and Hodge theory . . . . . . . . 544 C.2.1 Fundamental classes . . . . . . . . . . . . . . . . . 544 C.2.2 The Hodge decomposition . . . . . . . . . . . . . . 546 C.2.3 The Hodge diamond . . . . . . . . . . . . . . . . . 548 C.2.4 The Hodge conjecture . . . . . . . . . . . . . . . . 549 C.3 Comparison of rational equivalence with other cycle theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

12 xiv Contents 551 C.3.1 Algebraic equivalence . . . . . . . . . . . . . . . . C.3.2 Algebraic cycles modulo homological equivalence . 552 C.3.3 Numerical equivalence . . . . . . . . . . . . . . . . 552 C.3.4 Comparing the theories . . . . . . . . . . . . . . . . 553 C.4 The Lefschetz hyperplane theorem . . . . . . . . . . . 553 Applications to hypersurfaces and complete intersec- C.4.1 tions . . . . . . . . . . . . . . . . . . . . . . . . . . 554 C.4.2 Extensions and generalizations . . . . . . . . . . . . 557 C.5 The hard Lefschetz theorem and Hodge–Riemann bilinear relations . . . . . . . . . . . . . . . . . . . . . . 558 C.6 559 Chern classes in topology and differential geometry C.6.1 Chern classes and obstructions . . . . . . . . . . . . 560 561 C.6.2 Chern classes and curvature . . . . . . . . . . . . . Maps from curves to projective space 564 Appendix D D.1 What maps to projective space do curves have? . . . 565 D.1.1 The Riemann–Roch theorem . . . . . . . . . . . . . 567 D.1.2 Clifford’s theorem . . . . . . . . . . . . . . . . . . 569 D.1.3 Castelnuovo’s theorem . . . . . . . . . . . . . . . . 570 D.2 Families of divisors . . . . . . . . . . . . . . . . . . . . 571 D.2.1 The Jacobian . . . . . . . . . . . . . . . . . . . . . 571 D.2.2 Abel’s theorem . . . . . . . . . . . . . . . . . . . . 572 575 D.2.3 Moduli spaces of divisors and line bundles . . . . . The Brill–Noether theorem . . . . . . . . . . . . . . . 576 D.3 How to guess the Brill–Noether theorem and prove D.3.1 existence . . . . . . . . . . . . . . . . . . . . . . . 578 579 D.3.2 How the other half is proven . . . . . . . . . . . . . r 580 as a degeneracy locus . . . . . . . . . . . . . . . . W D.4 d D.4.1 The universal line bundle . . . . . . . . . . . . . . . 580 D.4.2 The evaluation map . . . . . . . . . . . . . . . . . . 582 D.5 Natural classes in the cohomology ring of the Jacobian 583 ́ D.5.1 Poincar 585 e’s formula . . . . . . . . . . . . . . . . . . D.5.2 Symmetric powers as projective bundles . . . . . . . 587 589 D.5.3 Chern classes from the symmetric power . . . . . . r W D.5.4 The class of . . . . . . . . . . . . . . . . . . . 590 d D.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 592 References 594 Index 602

13 Preface We have been working on this project for over ten years, and at times we have felt that we have only brought on ourselves a plague of locus. However, our spirits have been lightened, and the project made far easier and more successful than it would have been, by the interest and help of many people. First of all, we thank Bill Fulton, who created much of the modern approach to intersection theory, and who directly informed our view of the subject from the beginning. Many people have helped us by reading early versions of the text and providing comments and corrections. Foremost among these is Paolo Aluffi, who gave extensive and detailed comments; we also benefited greatly from the advice of Francesco Cavazzani and Izzet Co s ̧ kun. We would also thank Mike Roth and Stephanie Yang, who provided notes on the early iterations of a course on which much of this text is based, as well as students who contributed corrections, including Sitan Chen, Jun Hou Fung, Changho Han, Chi-Yun Hsu, Hannah Larson, Ravi Jagadeesan, Aaron Landesman, Yogesh More, Arpon Raksit, Ashvin Swaminathan, Arnav Tripathy, Isabel Vogt and Lynnelle Ye. Silvio Levy made many of the many illustrations in this book (and occasionally corrected our mathematical errors too!). Devlin Mallory then took over as copyeditor, and completed the rest of the figures. We are grateful to both of them for their many improvements to this text (and to Cambridge University Press for hiring Devlin!).

14 We are all familiar with the after-the-fact tone — weary, self-justificatory, aggrieved, apologetic — shared by ship captains appearing before boards of inquiry to explain how they came to run their vessels aground, and by authors composing forewords. –John Lanchester

15 Chapter 0 Introduction Es gibt nach des Verf. Erfarhrung kein besseres Mittel, Geometrie zu lernen, als das Studium des ̈ Kalk uls der abz ahlenden Geometrie . Schubertschen ̈ (There is, in the author’s experience, no better means of learning geometry than the study of Calculus of Enumerative Geometry .) Schubert’s –B. L. van der Waerden (in a Zentralblatt review of An Introduction to Enumerative Geometry by Hendrik de Vries). Why you want to read this book Algebraic geometry is one of the central subjects of mathematics. All but the most analytic of number theorists speak our language, as do mathematical physicists, complex analysts, homotopy theorists, symplectic geometers, representation theorists. . . . How else could you get between such apparently disparate fields as topology and number theory in one hop, except via algebraic geometry? And intersection theory is at the heart of algebraic geometry. From the very begin- nings of the subject, the fact that the number of solutions to a system of polynomial equations is, in many circumstances, constant as we vary the coefficients of those poly- nomials has fascinated algebraic geometers. The distant extensions of this idea still drive the field forward. At the outset of the 19th century, it was to extend this “preservation of number” that algebraic geometers made two important choices: to work over the complex numbers rather than the real numbers, and to work in projective space rather than affine space. (With these choices the two points of intersection of a line and an ellipse have somewhere to go as the ellipse moves away from the real points of the line, and the same for the point of intersection of two lines as the lines become parallel.) Over the course of the century, geometers refined the art of counting solutions to geometric problems — introducing the central notion of a parameter space , proposing the notions of an equivalence relation

16 2 Chapter 0 Introduction on cycles and a product on the equivalence classes and using these in many subtle calculations. These constructions were fundamental to the developing study of algebraic curves and surfaces. In a different field, it was the search for a mathematically precise way of describing ́ e’s study of what became algebraic topology. We intersections that underlay Poincar ́ owe Poincar e duality and a great deal more in algebraic topology directly to this search. ́ The difficulties Poincar e encountered in working with continuous spaces (now called manifolds) led him to develop the idea of a simplicial complex as well. Despite a lack of precise foundations, 19th century enumerative geometry rose to impressive heights: for example, Schubert, whose Kalk ul der abz ahlenden Geometrie ̈ ̈ (originally published in 1879, and reprinted 100 years later in [1979]) represents the summit of intersection theory in the late 19th century, calculated the number of twisted cubics tangent to 12 quadrics — and got the right answer (5,819,539,783,680). Imagine landing a jumbo jet blindfolded! At the outset of the 20th century, Hilbert made finding rigorous foundations for Schubert calculus one of his celebrated problems, and the quest to put intersection theory on a sound footing drove much of algebraic geometry for the following century; the search for a definition of multiplicity fueled the subject of commutative algebra in work of van der Waerden, Zariski, Samuel, Weil and Serre. This progress culminated, towards the end of the century, in the work of Fulton and MacPherson and then in Fulton’s landmark book Intersection theory [1984], which both greatly extended the range of intersection theory and for the first time put the subject on a precise and rigorous foundation. The development of intersection theory is far from finished. Today the focus includes virtual fundamental cycles, quantum intersection rings, Gromov–Witten theory and the extension of intersection theory from schemes to stacks. In a different direction, there are computer systems that can do many of the computations in this book and many more; see for example the package Schubert2 in Macaulay2 (Grayson and Stillman [2015]) and the library in S INGULAR (Decker et al. [2015]). Schubert A central part of a central subject of mathematics — of course you would want to read this book! Why we wrote this book Given the centrality of the subject, it is not surprising how much of algebraic geome- try one encounters in learning enumerative geometry. And that is how this book came to be written, and why: Like van der Waerden, we found that intersection theory makes for a great “second course” in algebraic geometry, weaving together threads from all over the subject. Moreover, the new ideas encountered in this setting are not merely more abstract definitions for the student to memorize, but tools that help answer concrete questions.

17 Introduction Chapter 0 3 This is reflected in the organization of the contents. A good example of this is Chap- ter 6 (“Lines on hypersurfaces”). The stated goal of the chapter is to describe the class, in n .1;n/ of lines in P of lines lying , of the scheme F the Grassmannian .X/  G .1;n/ G 1 n  P on a given hypersurface , as an application of the new technique of Chern classes. X F But this raises a question: how can we characterize the scheme structure on .X/ , and 1 what can we say about the geometry of this scheme? In short, this is an ideal time to intro- duce the notion of a Hilbert scheme , which gives a general framework for these questions; in the present setting, we can explicitly write down the equations defining F , and .X/ 1 prove theorems about its local geometry. In the end, a large part of the chapter is devoted to this discussion, which is as it should be: A reader may or may not have any use for the 4 knowledge that a general quintic hypersurface  P contains exactly 2875 lines, but a X functional understanding of Hilbert schemes is a fundamental tool in algebraic geometry. What’s with the title? The number in the title of this book is a reference to the solution of a classic problem in enumerative geometry: the determination, by Chasles, of the number of smooth conic plane curves tangent to five given general conics. The problem is emblematic of the dual nature of the subject. On the one hand, the number itself is of little significance: life would not be materially different if there were more or fewer. But the fact that the problem is well-posed — that there is a Zariski open subset of the space of 5-tuples .C of conics for which the number of conics tangent to all five is constant, ;:::;C / 5 1 and that we can in fact determine that number — is at the heart of algebraic geometry. And the insights developed in the pursuit of a rigorous derivation of the number — the recognition of the need for, and the introduction of, a new parameter space for plane conics, and the understanding of why intersection products are well-defined for this space — are landmarks in the development of algebraic geometry. The rest of the title is from “1066 & All That” by W. C. Sellar and R. J. Yeatman, a parody of English history textbooks; in many ways the number 3264 of conics tangent to five general conics is as emblematic of enumerative geometry as the date 1066 of the Battle of Hastings is of English history. What is in this book We are dealing here with a fundamental and almost paradoxical difficulty. Stated briefly, it is that learning is sequential but knowledge is not. A branch of mathematics [. . . ] consists of an intricate network of interrelated facts, each of which contributes to the understanding of those around it. When confronted with this network for the first time, we are forced to follow a particular path, which involves a somewhat arbitrary ordering of the facts. –Robert Osserman.

18 4 Chapter 0 Introduction Where to begin? To start with the technical underpinnings of a subject risks losing the reader before the point of all the preliminary work is made clear, but to defer the logical foundations carries its own dangers — as the unproved assertions mount up, the reader may well feel adrift. Intersection theory poses a particular challenge in this regard, since the development of its foundations is so demanding. It is possible, however, to state fairly simply and precisely the main foundational results of the subject, at least in the limited context of intersections on smooth projective varieties. The reader who is willing to take these results on faith for a little while, and accept this restriction, can then be shown what the subject is good for, in the form of examples and applications. This is the path we have chosen in this book, as we will now describe. Overture The first two chapters may be thought of as an overture to the subject, introducing the central themes that will play out in the remainder of the book. In the first chapter, we introduce rational equivalence, the Chow ring, the pullback and pushforward maps — the “dogma” of the subject. (In regard to the existence of an intersection product and pullback maps, we do not give proofs; instead, we refer the reader to Fulton [1984].) We follow this in the second chapter with a range of simple examples to give the reader a sense of the themes to come: the computation of Chow rings of affine and projective spaces, their products and (some) blow-ups. To illustrate how intersection theory is used in algebraic geometry, we examine loci of various types of singular cubic plane 9 P parametrizing plane cubics. curves, thought of as subvarieties of the projective space Finally, we briefly discuss intersection products of curves on surfaces, an important early example of the subject. Grassmannians The intersection rings of the Grassmannians are archetypal examples of intersection theory. Chapters 3 and 4 are devoted to them and their underlying geometry. Here we introduce Schubert cycles , whose classes form a basis for the Chow ring, and use them to solve a number of geometric problems, illustrating again how intersection theory is used to solve enumerative problems. Chern classes We then come to a watershed in the subject. Chapter 5 takes up in earnest a notion at the center of modern intersection theory, and indeed of modern algebraic geometry: Chern classes. As with the development of intersection theory, we focus on the classical characterization of Chern classes as degeneracy loci of collections of sections. This interpretation provides useful intuition and is basic to many applications of the theory.

19 Introduction Chapter 0 5 Applications, I: Using the tools We illustrate the use of Chern classes by taking up two classical problems: Chapter 6 deals with the question of how many lines lie on a hypersurface (for example, the fact that there are exactly 27 lines on each smooth cubic surface and 2875 lines on a general quintic threefold), and Chapter 7 looks at the singular hypersurfaces in a one-dimensional family (for example, the fact that a general pencil of plane curves of 2 has 3.d degree d singular elements). Using the basic technique of linearization , 1/ these problems can be translated into problems of computing Chern classes. These and the next few chapters are organized around geometric problems involving constructions of useful vector bundles and the calculation of their Chern classes. Parameter spaces Chapter 8 concerns an area in which intersection theory has had a profound influence on modern algebraic geometry: parameter spaces and their compactifications. This is illustrated with the five conic problem; there is also a discussion of the modern example of Kontsevich spaces, and an application of these. Applications, II: Further developments The remainder of the book introduces a series of increasingly advanced topics. Chapters 9, 10 and 11 deal with a situation ubiquitous in the subject, the intersection theory of projective bundles, and its applications to subjects such as projective duality and the enumerative geometry of contact conditions. Chern classes are defined in terms of the loci where collections of sections of a vector bundle become dependent. These can be interpreted as loci where maps from trivial vector bundles drop rank. The Porteous formula, proved and applied in Chapter 12, generalizes this, expressing the classes of the loci where a map between two general vector bundles has a given rank or less in terms of the Chern classes of the two bundles involved. Advanced topics Next, we come to some of the developments of the modern theory of intersections. In Chapter 13, we introduce the notion of “excess” intersections and the excess intersection formula , one of the subjects that was particularly mysterious in the 19th century but elucidated by Fulton and MacPherson. This theory makes it possible to describe the intersection class of two cycles, even if the dimension of their intersection is “too large.” Central to this development is the idea of specialization to the normal cone , a construction fundamental to the work of Fulton and MacPherson; we use this to prove

20 6 Chapter 0 Introduction Z X to the famous “key formula” comparing intersections of cycles in a subvariety  the intersections of those cycles in X , and use this in turn to give a description of the Chow ring of a blow-up. Chapter 14 contains an account of Riemann–Roch formulas, leading up to a descrip- tion of Grothendieck’s version. The chapter concludes with a number of examples and applications showing how Grothendieck’s formula can be used. Appendices The moving lemma The literature contains a number of papers proving various parts of the moving lemma (see below for a statement). We give a careful proof of the first half of the lemma in Appendix A. Cohomology and base change Many results in this book will be proved by constructing an appropriate vector bundle and computing its Chern classes. The theorem on cohomology and base change (Theorem B.5) is a key tool in these constructions: We use it to show that, under appropriate hypotheses, the direct image of a sheaf is a vector bundle. We present a complete discussion of this important result in Appendix B. Topology of algebraic varieties When we treat algebraic varieties over an arbitrary field we use the Zariski topology, where an open set is defined as the locus where a polynomial function takes nonzero values. But if the ground field is the complex numbers, we can also use the “classical” topology: With this topology, a smooth projective variety over C is a compact, complex manifold, and tools like singular homology can help us study its geometry. Appendix C explains some of what is known in this direction, and also compares some of the possible substitutes for the Chow ring. The Brill–Noether theorem Appendix D explains an application of enumerative geometry to a problem that is central in the study of algebraic curves and their moduli spaces: the existence of special linear series on curves. We give the Kempf/Kleiman–Laksov proof of this theorem, which draws upon many of the ideas and techniques of the book, plus a new one: the use of topological cohomology in the context of intersection theory. This is also a wonderful illustration of the way in which enumerative geometry can be the essential ingredient in the proof of a purely qualitative result.

21 Introduction Chapter 0 7 Intersection theory Relation of this book to [1984] is a great work. It sets up for the first time Fulton’s book Intersection theory a rigorous framework for intersection theory, and does so in a generality significantly extending and refining what was known before and laying out an enormous number of applications. It stands as an encyclopedic reference for the subject. By contrast, the present volume is intended as a textbook in algebraic geometry, a second course, in which the classical side of intersection theory is a starting point for exploring many topics in geometry. We describe the intersection product at the outset, but do not attempt to give a rigorous proof of its existence, focusing instead on basic examples. We use concrete problems to motivate the introduction of new tools from all over algebraic geometry. Our book is not a substitute for Fulton’s; it has a different aim. We do hope that it will provide the reader with intuition and motivation that will make reading Fulton’s book easier. Existence of the intersection product The was for most of a century the foundation on which intersection moving lemma theory was supposed to rest. It has two parts: Given classes , 2 A.X/ in the Chow group of a smooth, projective variety X (a) ̨;ˇ A and B intersecting generically transversely. we can find representative cycles (b) The class of the intersection of these cycles is independent of the choice of A and B . Using these assertions it is easy to define the intersection product on the Chow groups of a smooth variety: is defined to be the class of A \ B ̨ˇ A and B are , where cycles representing the classes ̨ and ˇ and intersecting generically transversely, and this is how intersection products were defined. The problem is that, while the first part can be and was proved rigorously, as far as we know there was prior to the publication of Fulton’s book in 1984 no complete proof of the second part. Of course, part (b) is an immediate consequence of the existence of a well-defined intersection product (Fulton [1984, Section 8.3]), and so we refer the reader to Fulton’s book for this key existence result. Nonetheless, we feel that part (a) of the moving lemma is useful in shaping one’s intuition about intersection products. Moreover, given the existence statement, part (a) of the moving lemma allows simpler and more intuitive proofs of a number of the basic assertions of the theory, and we will use it in that way. We therefore give a proof of part (a) in Appendix A, following Severi’s ideas. Keynote problems To highlight the sort of problems we will learn to solve, and to motivate the material we present, we will begin each chapter with some keynote questions .

22 8 Chapter 0 Introduction Exercises One of the wonderful things about the subject of enumerative geometry is the abundance of illuminating examples that are accessible to explicit computation. We have included many of these as exercises. We have been greatly aided by Francesco Cavazzani; in particular, he has prepared solutions, which appear on a web site associated to this book. Prerequisites, notation and conventions What you need to know before starting When it comes to prerequisites, there are two distinct questions: what you should know to start reading this book; and what you should be prepared to learn along the way. Of these, the second is by far the more important. In the course of developing and applying intersection theory, we introduce many key techniques of algebraic geome- try, such as deformation theory, specialization methods, characteristic classes, Hilbert schemes, commutative and homological algebra and topological methods. That is not to say that you need to know these things going in. Just the opposite, in fact: Reading this book is an occasion to learn them. So what do you need before starting? An undergraduate course in classical algebraic geometry or its equivalent, compris- (a) An invitation to algebraic ing the elementary theory of affine and projective varieties. (Smith et al. [2000]) contains almost everything required. Other books geometry that cover this material include Undergraduate algebraic geometry (Reid [1988]), Introduction to algebraic geometry (Hassett [2007]), Elementary algebraic geom- etry (Hulek [2012]) and, at a somewhat more advanced level, geometry, Algebraic (Mumford [1976]), Basic algebraic geometry, I I: Complex projective varieties Algebraic geometry: a first course (Harris [1995]). The (Shafarevich [1994]) and last three include much more than we will use here. (b) An acquaintance with the language of schemes. This would be amply covered by the first three chapters of The geometry of schemes (Eisenbud and Harris [2000]). (c) Faisceax An acquaintance with coherent sheaves and their cohomology. For this, alg (Serre [1955]) remains an excellent source (it is written in ebriques coh erents ́ ́ the language of varieties, but applies nearly word-for-word to projective schemes over a field, the context in which this book is written). In particular, Algebraic geometry (Hartshorne [1977]) contains much more than you need to know to get started.

23 Introduction Chapter 0 9 Language Throughout this book, a will be a separated scheme of finite type over scheme X of characteristic 0. (We will occasionally point out the an algebraically closed field k p ways in which the characteristic situation differs from that of characteristic 0, and how we might modify our statements and proofs in that setting.) In practice, all the integral to mean reduced schemes considered will be quasi-projective. We use the term and irreducible; by a we will mean an integral scheme. (The terms “curve” and variety “surface,” however, refer to one-dimensional and two-dimensional schemes; in particular,  X they are not presumed to be integral.) A subvariety Y will be presumed closed X k .X/ for the field of rational unless otherwise specified. If is a variety we write X sheaf on X will be a coherent sheaf unless otherwise noted. functions on . A point we mean a closed point. Recall that a locally closed subscheme U of a By a X scheme X . We use the is a scheme that is an open subset of a closed subscheme of term “subscheme” (without any modifier) to mean a closed subscheme, and similarly for “subvariety.” of X has a A consequence of the finite-type hypothesis is that any subscheme Y Y : locally, we can write the ideal of primary decomposition as an irredundant intersection of primary ideals with distinct associated primes. We can correspondingly write Y globally as an irredundant union of closed subschemes Y whose supports are distinct i X subvarieties of Y whose supports are maximal — . In this expression, the subschemes i corresponding to the minimal primes in the primary decomposition — are uniquely determined by ; they are called the irreducible components of Y . The remaining Y embedded components ; they are not determined by Y , though subschemes are called their supports are. If a family of objects is parametrized by a scheme B , we will say that a “general” member of the family has a given property P  B of members of the if the set U.P/ . When we say that a “very B family with that property contains an open dense subset of U.P/ contains the complement of general” member has this property we will mean that a countable union of proper subvarieties of B . projectivization of a vector space V , denoted P V , we will mean the scheme By the  V V . / (where by Sym V we mean the symmetric algebra of Sym ); this is the space Proj V . This is opposite whose closed points correspond to one-dimensional subspaces of to the usage in, for example, Grothendieck and Hartshorne, where the points of P V  V (that is, their P V is our P V correspond to one-dimensional quotients of ), but is in agreement with Fulton. n and Y  P If X join of X and Y , are subvarieties of projective space, we define the denoted X;Y , to be the closure of the union of lines meeting X and Y at distinct points. n Y D Ä  P If is a linear space, this is just the cone over X with vertex Ä ; if X and Y are both linear subspaces, this is simply their span.

24 10 Chapter 0 Introduction and X There is a one-to-one correspondence between vector bundles on a scheme X locally free sheaves on . We will use the terms interchangeably, generally preferring “line bundle” and “vector bundle” to “invertible sheaf” and “locally free sheaf.” When F on X at a point p 2 X , we will mean the we speak of the fiber of a vector bundle .p/ , where .p/ is the residue field at p . ̋ F (finite-dimensional) vector space , or linear series , on a scheme X , we will mean a pair D By a . L ;V / , linear system D 0 X and V  H L . L / a vector space of sections. Associating where is a line bundle on 0 L  H 2 . V / its zero locus V./ , we can also think of a linear system to a section  f V./ j  2 V g of divisors D  X parametrized by the projective space as a family V ; in this setting, we will sometimes abuse notation slightly and write 2 D . By P D of the linear series we mean the dimension of the projective space V dimension P the V 1 . Specifically, a one-dimensional linear system is parametrizing it, that is, dim , a two-dimensional system is called a net and a three-dimensional linear pencil called a system is called a web . O F for the local ring of X along Y , and, more generally, if We write is a sheaf X;Y for the corresponding -modules we write F of -module. O O X;Y X Y n n with A itself. If A We can identify the Zariski tangent space to the affine space n A affine tangent space is a subscheme, by the X to X at a point p we will mean the  n n C T projective X  A affine linear subspace . If X  P p is a subscheme, by the p n n , we will mean the closure in 2 , denoted T to X  P at X P p of the X tangent space p n n n for any open subset A X  P \ containing p . Concretely, A affine tangent space to n is the zero locus of polynomials F if (that is, X D V.I/  P X is the subscheme ̨ ;:::;Z D . f F defined by the ideal g /  k ŒZ ), the projective tangent space is the I ç n ̨ 0 common zero locus of the linear forms @F @F ̨ ̨ .Z/ L D .p/Z : .p/Z CC ̨ 0 n @Z @Z 0 n one-parameter family X ! B with B smooth By a we will always mean a family and one-dimensional (an open subset of a smooth curve, or spec of a DVR or power 0 2 B . In this context, “with parameter t ” series ring in one variable), with marked point t is a local coordinate on the curve, or a generator of the maximal ideal of the means DVR or power series ring. Basic results on dimension and smoothness There are a number of theorems in algebraic geometry that we will use repeatedly; we give the statements and references here. When X is a scheme, by the dimension of X we mean the Krull dimension, denoted X . If X is an irreducible variety and Y  X dim is a subvariety, then the codimension of Y in X , written codim Y Y (or simply codim X is any scheme X is clear from context), is dim X when dim Y ; more generally, if X

25 Introduction Chapter 0 11 Y is a subvariety, then codim Y denotes the minimum of and X 0 0 is a reduced irreducible component of X codim Y X g : j f X More on dimension and codimension can be found in Eisenbud [1995]. We will often use the following basic result of commutative algebra: Theorem 0.1 An ideal generated by n elements in a (Krull’s principal ideal theorem) .  . Noetherian ring has codimension n See Eisenbud [1995, Theorem 10.2] for a discussion and proof. We will also use the following important extension of the principal ideal theorem: Theorem 0.2 . If f W Y ! X is a morphism of (Generalized principal ideal theorem) X varieties and A  X , is smooth, then, for any subvariety 1 f A  codim A: codim A;B are subvarieties of X , and C is an irreducible component of A \ In particular, if , B then C  codim A C codim B . codim The proof of this result can be reduced to the case of an intersection of two subvari- eties, one of which is locally a complete intersection, by expressing the inverse image 1 f A as an intersection with the graph Ä . In this form it follows from  X  Y of f f Krull’s theorem. The result holds in greater generality; see Serre [2000, Theorem V.3]. Smoothness is necessary for this (Example 2.22). A module M is said to be of finite length if it has a finite maximal sequence of , and we will call the length composition series submodules. Such a sequence is called a of the sequence the of the module. The following theorem shows this length is length well-defined: ̈ M older theorem) . A module Theorem 0.3 (Jordan–H of finite length over a commutative local ring has a maximal sequence of submodules M © M 0 © R © M D  1 k Moreover, any two such maximal sequences are isomorphic; that is, they have the same length and composition factors (up to isomorphism). Theorem 0.4 (Chinese remainder theorem) . A module of finite length over a commutative ring is the direct sum of its localizations at finitely many maximal ideals. For discussion and proof see Eisenbud [1995, Chapter 2], especially Theorem 2.13. (Bertini) . If D is a linear system on a variety X in characteristic 0, the Theorem 0.5 D is smooth outside the base locus of D and the singular locus of X . general member of D Note that applying Bertini repeatedly, we see as well that if ;:::;D are general 1 k T D D then the intersection is smooth of dimension members of the linear system i . X k away from the base locus of D and the singular locus of X dim

26 12 Introduction Chapter 0 This is the form in which we will usually apply Bertini. But there is another version that is equivalent in characteristic 0 but allows for an extension to positive characteristic: n (Bertini) . If f W X ! P is any generically separated morphism from a Theorem 0.6 1 of a f smooth, quasi-projective variety to projective space, then the preimage .H/ X n general hyperplane H  P is smooth.

27 Chapter 1 Introducing the Chow ring Keynote Questions As we indicated in the introduction, we will preface each chapter of this book with a series of “keynote questions:” examples of the sort of concrete problems that can be solved using the ideas and techniques introduced in that chapter. In general, the answers to these questions will be found in the same chapter. In the present case, we will not develop our roster of examples sufficiently to answer the keynote questions below until the second chapter; we include them here so that the reader can have some idea of “what the subject is good for” in advance. F ;F (1) Let and F be three general homogeneous cubic polynomials 2 k ŒX;Y;Zç 2 0 1 t F F C C t in three variables. Up to scalars, how many linear combinations F t 1 0 2 0 2 1 factor as a product of a linear and a quadratic polynomial? (Answer on page 65.) ;F Let ;F ŒX;Y;Zç F k and F be four general homogeneous cubic polynomi- 2 (2) 2 3 0 1 t t C t als in three variables. How many linear combinations C F F C t F F 2 1 2 0 1 3 0 3 factor as a product of three linear polynomials? (Answer on page 65.) A;B;C are general homogeneous quadratic polynomials in three variables, for If (3) t do we have .t D ;t how many triples ;t / 2 0 1 D .t /‹ ;t .A.t/;B.t/;C.t// ;t 2 0 1 (Answer on page 55.) 3 3 S  P a general line. How many planes be a smooth cubic surface and L  P (4) Let containing are tangent to S ? (Answer on page 50.) L 3 3 L  P be surfaces of degrees be a line, and let S and T  P (5) Let s and t L containing . Suppose that the intersection S \ T is the union of L and a smooth curve C . What are the degree and genus of C ? (Answer on page 71.)

28 14 Chapter 1 Introducing the Chow ring 1.1 The goal of intersection theory Though intersection theory has many and surprising applications, in its most basic form it gives information about the intersection of two subvarieties of a given variety. An early incarnation, and in some sense the model for all of intersection theory, is the 2 ́ P theorem of B intersect transversely, then they intersect ezout: If plane curves  A;B A/. deg B/ points (see Figure 1.3 on page 18). . in deg is a line, this is a special case of Gauss’ fundamental theorem of algebra: If A f.x/ deg f roots, if the roots are counted A polynomial in one complex variable has with multiplicity. Late in the 19th century it was understood how to attribute multiplicities to the intersections of any two plane curves without common components (we shall ́ describe this in Section 1.3.7 below), so B ezout’s theorem could be extended: The intersection of two plane curves without common components consists a collection of . deg A/. points with multiplicities adding up to B/ . deg In modern geometry we need to understand intersections of subvarieties in much greater generality. In this book we will mostly consider intersections of arbitrary sub- varieties in a smooth ambient variety . The goal of this chapter is to introduce a ring X Chow ring of X , and to associate to every subscheme A  X a class A.X/ , called the 2 A.X/ ŒAç P in . In Section 1.3.7 we will explain a generalizing the degree of a curve in ́ ezout’s theorem: far-reaching extension of B ́ (B . If A;B  ezout’s theorem for dimensionally transverse intersections) Theorem 1.1 X are subvarieties of a smooth variety X and codim .A \ B/ D codim A C codim B , then we can associate to each irreducible component C of A \ B a positive integer i m .A;B/ in such a way that C i X  m ŒAçŒBç ç: .A;B/ D ŒC i C i m ; C .A;B/ is called the intersection multiplicity of The integer and B along A i C i giving a correct definition in this generality occupied algebraic geometers for most of the first half of the 20th century. Though Theorem 1.1 is restricted to the case where the subvarieties A;B meet only dimensionally proper intersection codim C codim in codimension (the case of A ), B there is a very useful extension to the case where the codimensions of the components of the intersection are arbitrary; this will be discussed in Chapter 13. Many important applications involve subvarieties defined as zero loci of sections of a vector bundle E on a variety X , and this idea has potent generalizations. It turns Chern classes out that there is a way of defining classes of . E / 2 A.X/ , called the c i E , and the theory of Chern classes is a pillar of intersection theory. The third and final section of this chapter takes up a special case of the general theory that is of particular importance and relatively easy to describe: the first Chern class of a line bundle. This allows us to introduce the canonical class , a distinguished element of the Chow ring of

29 The Chow ring Section 1.1 15 any smooth variety, and show how to calculate it in simple cases. The general theory of Chern classes will be taken up in Chapter 5. 1.2 The Chow ring We now turn to the definition and basic properties of the Chow ring. Then we introduce excision and Mayer–Vietoris theorems that allow us to calculate the Chow rings of many varieties. Most importantly we describe the functoriality of the Chow ring: the existence, under suitable circumstances, of pushforward and pullback maps. Chow groups form a sort of homology theory for quasi-projective varieties; that is, they are abelian groups associated to a geometric object that are described as a group of cycles modulo an equivalence relation. In the case of a smooth variety, the intersection product makes the Chow groups into a graded ring, the Chow ring. This is analogous to the ring structure on the homology of smooth compact manifolds that can be imported, ́ e duality, from the natural ring structure on cohomology. using Poincar k of Throughout this book we will work over an algebraically closed ground field characteristic 0. Virtually everything we do could be formulated over arbitrary fields ), and occasionally we p (though not every statement remains true in characteristic comment on how one would do this. 1.2.1 Cycles Let group of cycles on X be any algebraic variety (or, more generally, scheme). The Z.X/ , is the free abelian group generated by the set of subvarieties (reduced , denoted X X . The group Z.X/ irreducible subschemes) of Z .X/ is graded by dimension: we write k k for the group of cycles that are formal linear combinations of subvarieties of dimension P L Y ), so that D -cycles , where Z Z.X/ .X/ . A cycle Z D k (these are called n i i k k Y divisor are subvarieties, is effective the n are all nonnegative. A if the coefficients i i (sometimes called a Weil divisor ) is an .n 1/ -cycle on a pure n -dimensional scheme. is insensitive to It follows from the definition that D / ; that is, Z.X/ Z.X Z.X/ red whatever nonreduced structure X may have. Y  X we associate an effective cycle h Y i : If Y  X is To any closed subscheme Y ;:::;Y are the irreducible components of the reduced scheme Y , a subscheme, and 1 red s O then, because our schemes are Noetherian, each local ring has a finite composition Y;Y i ̈ series. Writing for its length, which is well-defined by the Jordan–H l older theorem i P l (Theorem 0.3), we define the cycle to be the formal combination Y h . (The Y i i i coefficient l along the irreducible component is called the multiplicity of the scheme Y i , and written mult Y ; we will discuss this notion, and its relation to the notion of .Y/ i Y i intersection multiplicity, in Section 1.3.8.) In this sense cycles may be viewed as coarse approximations to subschemes.

30 16 Chapter 1 Introducing the Chow ring X ! ! 0 0 ! 1 ! 1 1 0 1 P and ! Figure 1.1 Rational equivalence between two cycles on X . ! 1 0 1.2.2 Rational equivalence and the Chow group of X is the group of cycles of X modulo rational equivalence . The Chow group A ;A Informally, two cycles 2 Z.X/ are rationally equivalent if there is a rationally 0 1 1 P  X parametrized family of cycles interpolating between them — that is, a cycle on f t whose restrictions to two fibers g X and f t . Here is the g X are A A and 0 0 1 1 formal definition: Definition 1.2. Let Rat  Z.X/ be the subgroup generated by differences of .X/ the form ˆ \ ; f t h g X/ ih ˆ \ . f t g X/ i . 0 1 1 1 not contained in any fiber . 2 P where and ˆ is a subvariety of P ;t  X X f t g t 0 1 rationally equivalent if their difference is in We say that two cycles are .X/ , and we Rat say that two subschemes are rationally equivalent if their associated cycles are rationally equivalent — see Figures 1.1 and 1.2. Definition 1.3. The Chow group of X is the quotient A.X/ D Rat .X/; Z.X/= group of rational equivalence classes of cycles on the . If Y 2 Z.X/ is a cycle, we X write ŒYç 2 A.X/ for its equivalence class; if Y  X is a subscheme, we abuse notation i slightly and denote simply by the class of the cycle h Y ŒYç associated to Y . It follows from the principal ideal theorem (Theorem 0.1) that the Chow group is graded by dimension:

31 The Chow ring Section 1.2 17 ˆ X  X  t t 1 0 2 Figure 1.2 Rational equivalence between a hyperbola and the union of two lines in . P X is a scheme then the Chow group of X Proposition 1.4. If is graded by dimension; that is, M A.X/ D A .X/; k A k .X/ the group of rational equivalence classes of with -cycles. k 1 X P If   is an irreducible variety not contained in a fiber over X then, ˆ Proof: 1 ˆ \ ˆ in an appropriate affine open set  X/  ˆ , the scheme . \ . f t X/ g A 0 is defined by the vanishing of the single nonzerodivisor t t . It follows that the 0 components of this intersection are all of codimension exactly 1 in ˆ , and similarly for X/ . t \ g f . Thus all the varieties involved in the rational equivalence defined by ˆ ˆ 1 have the same dimension. X is equidimensional we may define the codimension of a subvariety Y  X When dim X dim Y , and it follows that we may also grade the Chow group by codimension. as c X is also smooth, we will write A When .X/ for the group A , and think of it as X dim c the group of codimension- c cycles, modulo rational equivalence. (It would occasionally X is singular, but this would conflict be convenient to adopt the same notation when with established convention — see the discussion in Section 2.5 below.) 1.2.3 Transversality and the Chow ring We said at the outset that much of what we hope to do in intersection theory is 2 ́ ezout theorem: that if plane curves A;B  P modeled on the classical B of degrees d and e intersect transversely then they intersect in de points. Two things about this

32 18 Chapter 1 Introducing the Chow ring B A B A Figure 1.3 Two conics meet in four points. result are striking. First, the cardinality of the intersection does not depend on the choice of curves, beyond knowing their degrees and that they meet transversely. Given this d invariance, the theorem follows from the obvious fact that a union of general lines meets a union of general lines in de points (Figure 1.3). e Second, the answer, de , is a product, suggesting that some sort of ring structure is present. A great deal of the development of algebraic geometry over the past 200 years is bound up in the attempt to understand, generalize and apply these ideas, leading to precise notions of the sense in which intersection of subvarieties resembles multiplication. What makes the Chow groups useful is that, under good circumstances, the rational equivalence class of the intersection of two subvarieties A;B depends only on the and B , and this gives a product structure on the Chow rational equivalence classes of A groups of a smooth variety. To make this statement precise we need some definitions. We say that subvarieties A;B X transversely at a point p A;B intersect and X are all smooth at p of a variety if A and the tangent spaces to B at p together span the tangent space to X ; that is, and T A C T X; B D T p p p or equivalently .T B: A \ T T B/ D codim T codim A C codim p p p p A;B We will say that subvarieties X are generically transverse , or that they  intersect generically transversely , if they meet transversely at a general point of each component \ B . The terminology is justified by the fact that the set of points of A of C B at which A and B are transverse is open. We extend the terminology to cycles A \ P P D n by saying that two cycles A are generically transverse if and B A D m B j i j i A B . each is generically transverse to each i j A More generally, we will say subvarieties  X intersect transversely at a smooth i  T P if p is a smooth point on each A , and codim point p T A A T 2 D X codim i p i i p and we say that they intersect generically transversely if there is a dense set of points in the intersection at which they are transverse. As an example, if and B have complementary dimensions in X (that is, if A dim A C dim B D dim X ), then A and B are generically transverse if and only if they are transverse everywhere; that is, their intersection consists of finitely many points and they intersect transversely at each of them. (In this case we will accordingly drop the

33 The Chow ring Section 1.2 19 codim A codim B > dim X , then A and B are generically modifier “generically.”) If C transverse if and only if they are disjoint. If is a smooth quasi-projective variety, then there is a X Theorem–Definition 1.5. satisfying the condition: unique product structure on A.X/ ) If two subvarieties of  X are generically transverse, then ( A;B ŒA \ Bç: ŒAçŒBç D This structure makes X dim M c D .X/ A.X/ A 0 c D into an associative, commutative ring, graded by codimension, called the Chow ring of X . Fulton [1984] gave a direct construction of the product of cycles on any smooth variety over any field, and proved that the products of rationally equivalent cycles are rationally equivalent. In a setting where the first half of the moving lemma (Theorem 1.6 below) holds, such as a smooth, quasi-projective variety over an algebraically closed .  / of Theorem–Definition 1.5. field, this product is characterized by the condition Even if is smooth and A;B are subvarieties such that every component of A \ B X has the expected codimension codim A C codim B , we cannot define ŒAçŒBç 2 A.X/ to be ŒA Bç , because the class ŒA \ Bç depends on more than the rational equivalence \ A B . This problem can be solved by assigning intersection multiplicities classes of and to the components; see Section 1.3.7. 1.2.4 The moving lemma Historically, the proof of Theorem–Definition 1.5 was based on the . moving lemma This has two parts: (Moving lemma) Theorem 1.6 Let X be a smooth quasi-projective variety. . (a) For every ̨;ˇ 2 A.X/ there are generically transverse cycles A;B 2 Z.X/ with ŒAç ̨ and ŒBç D ˇ . D (b) The class ŒA \ Bç is independent of the choice of such cycles A and B . A proof of the first part is given in Appendix A; this is sufficient to establish the uniqueness of a ring structure on A.X/ satisfying the condition (  ) of Theorem– Definition 1.5. The second part, which historically was used to prove the existence portion of Theorem–Definition 1.5, is more problematic; as far as we know, no complete proof existed prior to the publication of Fulton [1984].

34 20 Chapter 1 Introducing the Chow ring 2 P C L 0 L 1 L L Figure 1.4 The cycle , which can be “moved” to the rationally equivalent cycle 1 0 C is transverse to the given subvariety . The first half of the moving lemma is useful in shaping our understanding of intersection products and occasionally as a tool in the proof of assertions about them, and we will refer to it when relevant. 3 On a singular variety the moving lemma may fail: For example, if X is a  P X quadric cone then any two cycles representing the class of a line of meet at the origin, , and thus cannot be generically transverse (see Exercise 1.36). X a singular point of Further, the hypothesis of smoothness in Theorem 1.5 cannot be avoided: We will also see in Section 2.5 examples of varieties X where no intersection product satisfying the  ) of Theorem 1.5 can be defined. The news is not uniformly negative: basic condition ( can Intersection products be defined on singular varieties if we impose some restrictions on the classes involved, as we will see in Proposition 1.31. Kleiman’s transversality theorem There is one circumstance in which the first half of the moving lemma is relatively , we can use automor- easy: when a sufficiently large group of automorphisms acts on X phisms to move cycles to make them transverse. Here is a special case of a result of Kleiman: . Theorem 1.7 (Kleiman’s theorem in characteristic 0) Suppose that an algebraic group acts transitively on a variety X over an algebraically closed field of characteristic 0, G A and that X is a subvariety.  2 B  X is another subvariety, then there is an open dense set of g If G such that (a) gA is generically transverse to B . (b) More generally, if ' W Y ! X is a morphism of varieties, then for general g 2 G 1 the preimage .gA/ is generically reduced and of the same codimension as A . ' 2 G ŒgAç D ŒAç (c) If A.X/ for any g is affine, then G . 2 Proof: (a) This is the special case Y D B of (b). (b) Let the dimensions of X , A , Y and G be n , a , b and m respectively. If x 2 X , then the map ! X W g 7! gx is surjective and its fibers are the cosets of the stabilizer of x G in G . Since all these fibers have the same dimension, this dimension must be m n . Set  Ä .x;y;g/ 2 A  Y Df G j gx D '.y/ g :

35 The Chow ring Section 1.2 21 G acts transitively on , the projection  W Ä ! A  Y is surjective. Its fibers Because X , and hence have dimension n . It follows X m are the cosets of stabilizers of points in that Ä has dimension D a C b C m dim n: Ä 1 ! G is isomorphic to of the projection g Ä .gA/ . On the other hand, the fiber over ' g , or else it has dimension a C b n , Thus either this intersection is empty for general as required. Since is a variety it is smooth at a general point. Since G acts transitively, all X X look alike, so is smooth. Since any algebraic group in characteristic 0 is points of X smooth (see for example Lecture 25 of Mumford [1966]), the fibers of the projection to  are also smooth, so Ä A A Y  Y itself is smooth over . Since field extensions in sm sm .Ä n Ä G / ! characteristic 0 are separable, the projection is smooth over a nonempty sing open set of , where Ä G is the singular locus of Ä . That is, the general fiber of the pro- sing . If the projection of to jection of is smooth outside Ä is not dominant, Ä Ä G to G sing sing 1 ' then is smooth for general g . .gA/ To complete the proof of generic transversality, we may assume that the projection Ä G is dominant. Since G is smooth, the principal ideal theorem shows that ! sing Ä G has codimension  dim G , and thus every every component of every fiber of ! dim G in Ä . Since Ä G component of the general fiber has codimension exactly ! sing , so dim Ä dim G < dim Ä dim is dominant, its general fiber has dimension G sing 1 Ä . Thus no component of a general fiber can be contained in .gA/ is generically ' sing reduced for general g 2 G . (c) We will prove this part only for the case where GL , is a product of copies of G n as this is the only case we will use. For the general result, see Theorem 18.2 of Borel [1991]. G is an open set in a product M of vector spaces of matrices. Let L be In this case 1 to g in M . The subvariety the line joining 1 .g;x/ 2 .G \ L/  X j g Z Df x 2 A g gives a rational equivalence between and gA . A The conclusion fails in positive characteristic, even for Grassmannians; examples can be found in Kleiman [1974] and Roberts [1972b]. However, Kleiman showed that the conclusion holds in general under the stronger hypothesis that G acts transitively on nonzero tangent vectors to X (each tangent space to the Grassmannian is naturally identified with a space of homomorphisms — see Section 3.2.4 — and the automor- phisms preserve the ranks of these homomorphisms, so they do not act transitively on tangent vectors).

36 22 Chapter 1 Introducing the Chow ring c A A gA a d b 2 ŒAçŒgAç D Œa C b C ŒAç C dç D c A meets a general translate of itself generically transversely. Figure 1.5 The cycle 1.3 Some techniques for computing the Chow ring 1.3.1 The fundamental class X is a scheme, then the fundamental class of X If ŒXç 2 A.X/ . It is always is nonzero. We can immediately prove this and a little more, and these first results suffice to compute the Chow ring of a zero-dimensional scheme: Proposition 1.8. be a scheme. Let X D A.X/ / . (a) A.X red If X is irreducible of dimension k , then A and is generated by the .X/ Š Z (b) k fundamental class of X . More generally, if the irreducible components of X are X , then the classes ŒX ç generate a free abelian subgroup of rank m ;:::;X m i 1 A.X/ in . (a) Since both cycles and rational equivalences are generated by varieties we Proof: Z.X/ D Z.X . have and Rat .X/ D Rat .X / / red red (b) By definition the ŒX is generated ç are among the generators of A.X/ . Further, Rat .X/ i 1 1 X  X , each of which is contained in some P P  by varieties in . i Example 1.9 (Zero-dimensional schemes) . From Proposition 1.8 it follows that the Chow group of a zero-dimensional scheme is the free abelian group on the components. 1.3.2 Rational equivalence via divisors The next simplest case is that of curves, and it is not hard to see that the Chow group of 0-cycles on a curve is the divisor class group.

37 Some techniques for computing the Chow ring Section 1.3 23 z y z D x 0 ! 1 x Div .y=x/ Dh1ih 0 i 1 D y=x D 1 in P y , z on the open set Figure 1.6 Graph of the rational function 1 ŒV.x/ç D showing that in A. P ŒV.y/ç / . 0 X we can express the group Rat .X/ of cycles More generally, for any variety X is an affine rationally equivalent to 0 in terms of divisor classes: First, suppose that f variety. If O 2 is a function on X other than 0, then by Krull’s principal ideal X theorem (Theorem 0.1) the irreducible components of the subscheme defined by f are all of codimension 1, so the cycle defined by this subscheme is a divisor; we call it the Y Div divisor of . If , denoted is any irreducible codimension-1 subscheme of X , f .f / along .f / for the order of vanishing of f we write ord Y , so we have Y X : ord i Y h .f / .f / D Div Y irreducible X  Y If f;g are functions on X and ̨ D f=g , then we define the divisor Div . ̨/ D Div .f=g/ to be Div Div .g/ ; see Figure 1.6. This is well-defined because ord .f / .ab/ D Y for any functions defined on an open set. We denote by C ord ord .b/ .a/ Div . ̨/ 0 Y Y Div — in other words, the divisor of . ̨/ the positive and negative parts of Div . ̨/ and 1 zeros of and the divisor of poles of ̨ , respectively. ̨ We extend the definition of the divisor associated to a rational function to varieties X X is the same as the that are not affine as follows. The field of rational functions on U field of rational functions on any open affine subset X , so if ̨ is a rational function of on X then we get a divisor Div . ̨ j . These / on each open subset U  X by restricting ̨ U agree on overlaps, and thus define a divisor . ̨/ on X itself. We will see that the Div ̨ association Div . ̨/ is a homomorphism from the multiplicative group of nonzero 7! rational functions to the additive group of divisors on X . Proposition 1.10. If X is any scheme, then the group Rat .X/  Z.X/ is generated by all divisors of rational functions on all subvarieties of . In particular, if X is irreducible X of dimension n , then A . X is equal to the divisor class group of .X/ 1 n See Fulton [1984, Proposition 1.6] for the proof.

38 24 Chapter 1 Introducing the Chow ring C It follows from Proposition 1.10 that two 0-cycles on a curve (by Example 1.11. which we mean here a one-dimensional variety) are rationally equivalent if and only if they differ by the divisor of a rational function. In particular, the cycles associated 1 C C is birational to P , the are rationally equivalent if and only if to two points on isomorphism being given by a rational function that defines the rational equivalence. is an affine variety whose coordinate ring R does not have unique X Example 1.12. If factorization, then there may not be a “best” way of choosing an expression of a rational on X as a fraction, and Div function . ̨/ need not be the same thing as Div .f / for any ̨ 0 3 2 of ̨ . For example, on the cone Q D V.XZ D Y ̨ /  A f=g , one representation ̨ D X=Y has divisor L M , where L is the line X D Y D 0 the rational function M and D Z D 0 ; but, as the reader can check, ̨ cannot be written in any the line Y of with as a ratio ̨ D f=g .0;0;0/ Div .f / D L and Q neighborhood of the vertex D M . .g/ Div 1.3.3 Affine space Affine spaces are basic building blocks for many rational varieties, such as projective spaces and Grassmannians, and it is easy to compute their Chow groups directly: n n Proposition 1.13. / D Z  Œ A A. ç . A n Proof: A Let be a proper subvariety, and choose coordinates z  z ;:::;z Y on D n 1 n so that the origin does not lie in Y . We let A ı 1 n .t;tz/  . A W nf 0 g /  A Df j z 2 Y gD V. f f.z=t/ j f.z/ vanishes on Y g /: 1 ı The fiber of 2 A W nf 0 g is tY , that is, the image of Y under the t over a point n 1 n . Let W  P given by multiplication by  A t be the closure automorphism of A 1 n 1 ı ı  A in . Note that W of , being the image of . A P n 0/  Y , is irreducible, and W W . hence so is W over the point t D 1 is just Y . On the other hand, since the origin The fiber of n A in does not lie in Y there is some polynomial g.z/ that vanishes on Y and has a 1 n n D g.z=t/ on . A nonzero constant term . The function 0/  A G.t;z/ then extends c n 1 n on the fiber n 0/  A to a regular function on with constant value c . 1 A P . 1 Thus the fiber of t D 1 2 P over is empty, establishing the equivalence Y  0 W (see Figure 1.7). See Section 3.5.2 for a more systematic treatment of this idea. If you are curious D W over t about the fiber of 0 , see Exercise 1.34. 1.3.4 Mayer–Vietoris and excision We will use the next proposition in conjunction with Proposition 1.13 to find generators for the Chow groups of projective spaces and Grassmannians.

39 Some techniques for computing the Chow ring Section 1.3 25  z D 4Y W g 0 4  z 2Y g W D 0 2 W g.z/ D 0 Y O n A Figure 1.7 Scalar multiplication gives a rational equivalence between an affine variety not containing the origin and the empty set. Proposition 1.10 makes it obvious that, if Y X is a closed subscheme, then the  1 1  Y as cycles on P identification of the cycles on  P X Rat .Y/ ! induces a map Rat , and thus a map A.Y/ ! A.X/ (this is a special case of “proper pushforward;” .X/ X with the open set see Section 1.3.6). Further, the intersection of a subvariety of X n U is a subvariety of U (possibly empty), so there is a restriction homomorphism D Y ! Z.U/ . The rational equivalences restrict too, so we get a homomorphism of Z.X/ A.X/ A.U/ Chow groups (this is a special case of “flat pullback;” see Section 1.3.6.) ! Let be a scheme. Proposition 1.14. X (Mayer–Vietoris) If X , then there is a right exact (a) X are closed subschemes of ;X 2 1 sequence A.X 0: \ X ! / ! A.X / / ̊ A.X X / ! A.X [ 1 2 1 2 1 2 X (Excision) If X is a closed subscheme and U D  n Y is its complement, then (b) Y the inclusion and restriction maps of cycles give a right exact sequence ! A.X/ A.Y/ A.U/ ! 0: ! If is smooth, then the map A.X/ ! X is a ring homomorphism. A.U/ Before starting the proof, we note that we can restate the definition of the Chow group by saying that there is a right exact sequence 1 P  Z. X/ ! Z.X/ ! A.X/ ! 0; 1 X P where the left-hand map takes the any subvariety   to 0 if ˆ is contained in ˆ a fiber, and otherwise to h ˆ \ . : t i g X/ ih ˆ \ . f t X/ g f 1 0

40 26 Chapter 1 Introducing the Chow ring Proof of Proposition 1.14: (b) There is a commutative diagram 1 1 1 - - - - P P Z.Y / Z.U  P / /  0 Z.X  0 @ @ @ Y X U ? ? ? - - - - Z.U/ Z.Y/ 0 Z.X/ 0 ? ? ? - - A.X/ A.U/ A.Y/ ? ? ? 0 0 0 , where ! takes the class ŒAç 2 where the map Z.X/ A is a subvariety of Z.Y/ Z.Y/ 1 1 itself, considered as a class in X , and similarly for Z.Y  P Y / ! Z.X  P , to / . ŒAç Z.X/ The map takes each free generator ŒAç to the generator ŒA \ Uç , and ! Z.U/ 1 1 / Z.X ! Z.U  P  P . The two middle rows and all three columns similarly for / A.X/ ! A.U/ is surjective, are evidently exact. A diagram chase shows that the map and the bottom row of the diagram above is right exact, yielding part (b). X Y X (a) Let \ X . We may argue exactly as in . We may assume X D X D [ 1 2 1 2 part (b) from the diagram 1 1 1 1 - - - - Z.X Z.Y  P  / ̊ Z.X /  P P / 0 Z.X / P  0 2 1 @ @ @ ̊ @ ? ? ? - - - - 0 / Z.X ̊ Z.X/ / Z.Y/ Z.X 0 2 1 ? ? ? - - - A.X A.X / 0 A.Y/ A.X/ ̊ / 2 1 Z.Y/ Z.X Z.X to / ̊ where, for example, the map ! / takes a generator ŒAç 2 Z.Y/ 1 2 .ŒAç; ŒAç/ 2 Z.X is addition. / ̊ Z.X Z.X/ / and the map Z.X ! / ̊ Z.X / 2 1 1 2 A.Y/ A.Y/ of part (b) sends the class ŒZç 2 A.X/ of a subvariety Z of The map ! to the class Y 2 A.X/ of the same subvariety, now viewed as a subvariety of X . As ŒZç we will see in Section 1.3.6, this is a special case of the pushforward map f W A.Y/ !  A.X/ associated to any proper map f W Y ! X . The map A.X/ ! A.U/ , sending the Uç class A.X/ of a subvariety of X to the class ŒZ \ 2 2 A.U/ of its intersection ŒZç U , is a special case of a pullback map, also described in Section 1.3.6. with n If U  A Corollary 1.15. is a nonempty open set, then A.U/ D A . .U/ D Z  ŒUç n 1.3.5 Affine stratifications In general we will work with very partial knowledge of the Chow groups of a variety, but when X admits an affine stratification — a special kind of decomposition into a

41 Some techniques for computing the Chow ring 27 Section 1.3 union of affine spaces — we can know them completely. This will help us compute the Chow groups of projective space, Grassmannians, and many other interesting rational varieties. stratified by a finite collection of irreducible, locally We say that a scheme is X X is a disjoint union of the U and, in addition, the closure of U closed subschemes if i i meets is a union of U . The sets — in other words, if U any U U U , then U contains i i j j i j U are called the of the stratification, while the closures U are called the WD strata Y i i i closed strata . (If we want to emphasize the distinction, we will sometimes refer to the U of the stratification, even though they are not open in as the open strata strata X .) i The stratification can be recovered from the closed strata Y : we have i [ Y D : Y n U i j i Y ̈ Y i j X with strata U is: We say that a stratification of Definition 1.16. i k  A affine if each open stratum is isomorphic to some ; and k  . is isomorphic to an open subset of some quasi-affine if each U A i 0 1 n P P    P For example, a complete flag of subspaces gives an affine  i P stratification of projective space; the closed strata are just the and the open strata are i i 1 i affine spaces P U n Š A P . D i If a scheme has a quasi-affine stratification, then Proposition 1.17. is generated A.X/ X by the classes of the closed strata. We will induct on the number of strata U Proof of Proposition 1.17: . If this number i is 1 then the assertion is Corollary 1.15. Let U be a minimal stratum. Since the closure of U is a union of strata, U must 0 0 0 is stratified by the strata other than U n U WD X U . already be closed. It follows that 0 0 A.U/ is generated by the classes of the closures of these strata, and, by By induction, A.U Corollary 1.15, / is generated by ŒU ç . By excision (part (b) of Proposition 1.14) 0 0 the sequence Z ŒU 0 ç D A.U ! / !  ! A.X n U / A.X/ 0 0 0 is right exact. Since the classes in A.U/ of the closed strata in U come from the classes of (the same) closed strata in , it follows that A.X/ is generated by the classes of the X closed strata. In general, the classes of the strata in a quasi-affine stratification of a scheme X may 1 A.X/ ; for example, the affine line, with A. A be zero in / D Z , also has a quasi-affine stratification consisting of a single point and its complement, and we have already seen that the class of a point is 0. But in the case of an affine stratification, the classes are not only nonzero, they are independent:

42 28 Chapter 1 Introducing the Chow ring . Theorem 1.18 The classes of the strata in an affine stratification of a (Totaro [2014]) A.X/ form a basis of X . scheme We will often use results that are consequences of this deep theorem, although in our cases much more elementary proofs are available, as we shall see. 1.3.6 Functoriality A key to working with Chow groups is to understand how they behave with respect to morphisms between varieties. To know what to expect, think of the analogous situation with homology and cohomology. A smooth complex projective variety of (complex) dimension -manifold, so H n .X; Z / can be identified canoni- is a compact oriented 2n 2m 2n 2m H Z / (singular homology and cohomology). If we think of A.X/ cally with .X; Z / , then we should expect A H .X/ to be a covariant functor as being analogous to .X;  m from smooth projective varieties to groups, via some sort of pushforward maps preserv-  A.X/ as analogous to H .X; Z / , then we should expect ing dimension. If we think of to be a contravariant functor from smooth projective varieties to rings, via some A.X/ sort of pullback maps preserving codimension. Both these expectations are realized. Proper pushforward  W X is a proper map of schemes, then the image of a subvariety A ! Y is f Y If a subvariety X . One might at first guess that the pushforward could be defined f.A/  A to the class of f.A/ , and this would not be far off the mark. by sending the class of But this would not preserve rational equivalence (an example is pictured in Figure 1.8). Rather, we must take multiplicities into account. A  Y is a subvariety and dim A D dim f.A/ , then f j If W A ! f.A/ is A generically finite k .A/ is a finite extension , in the sense that the field of rational functions k .f.A// (this follows because they are both finitely generated fields, of the of the field dim A over the ground field). Geometrically the condition can same transcendence degree 1 2 f.A/ , the preimage y WD f j be expressed by saying that, for a general point x .x/ A in is a finite scheme. In this case the degree n WD Œ k .A/ W k .f.A//ç of the extension A y x for a dense open subset of of rational function fields is equal to the degree of over 2 f.A/ , and this common value n is called the degree of the covering of f.A/ by A . x f.A/ with multiplicity n in the pushforward cycle: We must count X (Pushforward for cycles) Let f W Definition 1.19 ! . be a proper map of schemes, Y and let A  Y be a subvariety. (a) If f.A/ has strictly lower dimension than A , then we set f . h A iD 0  (b) If dim D dim A and f j . has degree n f.A/ f i h A iD n h f.A/ , then we set  A (c) We extend f by linearity; that is, for any collection of subvarieties to all cycles on Y   P P f m . i A m D h A h i A f , we set Y  i i i i   i

43 Some techniques for computing the Chow ring Section 1.3 29 C a c  d C e C f  2g C h b C c d g X e b a f h 0 X 0 0 0 0 0 0 h g d c b a 0 0 0 0 0 0 C c 3d b  2g C C h a  Figure 1.8 Pushforwards of equivalent cycles are equivalent. With this definition, the pushforward of cycles preserves rational equivalence: If f W Y ! X is a proper map of schemes, then the map f ! W Z.Y/ Theorem 1.20.  defined above induces a map of groups f A W A . .Y/ ! Z.X/ k .X/ for each  k k For a proof see Fulton [1984, Section 1.4]. It is often hard to prove that a given class in A.X/ is nonzero, but the fact that the pushforward map is well-defined gives us a start: Proposition 1.21. If Spec k , then there is a unique map deg W A.X/ ! X is proper over Œpç of each closed point p 2 X to 1 and vanishing on the class of any Z taking the class . cycle of pure dimension > 0 As stated, Proposition 1.21 uses our standing hypothesis that the ground field is algebraically closed. Without this restriction we would have to count each (closed) point by the degree of its residue field extension over the ground field. A is We will typically use this proposition together with the intersection product: If k X and B is a k -codimensional -dimensional subvariety of a smooth projective variety a X such that A \ B is finite and nonempty, then the map subvariety of A .X/ ! Z W ŒZç 7! deg .ŒZçŒBç/ k sends to a nonzero integer. Thus no integer multiple mŒAç of the class A could be 0. ŒAç Pullback We next turn to the pullback. Let f W Y ! X be a morphism and A  X a  on cycles . A good pullback map f subvariety of codimension W Z.X/ ! Z.Y/ c 1 should preserve rational equivalence, and, in the nicest case, for example when f .A/ is generically reduced of codimension c , it should be geometric, in the sense that

44 30 Chapter 1 Introducing the Chow ring  ŒPç D ŒL f ç 1 C f L 2 f ç ŒCç D 2ŒL  2 p L 1  f .ŒPçŒCç/ Figure 1.9 D f 2Œpç .Œf D L . çŒCç/ D ŒL ç çf çŒ2L ŒCç D ŒL 1  2 1 1    1 iDh f A h .A/ i f A . This equality does not hold for all cycles, but does hold when is a Cohen–Macaulay variety. (Recall that a scheme is said to be Cohen–Macaulay if all its local rings are Cohen–Macaulay. For a treatment of Cohen–Macaulay rings see Eisenbud [1995, Chapter 18].) We start with a definition: f W Y ! X be a morphism of smooth varieties. We say a subvariety Definition 1.22. Let 1 is generically transverse to A if the preimage f  X .A/ is generically reduced f 1 and codim .f .A// . codim .A/ D X Y With that said, we have the following fundamental theorem: Theorem 1.23. Let f W Y ! X be a map of smooth quasi-projective varieties. c c  .Y/ .X/ ! A A W such that whenever A  X f There is a unique map of groups (a) we have is a subvariety generically transverse to f 1  .ŒAç/ Œf .A/ç: D f This equality is also true without the hypothesis of generic transversality as long 1  codim as .A// D codim is a .A/ and A is Cohen–Macaulay. The map f .f X Y A into a contravariant functor from the category of ring homomorphism, and makes smooth projective varieties to the category of graded rings. (Push-pull formula) The map is a map of graded modules over (b) W A.Y/ ! A.X/ f  k ˇ ̨ 2 A the graded ring .X/ and . More explicitly, if 2 A , then .Y/ A.X/ l  .X/: .f ̨ ̨  ˇ/ D f  f A ˇ 2   k l The last statement of this theorem is the result of applying appropriate multiplicities 1 f.f to the set-theoretic equality .A/ \ B/ D A \ f.B/ ; see Figure 1.9.

45 Some techniques for computing the Chow ring Section 1.3 31 One simple case of a projective morphism is the inclusion map from a closed subvariety Y  X . When X and Y are smooth, our definitions of intersections and i W is any subvariety of is represented , then ŒAçŒYç A pullbacks make it clear that, if X  — except that these are considered as classes in different i by the same cycle as .ŒAç/ varieties. More precisely, we can write  D .i ŒAç/: ŒAçŒYç i  In this case the extra content of Theorem 1.23 is that this cycle is well-defined as a Y cycle on X . Fulton [1984, Section 8.1] showed that it is even , not only as a cycle on X Y , and, more generally, he proved the existence of such a well-defined as a class on \ refined version of the pullback under a proper, locally complete intersection morphism (of which a map of smooth projective varieties is an example). The uniqueness statement in Theorem 1.23 follows at once upon combining the moving lemma with the following: Theorem 1.24. f W Y If X is a morphism of smooth quasi-projective varieties, then ! there is a finite collection of subvarieties is  X X A  X such that if a subvariety i generically transverse to each X . then A is generically transverse to f i (See Theorem A.6.) Note that this result depends on characteristic 0; it fails when f is not generically separable. Pullback in the flat case The flat case is simpler than the projective case for two reasons: first, the preimage is always of codimension k of a subvariety of codimension k ; second, rational functions on the target pull back to rational functions on the source. We will use the flat case to analyze maps of affine space bundles.  f W Y ! X be a flat map of schemes. The map f Theorem 1.25. W A.X/ ! A.Y/ Let defined on cycles by  1 h A i / WDh f f . .A/ i for every subvariety A  X preserves rational equivalence, and thus induces a map of Chow groups preserving the grading by codimension. X and Y When f is flat, the two pullback maps agree, as one sees are smooth and at once from the uniqueness statement in Theorem 1.23. 1.3.7 Dimensional transversality and multiplicities When two subvarieties A;B of a smooth variety X meet generically transversely, then we have ) D ŒA \ ŒAçŒBç 2 A.X/: (  Bç

46 32 Chapter 1 Introducing the Chow ring Does this formula hold more generally? Clearly it cannot hold unless the intersection A has the expected dimension. \ B A;B  be subvarieties of a smooth variety X such that every Let Theorem 1.26. X of the intersection \ B has codimension codim C D C irreducible component A .A;B/ m there is a positive integer . For each such component C , codim B C A codim C intersection multiplicity of and B along C , such that: called the A X ŒAçŒBç m .A;B/ŒCç 2 A.X/: (a) D C m (b) .A;B/ D 1 if and only if A and B intersect transversely at a general point of C . C (c) A and B are Cohen–Macaulay at a general point of C , then m In case .A;B/ is C the multiplicity of the component of the scheme \ B supported on C . In particular, A A and B are everywhere Cohen–Macaulay, then if D Bç: \ ŒAçŒBç ŒA at a general point of m .A;B/ depends only on the local structure of (d) and B . C A C For further discussion of this result see Hartshorne [1977, Appendix A], and for a full treatment see Fulton [1984, Chapter 7]. In view of Theorem 1.26, we make a definition: Definition 1.27. Two subschemes A and B of a variety X are dimensionally transverse if for every irreducible component C A \ B we have codim C D codim A C codim B . of The reader should be aware that what we call “dimensionally transverse” is often called “proper” in the literature. We prefer “dimensionally transverse” since it suggests the meaning (and “proper” means so many different things!). is smooth and C is a component of Recall that if \ B , then by Theorem 0.2 X A codim C  codim A C codim we have , so in this case the condition of dimensional B transversality is that and B intersect in the smallest possible dimension. (But note that A B \ A A and B are transverse.) may also be empty. In this case too, The Cohen–Macaulay hypothesis in part (c) is necessary: in Example 2.6 we will see a case where the intersection multiplicity is not given by the multiplicities of the components of the intersection scheme. Given that we sometimes have ŒA \ Bç ¤ ŒAçŒBç , it is natural to look for a correction term. This was found by Jean-Pierre Serre; we will describe it in Theorem 2.7, following Example 2.6. of the ŒAçŒBç Remarkably, it is often possible to describe the intersection product  X classes of subvarieties A;B geometrically even when they are not dimensionally transverse. See Chapter 13. P P Just as we say that cycles m A and B D A n are generically trans- Bj D i i j B verse if are and B A are generically transverse for all i;j , we say that A and j i dimensionally transverse if A i;j . B are dimensionally transverse for every and j i

47 Some techniques for computing the Chow ring Section 1.3 33 The following explains the amount by which generic transversality is stronger than dimensional transversality. A and B of a variety X are generically transverse if Proposition 1.28. Subschemes \ and only if they are dimensionally transverse and each irreducible component of B A A B is reduced. contains a point where \ X is smooth and A;B are dimensionally In particular, the proposition shows that, if X is smooth and A and B are generically , then C transverse subschemes that meet in a subvariety . The hypothesis that X is smooth cannot be dropped: For example, transverse along C 2 2 2 Œs ç the ideal .s ;st;t / defines a double line through the vertex k in the coordinate ring 2 .st;t in a reduced point. that meets the reduced line defined by / A If B are generically transverse, then each irreducible component C of Proof: and \ A contains a smooth point p 2 X such that A and B are smooth and transverse B at . It follows that C is smooth at p , and thus, in particular, C is reduced at p . p C be an irreducible component of \ B . Since the set of To prove the converse, let A is open, and since by hypothesis C contains one, the smooth points smooth points of X X that are contained in C form an open dense subset of C . Since A \ B of is generically reduced, the open set where is reduced is also dense, and it follows that the same is C C C p 2 C that is smooth on both true for the smooth locus of . Thus there is a point and . We must show that A and B are smooth at p . X The Zariski tangent space to C at p is the intersection of the Zariski tangent spaces . Since T T B in T X and C and X are smooth at p , A p p p dim C T C D dim T A C dim T dim B dim T X D p p p p dim T D A C dim T X: B dim p p By hypothesis, dim C dim A dim D B dim X: C Since dim T and A  dim A and dim T A B  dim B , we must have dim T dim A D p p p B T B D dim B , proving that A dim as well. Since the tangent are smooth at p and p spaces of A;B;X at p are equal to the corresponding Zariski tangent spaces, the equality dim T X C D dim T T A C dim T dim B p p p p above completes the proof. 1.3.8 The multiplicity of a scheme at a point In connection with the discussion of intersection multiplicities above, we collect here the basic facts about the multiplicity of a scheme at a point; for details, see Eisenbud [1995, Chapter 12]. We will also indicate, at least in some cases, how intersection multiplicities are related to multiplicities of schemes.

48 34 Chapter 1 Introducing the Chow ring 4 2 3 1 P P P P 3 and Figure 1.10 Ordinary double points of hypersurfaces of dimension 0;1;2 . Any discussion of the multiplicity of a scheme at a point begins with the case of a Z . In this case, we can be very explicit: hypersurface in a smooth n -dimensional variety is a hypersurface given locally around 2 X  Z and p as the zero locus of p If Z , we can choose local coordinates z D .z on ;:::;z a regular function / f Z in a 1 n neighborhood of and expand f around p , writing p f C C f f.z/ .z/ C f D .z/ 2 1 0 , .z/ homogeneous of degree k . The hypersurface X with p if f f D f.p/ D 0 contains 0 k has f and is then singular at D 0 . In general, we say that X if multiplicity m at p if p 1 p DD f . f at D 0 and f X ¤ 0 ; we write mult for the multiplicity of .X/ p 1 m m 0 m D 2 we say that p is a double point of X ; if m D 3 we say p is a triple point , and (If p tangent cone so on.) We define the X of X at TC to be the zero locus of f in the m p n A affine space with coordinates .z projectivized ;:::;z , and similarly we define the / n 1 1 n of X at p to be the scheme in P T X tangent cone defined by f . C m p We can say this purely in terms of the local ring O , without the need to invoke Z;p m  O is the maximal ideal, the multiplicity of local coordinates: If X at p is the Z;p m such that f 2 m largest . We can then take f in the quotient to be the image of m f m m m C 1 m = m . Note that since  m m 2 m C 1 m Sym D m = m m / D Sym = T . Z; m p m on the Zariski tangent space the vector space of homogeneous polynomials of degree , we can view the projectivized tangent cone as a subscheme of Z T P T . (Note that Z p p X itself is reduced at p , as the projectivized tangent cone may be nonreduced even though 2 3 in the case of a cusp, given locally as the zero locus of x .) The multiplicity can also y be characterized in these terms simply as the degree of the projectivized tangent cone. For example, the simplest possible singularity of a hypersurface X , generalizing the case of a node of a plane curve, is called an ordinary double point . This is a point 2 X such that the equation of p can be written in local coordinates with p D 0 X as above with f — that D f nondegenerate quadratic form D 0 and where f is a 2 0 1 X at p is a smooth quadric. Indeed, examples is, the projectivized tangent cone to are the cones over smooth quadrics — see Figure 1.10. (Here it is important that the n 1 P of its ideal, is smooth if the generator f characteristic is not 2: A quadric in 2

49 Some techniques for computing the Chow ring Section 1.3 35 f together with the derivatives of , is an irrelevant ideal; when the characteristic is not 2, 2 P 2f z Euler’s formula @f D [email protected] shows that it is equivalent to assume that the partial i 2 2 i are linearly independent, and this is the property we will often use. f derivatives of 2 no In characteristic 2 — where a symmetric bilinear form is also skew-symmetric — quadratic form in an odd number of variables has this property.) How do we extend this definition to arbitrary schemes X ? The answer is to start by defining the tangent cones. We can do this explicitly in terms of local coordinates .z z ;:::;z : We define the tangent ) on a smooth ambient variety Z containing D X n 1 n cone to be the subscheme of A defined by the leading terms of all elements of the I  O p of X at ideal , and the projectivized tangent cone to be the corresponding Z;p 1 n P subscheme of . As before, this can be said without recourse to local coordinates (or, for that matter, Z ). To start, we filter the local ring any ambient variety by powers of its maximal O X;p ideal m : 3 2 m  m   m  O : X;p We then form the associated graded ring 2 2 3 m = m D ̊ m k A m ̊ ̊ ; = and define the tangent cone and projectivized tangent cone to be Spec A and Proj A respectively. Note that since the ring A is generated in degree 1, we have a surjection 2  / D Sym .T X/ ! A; Sym m . = m p so that these can be viewed naturally as subschemes of the Zariski tangent space X T p and its projectivization, respectively. As we will see shortly, one important feature of these constructions is that we always have TC 1; X D dim X and dim T C dim X D dim X p p even though the dimension of the Zariski tangent space may be larger. We now define the multiplicity mult .X/ X at p to be the degree of the projec- of p tivized tangent cone T C . In X , viewed as a subscheme of the projective space P T X p p purely algebraic terms, we can express this directly in terms of the Hilbert polynomial of the graded ring : If we set A ; .m/ h dim A D k m A then for m  0 the function h of degree will be equal to a polynomial p .m/ A A dim X 1 , called the Hilbert polynomial of A . The multiplicity mult is then equal .X/ p . . dim X to times the leading coefficient of the Hilbert polynomial p .m/ 1/Š A

50 36 Chapter 1 Introducing the Chow ring mult of a scheme Y along an It follows from the theory that the multiplicity .Y/ Y i Y , as introduced in Section 1.2.1 in connection with the Y irreducible component of i definition of the cycle associated to a scheme, is equal to the multiplicity of Y at a . Y general point of i Tangent cones and blow-ups There is another characterization of the projectivized tangent cone that will be very useful to us in what follows. We start by recalling some basic facts about blow-ups. Blowing-up is an operation that associates to any scheme Y a morphism  W B D Bl Z .Z/ ! Z . and subscheme Y The general operation is described and characterized in Chapter 4 of Eisenbud and Harris is a [2000]; in the present circumstances, we will be concerned with the case where Y smooth point p Z . 2 1  B is defined to be  .Y/ The exceptional divisor , the preimage of Y in B . If E z  strict transform X X  B to be the closure in B Z is any subscheme, we define its 1 X .X Y \ X/  n away from Y . of the preimage of X is embedded in a smooth ambient variety Z of dimension n , and Suppose that Z p . In this case the exceptional divisor E is isomorphic to at consider the blow-up of n 1 P the projectivized tangent cone T C . Unwinding the definitions, we can see that X p n 1 z is the intersection of P X with E Š to X at p . This gives us immediately that dim T X D dim X 1 . C p X Z : Again, we can say this without having to choose an embedding of in a smooth Since blow-ups behave well with respect to pullbacks (see Proposition IV-21 of Eisenbud C and Harris [2000]), we could simply say that X T is the exceptional divisor in the p blow-up Bl .X/ . p Multiplicities and intersection multiplicities The notions of multiplicity (of a scheme at a point) and intersection multiplicities (of two subschemes meeting dimensionally transversely in a smooth ambient variety) n 2 X  P are closely linked: If is a point on a subscheme of pure dimension k p n k n ƒ  P P is a general .n Š k/ -plane containing p , then the intersection and m .X;ƒ/ is equal to mult multiplicity .X/ . p p This statement can be generalized substantially: X and Y be two subschemes of complementary dimension inter- Let Proposition 1.29. Z , and p 2 X \ Y any point of secting dimensionally properly in a smooth variety Y T X and T C C are disjoint in P T intersection. If the projectivized tangent cones Z , p p p then m .X;Y/ D mult .Y/: mult  .X/ p p p This proposition is proved in Section 2.1.10. In general, there is only the inequality see Fulton [1984, Chapter 12]. I .X;Y/  mult .Y/ .X/  mult m p p p

51 The first Chern class of a line bundle Section 1.3 37 1.4 The first Chern class of a line bundle Many of the most interesting and useful classes in the Chow groups come from vector bundles via the theory of Chern classes. The simplest case is that of the first Chern class of a line bundle, which we will now describe. We will introduce the theory in more generality in Chapter 5. If is a line bundle on a variety X and  is a rational section, then on an open affine L of a covering of X we may write  in the form f D =g set and define Div ./ j U U U U .f / . This definition agrees where two affine open sets overlap, and thus Div Div .g/ Cartier divisor (see Hartshorne [1977, Section II.6]). X defines a divisor on , which is a is another rational section of L then ̨ D = is a well-defined rational Moreover, if  function, so ./ Div Div ./ D Div . ̨/  0 mod Rat .X/: Thus for any line bundle on a quasi-projective scheme X we may define the first Chern L class c L / 2 A.X/ . 1 for any nonzero rational section  . (If to be the rational equivalence class of the divisor  we were working over an arbitrary scheme, we would have to insist that the numerator and denominator of our section were locally nonzerodivisors.) Note that there is no n c distinguished cycle in the equivalence class. As a first example, we see that O .d// . 1 P d ; in the notation of Section 2.1 it is d , where is the class of any hypersurface of degree is the class of a hyperplane.  Recall that the Picard group Pic .X/ is by definition the group of isomorphism 0 0 on X Œ L ç C Œ L L ç D Œ L ̋ L classes of line bundles ç . , with addition law If is a variety of dimension n , then c is a group homomorphism Proposition 1.30. X 1 .X/: W Pic .X/ ! A c 1 1 n X c is an isomorphism. If is smooth, then 1 Y If X is a divisor in a smooth variety X , then the ideal sheaf of Y is a line  bundle denoted O . The . Y/ , and its inverse in the Picard group is denoted O .Y/ X X c . above takes ŒYç to O inverse of the map .Y/ 1 X To see that c is a group homomorphism, suppose that Proof of Proposition 1.30: L 1 0 0 0 are line bundles on X . If  and  and are rational sections of L and L L respectively, 0 0 0 . then is a rational section of L ̋ L ̋ whose divisor is Div ./ C Div   / . Now assume that is smooth and projective. Since the local rings of X are unique X factorization domains, every codimension-1 subvariety is a Cartier divisor, so to any divisor we can associate a unique line bundle and a rational section. Forgetting the section, we get a line bundle, and thus a map from the group of divisors to Pic .X/ . By

52 38 Chapter 1 Introducing the Chow ring Proposition 1.10, rationally equivalent divisors differ by the divisor of a rational function, and thus correspond to different rational sections of the same bundle. It follows that the map on divisors induces a map on A . .X/ , inverse to the map c 1 1 n A c Pic .X/ ! W If .X/ is in general neither injec- X is singular, the map 1 n 1 is an irreducible plane cubic with a node, then X tive or surjective. For example, if W Pic .X/ ! c A .X/ is not a monomorphism (Exercise 1.35). On the other hand, if 1 1 3 X is a quadric cone with vertex p , then A P .X/ D Z and is generated by the class  1 .X/ W Pic .X/ ! A c of a line, and the image of is 2 Z (Exercise 1.36). 1 1 Another case when the moving lemma is easy is when the class of the cycle to be c . We also get a useful formula for the . moved has the form / for some line bundle L L 1 c L / : product of any class with . 1 Proposition 1.31. X L is a line Suppose that is a smooth quasi-projective variety and . If Y bundle on ;:::;Y X are any subvarieties of X , then there is a cycle in the class n 1 c is any . L / that is generically transverse to each Y X . If X of Y  is smooth and i 1 subvariety, then c . L /  ŒYç D c /: . L j 1 1 Y c j . L The class , / on the right-hand side of the formula is actually a class in A.Y/ 1 Y so to be precise we should have written i is the inclusion .c X . L j ! // , where i W Y , 1  Y i and the pushforward map, first encountered in Proposition 1.14 and defined in general  in Section 1.3.6. This imprecision points to an important theoretical fact: Even on a X singular variety (or scheme) one can form the intersection product of any class with the first Chern class of a line bundle, defined (when the class is the class of a subscheme) ŒYç . L /  via the prescription D c above. . L j / c 1 1 Y This intersection is actually defined by the formula as a class on Y , not just a class X . This is the beginning of the theory of “refined intersection products” defined in on Fulton [1984]. When we define other Chern classes of vector bundles we shall see that the same construction works in that more general case. We imposed the hypothesis of smoothness in Proposition 1.31 because we have only discussed products in this context. In fact, the formula could be used to define an action of a class of the form . L / c A.X/ much more generally. This is the point of view on 1 taken by Fulton. Sketch of proof of Proposition 1.31: X Since is quasi-projective, there is an ample 0 0 ̋ n X . For a sufficiently large integer L both the line bundles L bundle on and n n 0 ̋ n 0 ̋ 0 are very ample, so by Bertini’s theorem there are sections  H L . L ̋ L / and 2 0 0 ̋ n / L 2  ̋ L H whose zero loci Div ./ and Div ./ are generically transverse to . each Y , proving . The class c ./ . L / is rationally equivalent to the cycle Div ./ Div 1 i the first assertion. Moreover, c by Theorem– . L /ŒY ç ç D Œ Div ./ \ Y Y ç Œ Div ./ \ i i 1 i Definition 1.5. Since Div ./ \ Y j D Div . / , and similarly for  , we are done. i Y i

53 The first Chern class of a line bundle Section 1.4 39 deg .K / topology curvature dim Aut .X/ cover points genus X 1 < 0  > 0 3 CP infinite 0 > 0  D 0 0 1 C infinite 1 0 > 0  < 0 finite 2 Å finite  < 0 / < 0 0 .K .K / D , deg and deg .K / > 0 . Table 1.1 Behavior of curves for deg X X X 1.4.1 The canonical class Perhaps the most fundamental example of the first Chern class of a line bundle is the , which we will define here; in the following section, we will describe canonical class adjunction formula the , which gives us a way to calculate it in many cases. of be a smooth -dimensional variety. By the canonical bundle ! Let n X we X X V n of X of the cotangent bundle  mean the top exterior power ; this is the  X X line bundle whose sections are regular -forms. By the canonical class we mean the n 1 of this line bundle. Perhaps reflecting the German .! 2 / first Chern class A c .X/ 1 X language history of the subject, this class is commonly denoted by K . X The canonical class is probably the single most important indicator of the behavior of X , geometrically, topologically and arithmetically. For example, the only topological invariant of a smooth projective curve X is its genus g D g.X/ , over the complex field C and we have .K / deg 2g 2: D X and the arithmetic over Q of X are Virtually every aspect of the geometry over C deg K fundamentally different depending on whether is negative, zero or positive, X corresponding to 0;1 or g D 2 , as can be seen in Table 1.1. (Here the topology g  is represented by the topological Euler characteristic, the differential geometry by the curvature of a metric with constant curvature, the complex analysis by the isomorphism class as a complex manifold of the universal cover and the arithmetic by the number of rational points over a suitably large finite extension of Q .) (Projective space) . We can easily determine the canonical class of a Example 1.32 n n -form ! on P projective space. To do this, we have only to write down a rational and determine its divisors of zeros and poles. For example, if X ;:::;X are homogeneous n 0 n P and coordinates on X i D x ; i D 1;:::;n; i X 0 n n X Š  P are affine coordinates on the open set given by U ! ¤ 0 , we may take A 0 to be the rational n -form given in U by ! D dx ^^ dx : 1 n

54 40 Chapter 1 Introducing the Chow ring U is regular and nonzero in The form , so we have only to determine its order of zero ! n 0 be the at infinity. To this end, let or pole along the hyperplane  P / U V.X D H 0 0 , and take affine coordinates y ;:::;y on U X with y D X =X open set . 0 ¤ 1 n i 0 n n i We have  =y for y D 1;:::;n 1; i i 0 x D i for i D 1=y n; 0 so that  2 i for /dy 1; 1;:::;n D =y /dy .1=y .y 0 i 0 i 0 D dx i 2 /dy D for i n: .1=y 0 0 Taking wedge products, we see that n . 1/ ^^ D ! D dx dy ^^ dx ; dy 0 n 1 n 1 1 n C y 0 whence D .n Div 1/H; C .!/ so n K D .n C 1/; P n 1 2 A where . P  / is the class of a hyperplane. 1.4.2 The adjunction formula X n , and suppose that Y  X is a Let again be a smooth variety of dimension -dimensional subvariety. There is a natural way to relate the canonical .n smooth 1/ to that of X Y T of Y with the restriction class of : If we compare the tangent bundle Y X j , we get an exact sequence to Y of the tangent bundle T T of X X Y ! T ! T ! 0 0; ! N j Y Y X Y=X N where the right-hand term is called the normal bundle of Y in X . Taking exterior Y=X powers, this gives an equality of line bundles V V n n 1 j N Š . T ; / T ̋ Y X Y Y=X so that V V n n 1  T / T ; Š j . ̋ N Y X Y Y=X and dualizing we have Š : ! j ̋ N ! X Y Y Y=X N in another way. There is an exact sequence Moreover, we can compute Y=X ı 2 I ! = I 0; 0 ! !   j ! Y X Y Y=X Y=X

55 The first Chern class of a line bundle Section 1.4 41 ı sends the germ of a function to the germ of its differential (see, for where the map  example, Eisenbud [1995, Proposition 16.3]). This identifies N with the locally Y=X 2 is a Cartier divisor in = free sheaf . When Y I X , the case of primary interest I Y=X Y=X . of Y in X is the line bundle O for us, the ideal sheaf I Y/ , and the sheaf X Y=X 2 is its restriction to ; thus D O Y/ ̋ I . Y , denoted O =I I Y Y Y=X Y=X Y=X : N O j .Y/ Š X Y Y=X Combining this with the previous expression, we have what is commonly called the : adjunction formula .n Y  X Proposition 1.33 . 1/ -dimensional (Adjunction formula) If is a smooth -dimensional variety, then subvariety of a smooth n ! D j ! ̋ O .Y/ j ; Y X Y X Y which we usually write as ! .Y/ j Y . In particular, if is a smooth curve in a smooth Y X complete surface X , then the degree of K is given by an intersection product: Y  D K .K C ŒYç/ŒYç deg : deg X Y 1.4.3 Canonical classes of hypersurfaces and com- plete intersections We can combine the adjunction formula with the calculation in Example 1.32 to calculate the canonical classes of hypersurfaces, and more generally of complete n X be a smooth hypersurface of P  intersections, in projective space. To start, let . We have d degree n D ! ! .X/ 1/: n D O .d j X X X P Thus K .d n 1/; D X 1  D c . O . .1// 2 A where .X/ is the class of a hyperplane section of X 1 X More generally, suppose D Z X \\ Z 1 k Z . ;:::;Z ;:::;d is a smooth complete intersection of hypersurfaces d of degrees 1 1 k k Applying adjunction repeatedly to the partial intersections Z , we see that \\ Z i 1   X D O n 1 C ! d i X X and so   X : n 1 C D d K i X

56 42 Introducing the Chow ring Chapter 1 is assumed smooth the partial This argument is not complete, because even though X \\ Z Z intersections may not be. One way to complete it is to extend the definition i 1 of the canonical bundle to possibly singular complete intersections — the adjunction formula is true in this greater generality. Alternatively, if we order the hypersurfaces D V.F by a linear combination / so that d Z  d F and replace i i i 1 k k X 0 D F ; C F G F j j i i j 1 D i C 0 0 general of degree d will have intersection d / , the hypersurfaces Z with G D V.F j i j i i , and by Bertini’s theorem the partial intersections will be smooth. X 1.5 Exercises n Exercise 1.34. A be a subvariety not containing the origin, and let Let W Y   1 n  A be the closure of the locus P ı Df .t;z/ j z 2 t W Y g ;  W t D 0 is the cone with over as in the proof of Proposition 1.13. Show that the fiber of n n A \ over the intersection Y vertex the origin H 0 , where Y  P 2 is the closure of 1 n n n P and H is the hyperplane at infinity. D Y in n A P 1 Exercise 1.35. X is an irreducible plane cubic with a node, then c Show that if W 1 Pic A .X/ .X/ is not a monomorphism, as follows: Show that there is no biregular ! 1 1 P map from . Use this to show that if p to q 2 X are smooth points, then the X ¤ O are nonisomorphic. Show, however, that the zero loci of .p/ and line bundles .q/ O X X p q , are rationally equivalent. and their unique sections, the points 3 then X Show that if Exercise 1.36. is a quadric cone with vertex p  A Z .X/ D P 1 and is generated by the class of a line, and show that the image of c .X/ W Pic .X/ ! A 1 1 2 Z is by showing that the image consists of the subgroup of classes of curves lying on 3 P that have even degree as curves in X is not in X . In particular, the class of a line on the image. Hint: C  X of odd degree can be a Cartier divisor Do this by showing that no curve on X : If such a curve meets the general line of the ruling of X at ı points away from p and has multiplicity m at p , then intersecting C with a general plane through p we ; it follows that see that D 2ı C m .C/ m is odd, and hence that C cannot be Cartier deg . p . Thus, the class ŒMç of a line of the ruling cannot be c L . L / for any line bundle at 1

57 Chapter 2 First examples Keynote Questions (a) k Let ;F ;F 2 F ŒX;Y;Zç be three general homogeneous cubic polynomials in 1 2 0 C F F C t three variables. Up to scalars, how many linear combinations F t t 1 2 0 2 0 1 factor as a product of a linear and a quadratic polynomial? (Answer on page 65.) ;F Let ;F be four general homogeneous cubic polynomials in ;F ŒX;Y;Zç F k 2 (b) 3 2 0 1 t factor F three variables. How many linear combinations C t F F t C t C F 2 0 1 3 1 3 0 2 as a product of three linear polynomials? (Answer on page 65.) A;B;C be general homogeneous polynomials of degree d in three variables. Let (c) t D .t .A.t/;B.t/;C.t// ;t is ;t .0;0;0/ Up to scalars, for how many triples ¤ / 2 0 1 .t a scalar multiple of ;t / ? (Answer on page 55.) ;t 0 1 2 in two variables. Let S d denote the space of homogeneous polynomials of degree (d) d  S If and W  V S are general linear spaces of dimensions a and b with e d C b D d C 2 , how many pairs .f;g/ 2 V  W are there (up to multiplication of a j f g by scalars) such that f and g ? (Answer on page 56.) each of 3 3 S  P L be a smooth cubic surface and (e)  P Let a general line. How many planes containing are tangent to S ? (Answer on page 50.) L 3 3 L  P t be a line, and let S and T  P (f) be surfaces of degrees s and Let containing . Suppose that the intersection S \ T is the union of L and a smooth L curve C . What are the degree and genus of C ? (Answer on page 71.) In this chapter we illustrate the general theory introduced in the preceding chapter with a series of examples and applications. The first section is a series of progressively more interesting computations of Chow rings of familiar varieties, with easy applications. Following this, in Section 2.2 we see an example of a different kind: We use facts about the Chow ring to describe some geometrically interesting loci in the projective space of cubic plane curves.

58 44 Chapter 2 First examples Finally, in Section 2.4 we briefly describe intersection theory on surfaces, a setting in which the theory takes a particularly simple and useful form. As one application, we , a tool used classically to understand the describe in Section 2.4.3 the notion of linkage 3 P . geometry of curves in n and some related 2.1 The Chow rings of P varieties So far we have not seen any concrete examples of the intersection product or pullback. The first interesting case where this occurs is projective space. n is Theorem 2.1. The Chow ring of P n C n 1 Z D / P /; A. Œç=. n 1 2 . P /  is the rational equivalence class of a hyperplane; more generally, the where A k k d is d . class of a variety of codimension and degree n m . P In particular, the theorem implies that / Š Z for 0  m  n A , generated by the .n class of an m/ -plane. The natural proof, given below, uses the intersection product. 1 n be a complete flag of subspaces. Applying Propo- P Let    P g  p f Proof: i i 1 n k P U / D , we see that A P . P sition 1.17 to the affine stratification with strata n i n k n , and thus by the class of any .n k/ -plane L  P is generated by the class of . P n n . Since a general P Using Proposition 1.21, we get / D Z . .n k/ -plane L intersects A a general k -plane M transversely in one point, multiplication by ŒMç induces a surjective n n n n k k map A . . P P / D Z , so A A . P / / D Z for all k . ! n .n An -plane L  P k/ is the transverse intersection of k hyperplanes, so k D ; ŒLç  n n 1 where .  2 / is the class of a hyperplane. Finally, since a subvariety X  P A of P dimension n k and degree d intersects a general k -plane transversely in d points, we k n n k D d . Since deg . have deg / D 1 , we conclude that ŒXç / d D . .ŒXç Here are two interesting qualitative results that follow from Theorem 2.1: n Corollary 2.2. P to a quasi-projective variety of dimension strictly A morphism from n is constant. less than n m ' W P Proof: ! X  P Let be the map, which we may assume is surjective onto X . The preimage of a general hyperplane section of is disjoint from the preimage of a X general point of X . But if 0 < dim X < n then the preimage of a hyperplane section of X has dimension n 1 and the preimage of a point has dimension > 0 . Since any two n must meet, this is a contradiction. such subvarieties of P

59 n The Chow rings of and some related varieties Section 2.1 45 P n n If Corollary 2.3. is a variety of dimension m and degree d then A X . P P n X/ Š  m n 0 0 =.d/ n then A m < m are . P  n X/ D Z . In particular, m and d Z , while if m n P n X . determined by the isomorphism class of A Proof: .X/ ! Part (b) of Proposition 1.14 shows that there are exact sequences i n n / ! A A . P ! P X/ . 0 . Furthermore A .X/ D Z by part (b) of Proposition 1.8, n m i i n 0 0 .X/ 0 D for m < m A  n . By Theorem 2.1, we have A for . P while / D Z i m n i n , and the image of the generator of A 0 .X/ in   . P A / is d times the generator m m n P A / . The results of the corollary follow. of . i n n ́ P Theorem 2.1 implies the analog of Poincar / : A e duality for . P A. / is dual to k n k A . / via the intersection product. The reader should be aware that in cases where the P ́ not Chow groups and the homology groups are different, Poincar e duality generally does X is a variety, A .X/ hold for the Chow ring; for example, when .X/ Š Z , but A 0 dim X need not even be finitely generated. One aspect of Theorem 2.1 may, upon reflection, seem strange: why is it that only n P the dimension and degree of a variety are preserved under rational equivalence, X  a curve) the arithmetic genus? X and not other quantities such as (in the case of First of all, to understand why this may appear curious, we recall from Eisenbud and Harris [2000, Proposition III-56] (see also Corollary B.12) that, if B is reduced n B  P B is flat over and connected, then a closed subscheme if and only if the Y  1 n Z  P  is an fibers all have the same Hilbert polynomial. Thus, for example, if P 1 , then the fibers Z and Z irreducible surface dominating will be one-dimensional P 0 1 n P subschemes of having not only the same degree, but also the same arithmetic genus. 3 0 and  C P C Why does this not contradict the assertion of Theorem 2.1 that curves but different genera are rationally equivalent? d of the same degree 1 P The explanation is that both can be deformed, in families parametrized by , 3 0 , C to schemes , so that C L  P supported on a line and having multiplicity d 0 0 0 C i  h C . The difference in the genera i D d h L i as cycles, and likewise for C h 0 0 C will be reflected in two things: the scheme structure along the line in C and of 0 and the flat limits C C , and the presence and multiplicity of embedded points in 0 0 these limits. 0 2 For an example of the former, note that the schemes V..x;y/ C / and C D D 0 0 3 / are both supported on the line V.x;y D V.x;y/ , and both have multiplicity 3, but L 0 C is 1 (after all, it is a plane cubic!). But is 0, while that of C the arithmetic genus of 0 0 the mechanism by which we associate a cycle to a scheme does not see the difference in 0 . Similarly, a twisted cubic curve iDh C h the scheme structure; we have C 3 h L i iD 0 0 3 0 P C can be deformed to a scheme generically isomorphic to either C ; the or C  0 0 difference in the arithmetic genus is accounted for by the fact that in the latter case the limiting scheme will necessarily have an embedded point. But again, rational equivalence does not “see” the embedded point; we have ŒCç D 3ŒLç regardless.

60 46 Chapter 2 First examples ́ ezout’s theorem 2.1.1 B ́ As an immediate consequence of Theorem 2.1, we get a general form of B ezout’s theorem: n ́ If (B X ezout’s theorem) are subvarieties of codimen-  P . ;:::;X Corollary 2.4 1 k P n ;:::;c c c sions  , with , and the X intersect generically transversely, then i i 1 k Y deg X deg / D .X \\ .X /: i 1 k n  P In particular, two subvarieties having complementary dimension and inter- X;Y deg secting transversely will intersect in exactly deg .Y/ points. .X/  Using multiplicities we can extend this formula to the more general case where we assume only that the varieties intersect dimensionally transversely (that is, all components T P c D have codimension equal to of the intersection Z X ), as long as the X are i i i generically Cohen–Macaulay along each component of their intersection. In this case, m the intersection multiplicity of their Z along a component X of the / ;:::;X .X i ̨ 1 Z k ̨ Z at intersection, as described in Section 1.3.7, is equal to the multiplicity of the scheme Z . a general point of ̨ n Corollary 2.5. X ;:::;c ;:::;X c  P Suppose are subvarieties of codimensions 1 1 k k P n whose intersection is a scheme c , with irreducible com- of pure dimension Z i ;:::;Z ponents Z . If the X , are Cohen–Macaulay at a general point of each Z ̨ i t 1 then X Y ŒZ D ç D ŒZç ŒX I ç i j equivalently, X Y : deg deg D Z deg X D Z i j n Note that by the degree of a subscheme Z of dimension m we mean mŠ times P  is irreducible this will be Z the leading coefficient of the Hilbert polynomial; in case equal to the degree of the reduced scheme Z times the multiplicity of the scheme, and red more generally it will be given by X D deg mult .Z .Z/ .Z/ deg /; red Z i where the are the irreducible components of Z of maximal dimension m . Z i The Cohen–Macaulay hypothesis is satisfied if, for example, the X are all hyper- i 2 surfaces; thus the classical case of two curves intersecting in P is covered. There is a standard example that shows that the Cohen–Macaulay hypothesis is necessary:

61 n The Chow rings of and some related varieties Section 2.1 47 P V V 1 2 p B 4 A [ V Figure 2.1 Let  P V , where the V D are general 2-planes, and let B be a i 1 2 4 V 2-plane passing through the point . The degree of the product ŒBçŒAç V A. P \ / in 2 1 0 B to a plane transverse to A , but the length of the local is 2, as one sees by moving B \ is 3. B ring of A 4 4 P V , let Example 2.6. Let ;V . X P D be general 2-planes and let A D V V [  1 2 2 1 V be a 2-plane that and V Since are general, they meet in a single point p . Let B 2 1 0 and does not meet A anywhere else, and let B p be a 2-plane that does passes through p and meets each of not pass through ;V V in a single (necessarily reduced) point. The 1 2 4 0 0 A h and i are rationally equivalent in P i . The intersection B h \ B consists of B cycles 0 .B \ A/ D 2 (see Figure 2.1). two reduced points, so deg B \ A is strictly greater than 2: Since the Zariski However, the degree of the scheme 4 A V D [ V , the tangent at the point p is all of T / . P tangent space to the scheme p 1 2 B \ A at p must be all of T must space to the intersection . In other words, B \ A .B/ p contain the “fat point” at p in the plane B (that is, the scheme defined by the square of the ideal of p in B ), and so must have degree at least 3. In fact, we can see that the degree of the scheme B A is equal to 3 by a local \ B meets only at the point p , we have to show that the calculation, as follows. Since A is 3. Let O I .B/ C I .A// O ç ;:::;x =. S D k Œx length of the Artinian ring 4 4 0 4 ;p P P ;p 4 V . We may choose be the homogeneous coordinate ring of to have ;V P and B 1 2 homogeneous ideals x .x /; ;x x / \ .x ;x ;x x / D .x I.A/ D ;x ;x x 2 0 2 1 3 2 3 1 1 3 0 0 I.B/ D .x /: x x ;x 1 2 0 3 Modulo I.B/ , we can eliminate the variables and x x and the ideal I.A/ becomes 3 2 2 2 .x x ;x ;x , this is the square of the / . Passing to the affine open subset where x ¤ 0 4 0 1 1 0 maximal ideal corresponding to the origin in B . Therefore O O .A// I =. I .B/ C 4 4 P ;p ;p P f =x , and hence its length is 3. has basis g ;x 1;x =x 4 4 0 1 Given that we sometimes have ŒA \ Bç ¤ ŒAçŒBç , it is natural to look for a correction term. In the example above, the set-theoretic intersection is a point, so this comes down to 3 2 D 1 . Of course looking for a formula that will predict the difference in multiplicities the correction term should reflect nontransversality, and one measure of nontransversality is the quotient I.A/ \ I.B/=.I.A/  I.B// . In the case above one can compute this, finding

62 48 Chapter 2 First examples that the quotient is a finite-dimensional vector space of length 1 — just the correction J/ .I \ J/=.I  , the quotient I;J in any ring R term we need. Now for any pair of ideals R .R=I;R=J/ (see Eisenbud [1995, Exercise A3.17]). With this is isomorphic to Tor 1 information, knowing a special case proven earlier by Auslander and Buchsbaum, Serre [2000] produced a general formula (originally published in 1957): (Serre’s formula) . Suppose that  X are dimensionally transverse Theorem 2.7 A;B and Z is an irreducible component of A \ B . The subschemes of a smooth scheme X and Z is intersection multiplicity of B A along dim X X  O X;Z i D .A;B/ m : length ; / . Tor O O . 1/ Z A;Z B;Z O i A \ B;Z 0 i D The first term of the alternating sum in Serre’s formula is O X;Z length /; . O I ; O C / D Tor I length O =. X;Z A B;Z A;Z B O O 0 \ \ B;Z B;Z A A Z A \ B ; the remaining terms, in the subscheme which is precisely the multiplicity of involving higher Tors, are zero in the Cohen–Macaulay case and may be viewed as correction terms. We note that this formula is used relatively rarely in practice, since there are many alternatives, such as the one given by Fulton [1984, Chapter 7]. 2.1.2 Degrees of Veronese varieties Let   C n d n N 1; ; with N D ! P D P   W n;d n be the Veronese map I ;:::ç; ç ;:::;Z Œ:::;Z ŒZ 7! n 0 I ˆ ranges over all monomials of degree d in n C 1 variables. The image where D Z N n  ˆ P D  is called the of the Veronese map d -th Veronese variety of P  , as n;d n;d N P is any subvariety of projectively equivalent to it. This variety may be characterized, N P , as the image of the map associated to the up to automorphisms of the target n j complete linear system .d/ j ; in other words, by the property that the preimages O P n N n 1 in P  P .H/ comprise all hypersurfaces of degree d H P  . of hyperplanes  n P D There is another attractive description, at least in characteristic 0: writing V , P  V .n C 1/ -dimensional vector space, is an where is projectively equivalent to the n;d d d V ! P Sym map taking V by Œvç 7! Œv P ç ; for if the coordinates of v are v ;:::;v 0 n d v are then the coordinates of dŠ d d 0 n Q .v v /:  n 0 d Š i i

63 n The Chow rings of Section 2.1 49 P and some related varieties If the characteristic is 0 then the coefficients are nonzero, so we may rescale by an N P automorphism of to get the standard Veronese map above. We can use Corollary 2.4 to compute the degrees of Veronese varieties: n ˆ d . The degree of is Proposition 2.8. n;d is the cardinality of its intersection with n general hyperplanes The degree of ˆ Proof: N H  P is one-to-one, this is in turn the cardinality of the ; since the map  ;:::;H n 1 n 1 1 \\ intersection / .H .H f /  P f . The preimages of the hyperplanes H i 1 n n ́ d in P general hypersurfaces of degree . By B ezout’s theorem, the cardinality of n are n . their intersection is d 2.1.3 Degree of the dual of a hypersurface The same idea allows us to compute the degree of the dual variety of a smooth n n   d , that is, the set of points X   P X of degree P hypersurface corresponding n P that are tangent to X . (In Chapter 10 we will generalize this to hyperplanes of notion substantially, discussing the duals of varieties of higher codimension and singular varieties as well.)  G is a variety because it is the image of X under the Gauss map The set X W X n  X , a morphism that sends a point p 2 ; in to its tangent hyperplane T X P ! X p X is the zero locus of the homogeneous polynomial F.Z coordinates, if ;:::;Z , then / n 0 G is given by the formula X   @F @F 7! G : W .p/ .p/;:::; p X @Z @Z n 0 is smooth, the partials of X To see that this map is well-defined, note first that, since X (and this implies, by Euler’s relation, that they do not F have no common zeros on n n  n have any common zeros in defines a morphism P P ! P X ). Thus . When p 2 G , X Euler’s relation shows that the vector .p/ is orthogonal to the vector Q p representing G X p ; thus the linear functional represented by G .p/ induces a functional on the point X n P , and the zero locus of this functional is the tangent space to the tangent space to X at p .  d D 1 , the map G , then the fact is constant and X If is a point. But if d > 1 X that the partials of have no common zeros says that the map G F is finite: If G were X X C  X , the restrictions to constant on a complete curve of the partials of F would be C scalar multiples of each other, and so would have a common zero. n X  P 2 is a smooth hypersurface of degree d  In particular, if , the dual n   variety  P is again a hypersurface, though not usually smooth. The smoothness X  Q D of the quadric cone Q hypothesis is necessary here; for example, the dual 3 3  2 . is a conic curve in P Y V.XZ /  P

64 50 Chapter 2 First examples Figure 2.2 Six of the lines through a general point are tangent to a smooth plane cubic (but often not all the lines are defined over R ). is a smooth hypersurface the map X is G We will see in Corollary 10.21 that when X birational onto its image as well as finite. (This requires the hypothesis of characteristic 0; p strangely enough, it may be false in characteristic , where for example a general tangent line to a smooth plane curve may be bitangent!) We will use this now to deduce the degree of the dual hypersurface: n X  If is a smooth hypersurface of degree d > 1 , then the dual of Proposition 2.9. P n 1 is a hypersurface of degree 1/ . X d.d  n  is the number of points of intersection P X The degree of the dual variety  Proof: n    1 general hyperplanes H X of P and n . Since by Corollary 10.21 the map i n   ! G is birational, this is the same as the number of points of  P W X X X 1 / is given by the partial derivatives of .H intersection of the preimages G . Since G i X X the defining equation F of X , the preimages of these hyperplanes are the intersections n n with the hypersurfaces  P of of degree d 1 in P Z given by general linear X i combinations of these partial derivatives. Inasmuch as the partials of F have no common zeros, Bertini’s theorem (Theorem 0.5) tells us that the hypersurfaces given by n 1 ́ . By B general linear combinations will intersect transversely with ezout’s theorem the X number of these points of intersection is the product of the degrees of the hypersurfaces, n 1 that is, 1/ . d.d 2 is a smooth cubic curve in P For example, suppose that . By the above formula, X 2   is 6. Since a general line in P X the degree of corresponds to the set of lines through 2 2 p a general point , there will be exactly six lines in P P through 2 tangent to X , as p shown in Figure 2.2. Proposition 2.9 gives us the answer to Keynote Question (e): Since the planes 3  form a general line in the dual projective space P containing the line L , the number of 3 2 3 D . is 12  2 S P  such planes tangent to a smooth cubic surface

65 n The Chow rings of and some related varieties Section 2.1 51 P 2.1.4 Products of projective spaces Though the Chow ring of a smooth variety behaves like cohomology in many ways, there are important differences. For example the cohomology ring of the product of two    ̈ Y/ D H unneth formula .X/ ̋ H H .Y/ spaces is given modulo torsion by the K .X  , ̈ but in general there is no analogous K unneth formula for the Chow rings of products of D of genera varieties. Even for a product of two smooth curves  1 we have and C g;h 1 2  A A .C .C  D/ looks like, , and no idea at all what no algorithm for calculating D/ beyond the fact that it cannot be in any sense finite-dimensional (Mumford [1962]). However the Chow ring of the product of a variety with a projective space does ̈ unneth formula, as we will prove in a more general context in Theorem 9.6 obey the K (Totaro [2014] proved it for products of any two varieties with affine stratifications). For the moment we will content ourselves with the special case where both factors are projective spaces: r s is given by the formula P The Chow ring of P Theorem 2.10.  r s r s A. / Š A. P  / ̋ A. P P /: P r s 1 Equivalently, if . P 2  P A / denote the pullbacks, via the projection maps, of ̨;ˇ s r the hyperplane classes on P respectively, then P and s r C 1 r s C 1 Œ ̨;ˇç=. ̨ /  P A. P Š Z /: ;ˇ Moreover, the class of the hypersurface defined by a bihomogeneous form of bidegree s r on  P .d;e/ P d ̨ C eˇ . is Proof: We proceed exactly as in Theorem 2.1. We may construct an affine stratification s r of P P by choosing flags of subspaces  s r Ä ;  ƒ  P D  ƒ Ä D P ƒ and Ä  ƒ Ä   1 0 1 0 r s 1 r 1 s ƒ D i D dim Ä , and taking the closed strata to be with dim i i s r P D ƒ  Ä : „ P   a r a;b b s The open strata z „ WD „ / n .„ „ [ 1;b a a;b 1 a;b a;b of this stratification are affine spaces. Invoking Proposition 1.17, we conclude that the s s r r C b a ' . D Œ„ / ç 2 A are generated by the classes  P . P Chow groups of  P P a;b a;b r hyperplanes in is the transverse intersection of the pullbacks of a Since P „ and b a;b s P hyperplanes in , we have a b ' D ̨ ˇ ; a;b

66 52 Chapter 2 First examples r s r 1 s C 1 C D 0 . This shows that A. P ̨  P D / is a homomorphic and in particular ˇ image of r 1 s C 1 C C 1 s C 1 r Œ ̨;ˇç=. ̨ Z / ̋ / Z Œˇç=.ˇ D Z Œ ̨ç=. ̨ /: ;ˇ Z „ is a single point, so deg ' On the other hand, D 1 . The pairing r;s r;s r p r C s C p q r s q s A  P deg / . P   P A / ! Z ; .ŒXç;ŒYç/ ! . .ŒXçŒYç/ P q m n p . ̨ ˇ ˇ sends / to 1 if p C m D r and q C n D s , because in this case the ; ̨ intersection is transverse and consists of one point, and to 0 otherwise, since then the intersection is empty. This shows that the monomials of bidegree 0  p  r .p;q/ , for 0 q  s , are linearly independent over Z , proving the first statement. and  F.X;Y/ is a bihomogeneous polynomial with bidegree .d;e/ , then, because If r s d e Y X is a rational function on P F.X;Y/=X , the class of the hypersurface  P 0 0 F D 0 is d times the class of the hypersurface X defined by D 0 plus e times the class 0 D Y D 0 ; that is, of the hypersurface . d ̨ C eˇ ŒXç 0 2.1.5 Degrees of Segre varieties 1 1/ C 1/.s r C .r s in P  P is by definition the image of P † The Segre variety under r;s the map  ŒX W ;:::;X ;:::ç: ç;ŒY Y ;:::;Y Œ:::;X ç  7! s i 0 j r 0 r;s X  and one of is an embedding because on each open set where one of the The map i r;s are nonzero the rest of the coordinates can be recovered from the products. Y the j V and W are vector spaces of dimensions r C 1 and s If 1 , we may express  C r;s without choosing bases by the formula ! P V  P W W P .V ̋ W /;  r;s 7! v ̋ w: .v;w/ ,  is defined by the four forms a D X , Y Y X b D For example, the map 1;1 0 0 1 0 c D X ; thus the Segre Y 0 , d D X D Y bd , and these satisfy the equation ac 1 1 1 0 3 † P . variety is the nonsingular quadric in 1;1 s r P The degree of the Segre embedding of  P is Proposition 2.11.   r s C D † deg : r;s r Proof: † The degree of is the number of points in which it meets the intersection of r;s .r 1/.s C 1/ 1 C C s hypersurfaces in P . Since r  is an embedding, we may compute this r;s r s  P and computing in the Chow ring P number by pulling back these hypersurfaces to r s r C s P  deg † , which gives the desired formula because D deg . ̨ C of P ˇ/ . Thus r;s  s C r s C r r s . ̨ ˇ/ D C . ˇ ̨ r

67 n The Chow rings of and some related varieties Section 2.1 53 P 3 P Figure 2.3 A tangent plane to a quadric in meets the quadric in two lines, one from each ruling. 1 1 r C 2r C P  has degree r  1 . These varieties For instance, the Segre variety P P (see Section 9.1.1). The simplest of are among those called rational normal scrolls 1 3 1 , which is the Segre image of P   P P ; the Q these is the smooth quadric surface ̨ and ˇ of the point classes via the two projections are the classes of the lines pullbacks Q , and we have  D ̨ C ˇ , where  of the two rulings of is the hyperplane class on 3 restricted to Q — a fact that is apparent if we look at the intersection of Q with any P tangent plane, as in Figure 2.3. This discussion can be generalized to arbitrary products of projective spaces (see Exercise 2.30). 2.1.6 The class of the diagonal r r  P P  Next we will find the class ı in the Chow group of the diagonal Å r r r s r A P / , and more generally the class . Apart of the graph of a map f W P . ! P  P f from the applications of such a formula, this will introduce the method of undetermined coefficients , which we will use many times in the course of this book. (Another approach to this problem, via specialization, is given in Exercise 2.31.) By Theorem 2.10, we have C r r 1 C r 1 r P Œ ̨;ˇç=. ̨ ;ˇ Z D / /;  A. P r r 1 P P 2  A . / are the pullbacks, via the two projection maps, of the ̨;ˇ where r 1 . hyperplane class in A / . The class ı D ŒÅç of the diagonal is expressible as a P linear combination r r r 1 r 2 2 ı C D C c ̨ c c ̨ ˇ ̨ CC c ˇ ˇ 0 r 2 1 . We can determine the coefficients for some ;:::;c by taking the product 2 Z c c 0 i r of both sides of this expression with various classes of complementary codimension: i i r ˇ Specifically, if we intersect both sides with the class and take degrees, we have ̨ i i r c  ̨ /: ˇ D deg .ı i

68 54 Chapter 2 First examples 1 L M Š P Š L f1g 1 1  P P f M g 0 Å L f 0 g g f L 0 p L f 0 g ç D 1 D Figure 2.4 f 0 g Mç , so ŒÅç D Œ f 0 g Mç C ŒL f 0 g ç , as one ŒÅçŒL ŒÅçŒ also sees from the degeneration in the figure. i r i ˇ We can evaluate the product ı  ̨ ƒ directly: If Ä are general linear and i r i i , respectively, then Œƒ  Äç D ̨ subspaces of codimension ˇ and r . Moreover, i  Ä/ Å Š ƒ \ Ä \ .ƒ is a reduced point, so i r i  ̨ c ˇ 1: D deg / D .ı .Å \ .ƒ  Ä// D # .ƒ \ Ä/ D # i Thus r r r 1 r 1 ̨ ˇ CC ̨ˇ : C ̨ C ˇ ı D See Figure 2.4. (This formula and its derivation will be familiar to anyone who has had a course in algebraic topology. As partisans we cannot resist pointing out that algebraic geometry had it first!) 2.1.7 The class of a graph r s .s P P Let W C 1/ homogeneous polynomials F f ! be the morphism given by i that have no common zeros: of degree d .X/;F ŒX ;:::;X f ç 7! ŒF W .X/ç: .X/;:::;F 1 s 0 0 r r s r . Let Ä   P By Corollary 2.2, we must have s P  be the graph of f . What is f r s s ŒÄ ? ç 2 A its class . P  P D / f f As before, we can write 2 r s r r 1 s s r C 1 r 2 s r C ˇ C c C c CC ̨ c ̨ ̨ ˇ D ˇ ˇ c 2 0 1 r f c Z ;:::;c in this 2 for some , and as before we can determine the coefficients c i r 0 expression by intersecting both sides with a cycle of complementary dimension: i r i . .ƒ  ̨ c ˇ  ˆ// deg / D # .Ä \ D i f f

69 n The Chow rings of and some related varieties Section 2.1 55 P r i s r C i s ƒ and ˆ Š P  P for general linear subspaces . By Theorem 1.7 the Š P .ƒ  ˆ/ is generically transverse. intersection Ä \ f ƒ \ .ƒ  ˆ/ is the zero locus in Finally, of r i general linear combinations Ä f F ;:::;F of the polynomials . By Bertini’s theorem, the corresponding hypersurfaces s 0 r i ́ ezout’s theorem the intersection will consist of will intersect transversely, and by B d points. Thus we arrive at the formula: s r Proposition 2.12. W ! P f is a regular map given by polynomials of degree d If P r is given by , the class on of the graph of f P f r X i s i r s i s ˇ P d /: 2 A ̨ . P D  f 0 D i Using this formula, we can answer a general form of Keynote Question (c). A r ;:::;F F of general homogeneous polynomials of degree sequence in variables C 1 d 0 r r r P defines a map ! P f , and we can count the fixed points W r : f D ;:::;t g ç 2 P t j f.t/ D t Œt r 0  F are general, we can take them to be general translates under GL GL Since the 1 i r r C 1 C of arbitrary polynomials, so the cardinality of this set is the degree of the intersection of r r P f Å  P  with the diagonal the graph . This is of f  1 r 1 r r r r r r 1 r / CC ˇ /  .d ̨ ̨ C C d . ̨ deg D ̨ / deg ˇ CC ˇ  ˇ .ı f r r 1 C C d CC d d 1 I D A;B;C are general forms of degree d in three variables then there are in particular, if 2 2 2 C d C 1 points t D Œt , ;t ç ;t ;t ç exactly P d such that ŒA.t/;B.t/;C.t/ç D Œt ;t 0 1 1 0 2 2 and this is the answer to Keynote Question (c). d D Note that in the case and s D r , Proposition 2.12 implies that a general 1 C 1/  .r C 1/ matrix has r C 1 eigenvalues. It also follows that an arbitrary matrix .r has at least one eigenvalue. 1 2.1.8 Nested pairs of divisors on P We consider here one more example of an intersection theory problem involving products of projective spaces; this one will allow us to answer Keynote Question (d). To d 0 be the projectivization of the space of homogeneous P H set this up, let . O P .d// D 1 P 1 polynomials of degree d on (equivalently, the space of effective divisors of degree d P 1 P d ). For any pair of natural numbers on and e with e  d , we consider the locus d e P 2 P ˆ  Df .f;g/ j f j g g :

70 56 Chapter 2 First examples d 1 on P Alternatively, if we think of , we can as parametrizing divisors of degree P d write this as d e .D;E/  P P j E  D Df : g ˆ 2 d e d has fibers isomorphic to P P ! ˆ , we see that ˆ is  Since the projection map W e d , in  d , or codimension P e . We ask: What is the class of irreducible of dimension P e d d A  P . / ? ˆ P in d d e e 1 P  P / A be the pullbacks of the hyperplane classes in P 2 and P ; , . Let respectively. A priori, we can write X i d i c Œˆç  D ;  i d i e d C i c  where each coefficient Œˆç   ; that is, is given by the degree of the product i ˆ ƒ  Ä of general linear spaces the number of points of intersection of with the product d i e i d and Ä Š P Š P . This is exactly the number asked for in Keynote  P  P ƒ Question (d), but it may not be clear at first glance how to evaluate it. The key to doing this is the observation is that, abstractly, the variety ˆ is isomorphic d e e d d d P   P P to a product P under the map : Specifically, it is the image of d e d d e P P ! P W  P  ; ̨ 0 0 / 7! .D;D C D /: .D;D d e d e d  W / ! A. P A.  P P ̨  / is readily described. Furthermore, the pullback map P d e d d e d 1 P 2 / be the pullbacks of the hyperplane classes from ;  and P P . P , Let A ̨ commutes with the projection on the first factor, we see that respectively. Since e e d d e d  ! ; since the composition  ! P D  P ./ P P  is given by bilinear P ̨ e d d  P  , we have ̨  ./ D  C P . To evaluate the coefficient c forms on , we write i i i e d C i d d i e d C  deg   // . / D . ̨ .Œˆç   deg d i e d C i D . C / deg / .   C i d e D I i thus   X i e d C i d i   Œˆç D ; i   e d C a 1 and correspondingly the answer to Keynote Question (d) is . a 1 n P 2.1.9 The blow-up of at a point We will see in Chapter 13 how to describe the Chow ring of a blow-up in general. In this chapter, both to illustrate some of the techniques introduced so far and because the formulas derived will be useful in the interim, we will discuss two special cases: here the

71 n The Chow rings of Section 2.1 57 P and some related varieties E E ƒ ƒ z z L L 0 ƒ p L 2 Figure 2.5 Blow-up of P . n blow-up of n  P and in Section 2.4.4 the blow-up of any smooth at a point for any 2 surface at a point. n n Recall that the at a point p is the morphism blow-up W B ! P of , where P  1 n n n n 1 P is the closure of the graph of the projection P P nf p g ! W  P   B p , and  is the projection on the first factor: from p B ̨  -  p  n n 1 - ... P P n B nf p g , Since the graph of the projection is isomorphic to the source is irreducible. P B and to show that it is smooth as well; see, It is not hard to write explicit equations for for example, Section IV.2 of Eisenbud and Harris [2000]. 1 E  B is defined to be  The exceptional divisor .p/ , the preimage of p in B , n 1 n 1 n P p g  P which, as a subset of , is . Some other obvious divisors on B are f P n n . If the hyperplane H  P P contains p , then its the preimages of the hyperplanes of z H E and ; the latter is called the strict preimage is the sum of two irreducible divisors, n , or proper transform , of H . More generally, if Z  P transform is any subvariety, we 1 .Z of the preimage  define the strict transform of B to be the closure in nf p g / . Z See Figure 2.5. To compute the Chow ring of B , we start from a stratification of B , using the n 1 to the second factor. We do this by first ! P geometry of the projection map W B ̨ n 1 choosing a stratification of the target P , and taking the preimages in B of these strata.

72 58 Chapter 2 First examples Ä 1 E ƒ D ƒ 1 Ä 0 ̨ 0 1 Ä P D 1 0 Ä 0 2 1 P -bundle. P Figure 2.6 Blow-up of as 1 n ̨ Then we choose a divisor P that maps isomorphically by B  — a section ƒ to — and take, as additional strata, the intersections of these preimages with ƒ . of ̨ 1 0 0 We will choose as our section the preimage .ƒ ƒ D of a hyperplane ƒ  Š / n 1 n P p . (There are other possible choices of a section,  not containing the point P such as the exceptional divisor B ; see Exercise 2.37.) E  To carry this out, let 0 1 n 0 0 0 P Ä Ä  Ä  Ä  D n 1 2 0 1 n k D 1;2;:::;n , let be a flag of linear subspaces and, for 1 0 ̨ Ä B: .Ä D  / k 1 k n 1 W k ! P are projective lines, the dimension of Ä Since the fibers of is ̨ . Next, B k k 0;1;:::;n 1 , we set D for Ä D \ ƒ; ƒ 1 k k C 1 n 0 ! so that under the isomorphism ̨ j . W ƒ is the preimage of P Ä ƒ ƒ k k The subvarieties Ä ;:::;Ä ;ƒ ;:::;ƒ are the closed strata of a stratification 0 n n 1 1 of B , with inclusion relations  ƒ ƒ ƒ ƒ 1 2 0 n 1 n Ä B Ä D  Ä Ä n 2 1 n 1 As we will soon see, this is an affine stratification, so that the classes of the closed A.B/ . (In fact, the open strata are isomorphic to affine strata generate the Chow group A.B/ freely; we will verify spaces, and it follows from Totaro [2014] that they generate this independently when we determine the intersection products.) 1 B as the total space of a P To visualize this, we think of the blow-up -bundle over 1 n ; for example, this is the picture that arises if we take the via the projection map ̨ P 2 standard picture of the blow-up of P at a point (shown in Figure 2.5) and “unwind” it as in Figure 2.6.

73 n The Chow rings of P and some related varieties Section 2.1 59 n 2 be the blow-up of n  P . With notation B Let Proposition 2.13. at a point, with Œƒ as above, the Chow ring ç D A.B/ is the free abelian group on the generators k k n n k for k D 0;:::;n 1 and ŒÄ k ç D ŒÄ for . The class of 1;:::;n ç D ç Œƒ 1 n 1 n k is Œƒ and the exceptional divisor , then ç ŒÄ ŒEç D e ç . If we set  D Œƒ E ç 1 1 n 1 n n Œ;eç Z Š A.B/ n n n .e; 1/ e C / . as rings. ı ı ı ı ;:::;ƒ Proof: ;:::;Ä ;ƒ Ä of the We start by verifying that the open strata n n 1 0 1 B Ä ;ƒ are isomorphic to affine spaces. This is with closed strata stratification of k k ı ı . For the strata ƒ immediate for the strata , we choose coordinates .x ;:::;x Ä / on n 0 k k n n 0 D .1;0;:::;0/ and ƒ P  P D is the hyperplane x so that p 0 . By definition, 0 ̊ n n 1 j ..x ;:::;y 1 // 2 P B  P ;:::;x : /; .y D x  y i;j D x for all y n i j i n 1 j 0 n 1 0 -plane Ä D . We can write 0 Say the  P .k 1/ is given by y y DD 1 n k 1 k  ı 0 0 1 Ä Ä Ä D the open stratum n ̨ .B \ n ƒ/ as k k 2 k 1 ̊ ı ;:::;y : /; .0;:::;0;1;y ..1;0;:::;0;;y ;:::;y D // Ä n n k C 2 C 2 k n n k k ı ;y A with The functions ;:::;y Ä give an isomorphism of . n k n C 2 k It follows that the classes .B/ D Œƒ in ç and  D ŒÄ ç A k k k k k B . generate the Chow groups of ƒ is the preimage of a We next compute the intersection products. Since k -plane in k n n not containing p , and any two such planes are linearly equivalent in P , the classes P of their pullbacks are all equal to  . Similarly, the class of the proper transform of any k n P k containing p is -plane in . Having these representative cycles for the classes  k k and makes it easy to determine their intersection products. k n k -plane in For example, a general intersects a general l -plane transversely in a P .k C l n/ general -plane; thus   n: D   l C k for all n k l C k l n k Similarly, the intersection of a general containing p with a general l -plane P -plane in p is a general .k C l n/ -plane not containing p , so that not containing  D  for all k C l  n; n k l k C l and likewise 1: C n D  l C k for all n l C k l k

74 60 Chapter 2 First examples k l n C 1 on the last set of products: In the case k C l D n , Note the restriction C  p l -plane through p -plane through the proper transforms of a general and a general k are disjoint. A This determines the Chow ring of .B/  A B . The pairing Z .B/ ! A Š .B/ 0 k k n is given by   0: D D  and 1 D D  n n k k n k k k k k k n and ;:::; This is nondegenerate, so the classes . A.B/  freely generate ;:::; n 1 1 n 0 It follows that we can express the class of the exceptional divisor E in terms of the generators . The most geometric way to do this is to and Ä of A .B/ ƒ 1 n n 1 n 1 n n 0 , so p is linearly equivalent in P observe that to a hyperplane †  P ƒ containing 1 n is linearly equivalent to the union of the exceptional divisor and † E the pullback of 1 n P , it is contained in the preimage D . Since D a divisor projects to a hyperplane of 1 Ä of such a hyperplane. Since P -bundle over its image, it is irreducible. We Ä is a n 1 . Since any two hyperplanes in P see upon comparing dimensions that D D Ä are rationally equivalent, so are their preimages in ; thus ƒ , E B  D C E  Ä C n 1 1 n ŒEç  or . D 1 n 1 n We now turn to the ring structure of A.B/ . Let  D Œƒ . ç and e D ŒEç D  1 n 1 n ƒ \ E D ¿ , we have Since 1 n 0: D e Also, k n D  for k D 0;:::;n 1;  k  and, since , D e 1 n k n n k k n k n k n D e/  . D C . 1/ 1;:::;n: D k e for D k n 1  and e generate A.B/ as a ring. In addition to the relation e D 0 , they It follows that satisfy the relation n n n n n D e D . e/ D  0 C . 1/ : 1 n Thus the Chow ring is a homomorphic image of the ring n n n 0 A C . 1/ WD e Z Œ;eç=.e; /: 0 D 1;:::;n 1 m m in A For is a , it is clear that every homogeneous element of degree m m m is a free e -linear combination of . Since for 0 < m < n the group A Z .B/  and 0 Z -module of rank 2, this implies that the map A  A is an isomorphism. We have computed the intersection products of the ƒ by taking represen- and Ä k k tatives that meet transversely (indeed, the possibility of doing so motivated our choice of ƒ as a cross section of ̨ above). Since E is the only irreducible variety in the 2 class ŒEç we cannot give a representative for e quite as easily. But as we have seen,

75 n The Chow rings of and some related varieties Section 2.1 61 P D Œƒ ŒEç (this illustrates ç ŒÄ E are transverse to Ä ç and both ƒ and n 1 n 1 n 1 n 1 the conclusion of the moving lemma!). It follows that 2 ŒE \ .ƒ Ä/ç D ŒE \ e Ä D ç: 1 n n 1 n 1 Ä projects to a hyperplane in P E projects isomorphically to P , we Since and 2 ; that is, E of the class of a negative is a hyperplane in E Ä ŒEç \ is the see that 1 n E . hyperplane in 3 The Chow ring of the blow-up of along a line is worked out in Exercises 2.38– P 2.40. More generally, we will see how to describe the Chow ring of a general projective bundle in Chapter 9, and the Chow ring of a more general blow-up in Chapter 13. 2.1.10 Intersection multiplicities via blow-ups n B P at a point of We can use the description of the Chow ring of the blow-up to prove Proposition 1.29, relating the intersection multiplicity of two subvarieties n P of complementary dimension at a point to the multiplicities of X and Y  X;Y . (The same argument will apply to subvarieties of an arbitrary smooth variety p at once we have described the Chow ring of a general blow-up in Section 13.6.) The idea n n Y Y \ of X and P in P X with the intersection is to compare the intersection  z z \  B of their proper transforms in the blow-up. X Y We start by finding the class of the proper transforms: n z  P Proposition 2.14. be a k -dimensional variety and Let X its proper B X  n . If P B transform in the blow-up p of X has degree d and multiplicity at a point m D mult , then the class of the proper transform is .X/ at p p z D .d m/ Œ C m Xç 2 A.B/: k k Proof: p as the This follows from two things: the definition of the multiplicity of X at C X (Section 1.3.8), and the identification degree of the projectivized tangent cone T p C X to X at p with the intersection of the proper T of the projectivized tangent cone p n 1 z  B (on page 36). X transform E Š P with the exceptional divisor i E , ! B is the Given these, the proposition follows from the observation that if W  ( . inclusion, then / D 0 i  ) is represented by the cycle ƒ E , which is disjoint from k k k n 1  is represented by the cycle / is the class of a and 1/ -plane in E Š P i .k . ( k k , which intersects E transversely in a .k Ä 1/ -plane). This says that the coefficient k z Xç in the expression above for Œ of must be the multiplicity m D mult ; the .X/ p k coefficient of  similarly follows by restricting to a hyperplane not containing p . k n X;Y  P Now suppose we are in the setting of Proposition 1.29: are dimensionally transverse subvarieties of complementary dimensions n k , having multiplicities and k 0 m m respectively at p . If, as we supposed in the statement of the proposition, the and z z and Y at p are disjoint (that is, projectivized tangent cones to X \ ), then Y \ E D ¿ X

76 62 First examples Chapter 2 p m X and Y at of is simply the difference between the intersection multiplicity .X;Y/ p n of X and Y in P deg and the intersection number the intersection number .ŒXçŒYç/ z z XçŒ deg of their proper transforms in B ; by Proposition 2.14 and our description of .Œ Yç/ 0 A.B/ mm . the Chow ring , this is just 2.2 Loci of singular plane cubics This section represents an important shift in viewpoint, from studying the Chow rings parameter spaces . It of common and useful algebraic varieties to studying Chow rings of is a hallmark of algebraic geometry that the set of varieties (and more generally, schemes, morphisms, bundles and other geometric objects) with specified numerical invariants may often be given the structure of a scheme itself, sometimes called a parameter space. Applying intersection theory to the study of such a parameter space, we learn something about the geometry of the objects parametrized, and about geometrically characterized enumerative geometry , and classes of these objects. This gets us into the subject of was one of the principal motivations for the development of intersection theory in the 19th century. 2 P By way of illustration, we will focus on the family of curves of degree 3 in : plane cubics. Plane cubics are parametrized by the set of homogeneous cubic polynomials 9 P F.X;Y;Z/ . in three variables, modulo scalars, that is, by There is a continuous family of isomorphism classes of smooth plane cubics, parametrized naturally by the affine line (see Hartshorne [1977]), but there are only a finite number of isomorphism classes of singular plane cubics:  irreducible plane cubics with a node;  irreducible plane cubics with a cusp;  plane cubics consisting of a smooth conic and a line meeting it transversely;  plane cubics consisting of a smooth conic and a line tangent to it;  plane cubics consisting of three nonconcurrent lines (“triangles”);  plane cubics consisting of three concurrent lines (“asterisks”);  cubics consisting of a double line and a line; and finally  cubics consisting of a triple line. These are illustrated in Figures 2.7–2.9, where the arrows represent specialization, as explained below. 9 The locus in of points corresponding to singular curves of each type is an orbit P 9 and a locally closed subset of of P PGL . These loci, together with the open subset 3 9 9 P U of smooth cubics, give a stratification of P  . We may ask: What are the closed strata of this stratification like? What are their dimensions? What containment relations hold among them? Where is each one smooth and singular? What are their tangent 9 ? spaces and tangent cones? What are their degrees as subvarieties of P

77 Loci of singular plane cubics 63 Section 2.2 Œ2ç Œ3ç 9 8 7 6 5 4 2 Dimension: Figure 2.7 Hierarchy of singular plane cubic curves. Figure 2.8 Nodal cubic about to become the union of a conic and a transverse line:    2 2 1 1 1 2 2 x 1/ C .x . x y y/ 100.x C y C 2 2 2 Some of these questions are easy to answer. For example, the dimensions are given in Figure 2.7, and the reader can verify them as an exercise. The specialization relationships (when one orbit is contained in the closure of another, as indicated by arrows in the chart) are also easy, because to establish that one orbit lies in the closure of another it 2 f C of plane cubics with an open set of  P suffices to exhibit a one-parameter family g t t corresponding to one type and a point corresponding to the other. The parameter values noninclusion relations are subtler — why, for example, is a triangle not a specialization of a cuspidal cubic? — but can also be proven by focusing on the singularities of the curves. Figure 2.9 Cuspidal cubic about to become the union of a conic and a tangent line: 2 2 2 3 x 7y.x C .y C 1/ y 1/ .

78 64 Chapter 2 First examples The tangent spaces require more work; we will give some examples in Exercises 2.42– 2.43, in the context of establishing a transversality statement, and we will see more of these, as well as some tangent cones, in Section 7.7.3. In the rest of this section we will focus on the question of the degrees of these loci; we will find the answer in the case of the loci of reducible cubics, triangles and asterisks. In the exercises we indicate how to compute the degrees of the other loci of plane cubics, except for the loci of irreducible cubics with a node and of irreducible cubics with a cusp; these will be computed in Section 7.3.2 and Section 11.4 respectively. The calculations here barely scratch the surface of the subject; see for example Aluffi [1990; 1991] for a beautiful and extensive treatment of the enumerative geometry of plane cubics. Moreover, the answers to analogous questions for higher-degree curves or hypersurfaces of higher dimension — for example, about the stratification by singularity type — remain mysterious. Even questions about the dimension and irreducibility of these loci are mostly open; they are a topic of active research. See Greuel et al. [2007] for an introduction to this area.  d ı  For example, it is known that for 0 the locus of plane curves of degree d  2 N having exactly nodes is irreducible of codimension ı in the projective space P of all ı d (see, for example, Harris and Morrison [1998]), and its degree plane curves of degree has also been determined (Caporaso and Harris [1998]). But we do not know the answers to the analogous questions for plane curves with nodes and  cusps, and when it comes ı to more complicated singularities even existence questions are open. For example, for 2 d P d > 6 of degree C  it is not known whether there exists a rational plane curve whose singularities consist of just one double point. 2.2.1 Reducible cubics 9  P Let Ä be the closure of the locus of cubics consisting of a conic and a transverse line (equivalently, the locus of reducible and/or nonreduced cubics). We can Ä as the image of the map describe 5 2 9 P P W ! P   5 2 P of homogeneous linear forms and the space of P from the product of the space 9 .F;G/ 7! P homogeneous quadratic polynomials to , given simply by multiplication: FG FG F . Inasmuch as the coefficients of the product are bilinear in the coefficients of 9  1 of the hyperplane class ./ and  2 A G . P , the pullback / is the sum   ˇ; D ̨ C ./  2 5 2 5 are the pullbacks to P where  P ̨ of the hyperplane classes on P and and P ˇ . By unique factorization of polynomials, the map is birational onto its image; it  follows that the degree of Ä is given by  7 7 deg . 21; deg .Ä/ / D deg .. ̨ C ˇ/ D / D ./

79 Loci of singular plane cubics Section 2.2 65 and this is the answer to Keynote Question (a). Another way to calculate the degree of Ä is described in Exercises 2.42–2.44. 2.2.2 Triangles A similar analysis gives the answer to Keynote Question (b) — how many cubics in a three-dimensional linear system factor completely, as a product of three linear forms. 9 of the locus of such totally reducible cubics, P  † Here, the key object is the closure ; the keynote question asks us for the number of points of which we may call triangles intersection of . † with a general 3-plane. By Bertini’s theorem this is the degree of † is the image of the map † Since 2 2 2 9  W P P ! P  ;  P L ç;ŒL ç;ŒL .ŒL ç/ 7! ŒL ç; L 1 2 3 1 3 2 we can proceed as before, with the one difference that the map is now no longer birational, 2 2 2 1 / , but rather is generically six-to-one. Thus if ̨ 2 A are the . P ,  P ̨  P ̨ 3 1 2 2 P pullbacks of the hyperplane classes in the factors via the three projections, so that  D ̨  C ̨ C ./ ̨ ; 3 2 1 we get   6 6 1 1 / . D ̨ 15: C ̨ D ̨ C deg .†/ D deg 1 3 2 6 6 2;2;2 This is the answer to Keynote Question (b): In a general three-dimensional linear system of cubics, there will be exactly 15 triangles. 2.2.3 Asterisks By an asterisk , we mean a cubic consisting of the sum of three concurrent lines. To 9 and to calculate its degree, let P see that the closure of this locus is indeed a subvariety of 9 2 2 2   P P ! P P W  be as in Section 2.2.2, and consider the subset 2 2 2 ;L ;L ˆ ¿ / 2 P Df  P .L  P gI j L ¤ \ L L \ 2 3 1 3 1 2 9 A  P under the map of asterisks is then the image .ˆ/ of ˆ the locus  . If we write the line L as the zero locus of the linear form i C X a a Y C a Z; i;3 i;1 i;2 L L \ then the condition that is equivalent to the equality \ L ¿ ¤ 3 2 1 ˇ ˇ ˇ ˇ a a a 1;1 1;2 1;3 ˇ ˇ ˇ ˇ 0: D a a a 2;2 2;1 2;3 ˇ ˇ ˇ ˇ a a a 3;3 3;2 3;1

80 66 Chapter 2 First examples 2 2 2 P P The left-hand side of this equation is a homogeneous trilinear form on ,   P 2 2 2 P P  P is a closed subset of and A is a closed subset from which we see that ˆ  9 . Moreover, we see that the class of ˆ of P is 1 2 2 2 P C ̨ D 2 A ̨ . /; C  P ̨  P Œˆç 1 3 2 9 of five general hyperplanes in so that the pullback via  will intersect ˆ in P   6 5 6 C ̨ D / deg / D deg . ̨ ̨ C ̨ 90 C ̨ .Œˆç. ̨ / C D 3 1 1 3 2 2 2;2;2  A j W ˆ ! points. Since the map has degree 6, it follows that the degree of the locus ˆ 9 A  of asterisks is 15. P 2.3 The circles of Apollonius Apollonius posed the problem of constructing the circles tangent to three given ́ circles. Using B ezout’s theorem we can count them. Theorem 2.15. If D are three general circles, there are exactly eight ;D D and 2 3 1 circles tangent to all three. 2.3.1 What is a circle? We first need to say what we mean by a circle in complex projective space. While circles are usually characterized in terms of a metric, in fact they have a purely algebro- geometric definition. Starting from the affine equation 2 2 2 .y b/ .x D a/ r C (2.1) 2 .a;b/ in A r centered at a point of a circle of radius , and homogenizing with respect to 2 2 2 2 az/ a new variable C .y bz/ z D r , we get z .x : We think of the line z D 0 as the “line at infinity,” and we see that the circle passes through the two points ı D .1;i;0/ and ı WD .1; i;0/ C on the line at infinity; these are called the . Conversely, it is an easy circular points exercise to see that the equation of any smooth conic passing through the two circular points can be put into put into the form (2.1). 2 to be a conic in P x;y;z with coordinates We thus define a passing through circle z D 0 ; equivalently, a circle is a conic the two circular points on the line at infinity 2 2 2  P C whose defining equation f lies in the ideal .z;x D C y V.f / / . (This formulation makes sense over any field of characteristic 2 .) We see from this that the ¤ 5 2 set of circles is a three-dimensional linear subspace in the space of all conics in P . P

81 The circles of Apollonius Section 2.3 67 Much geometry can be done in this context. For example, a direct calculation shows that the center of the circle is the point of intersection of the tangent lines to the circle at the circular points; in particular, the coordinates of the center are rational functions of the coefficients of its defining equation. Note that when we characterize circles as conics containing the circular points p;q at infinity, we are including singular conics that pass through these points, and we see ; with ı that there are two kinds of singular circles: unions of the line at infinity ı C 2 , and unions L [ M of lines with ı another line in 2 L P ı and 2 M . It is easy to C as r !1 see from the equations that these are the limits of smooth circles of radius r ! , respectively. (When the radius of a circle goes to 0, we may think the circle r 0 and 2 R C , the conic shrinks to a point, but that is because we are seeing only points in : over 2 2 y x D 0 consists of the two lines C D ̇ iy .) x 2.3.2 Circles tangent to a given circle Next, we have to define what we mean when we say two circles are tangent. Let 2 D be a smooth circle. If C is any other circle, we can write the intersection C \ D ,  P , as the sum viewed as a divisor on D D Dı C Cı \ C p C q: C In these terms, we make the following definition: We say that the circle Definition 2.16. is tangent to the circle D if p D q . C C D In other words, are tangent if they have two coincident intersections in and C;D addition to their intersection at the circular points; this includes the case where have intersection multiplicity 3 at or q . Let Z be the variety of circles tangent to a p D 3 D . We will show that is a quadric cone in the P given smooth circle of circles. Z D 2 R tangent to a given circle is It is visually obvious that the family of circles in two-dimensional. To prove this algebraically we consider the incidence correspondence 3 r at D ; is tangent to j C g P D 2 .r;C/ Df  ˆ r is a circular point the condition should be interpreted as saying the where when . The condition that a curve m is intersection multiplicity 3 .C;D/ f D 0 meet a curve  r f with multiplicity r 2 D means that the function at a smooth point j m vanishes D D m at r ; it is thus m linear conditions on the coefficients of the equation f . This to order 1 r 2 D , the fiber of ˆ over r is a P shows that, for each point , cut out by two linear equations in the space of circles. It follows that is irreducible of dimension 2. Since ˆ 3 ˆ ! almost all circles tangent to P sending D are tangent at a single point, the map 3 C is birational. Thus the image Z .r;C/ of ˆ in P to also two-dimensional. D 3 3 be a general line, corresponding to  P To show that is a quadric, let L  P Z D , the f C C g and C are the defining equations of g a pencil of circles f and . If 1 0 t 1 P 2 t

82 68 Chapter 2 First examples f=g has two zeros (where rational function meets D , aside from the circular points) C 0 C D , aside from the circular points), so f=g gives a map meets and two poles (where 1 1 of degree 2. P D ! The circles tangent to C correspond to the branch points of this map; by the D t classical Riemann–Hurwitz formula, there will be two such points. Thus the degree of is a quadric surface. On the other hand, if Z is 2, and we see that Z L \ C ¤ D D D D at r is tangent to D , then every member of the linear space of circles jointing C 2 3 satisfies the linear condition for tangency at r , so Z D  P to is a cone with vertex D corresponding to , as claimed. D 2.3.3 Conclusion of the argument D is ;D WD ;D Z be three circles. If the intersection A Now let Z \ Z \ 3 1 2 D D D 1 2 3 3 ́ deg D 2 finite, then B D 8 A ezout’s theorem implies that . To prove that the intersection D , we consider the incidence correspondence is finite for nearly all triples of circles i 4 3 g / j C is tangent to each of the D : ‰ ;D ;C/ 2 . P .D ;D WDf i 2 1 3 3 9 If we project onto the last factor, the fiber is , and thus has dimension 6, so dim ‰ D Z . C Thus the projection to the nine-dimensional space consisting of all triples .D ;D ;D / 2 3 1 cannot have generic fiber of positive dimension. We have now shown that, counting with multiplicity, there are eight circles tangent D ;D . To prove that there are really eight distinct circles, ;D to three general circles 2 3 1 we would need to prove that the intersection Z \ Z is transverse. In \ Z D D D 3 2 1 Section 8.2.3 we will see how to do this directly, by identifying explicitly the tangent spaces to the loci . For now we will be content to give an example of the situation Z D where the eight circles are distinct: it is shown on the cover of this book! Another approach to the circles of Apollonius, via the notion of theta-characteristics , 3 sphere in P is given in Harris [1982]. There is also an analogous notion of a ; see for example Exercise 13.32. 2.4 Curves on surfaces Aside from enumerative problems, intersection products appeared in algebraic geometry as a central tool in the theory of surfaces, developed mostly by the Italians in the late 19th and early 20th centuries. In this section we describe some of the basic ideas. This will serve to illustrate the use of intersection products in a simple setting, and also provide us with formulas that will be useful throughout the book. A different treatment of some of this material is in the last chapter of Hartshorne [1977]; and much more can be found, for example, in Beauville’s beautiful book on algebraic surfaces [1996].

83 Curves on surfaces Section 2.4 69 S Throughout this section we will use some classical notation: If is a smooth 1 for the degree .S/ , we will write ̨  ˇ 2 deg . ̨ˇ/ of ̨;ˇ projective surface and A 2 A of the two .S/ , and we refer to this as the intersection number their product ̨ˇ 2  C is a curve we will abuse notation and write C for the class classes. Further, if S 1 are two curves, we will write .S/ 2 C;D  S A C  D in ŒCç . Thus, for example, if 2 2  / and we will write C .ŒCç for deg .ŒCç deg place of . The reader should not be ŒDç/ 2 A D Z — as we have already remarked, misled by this notation into thinking that .S/ 2 need not even be finite-dimensional in any reasonable sense. A the group .S/ 2.4.1 The genus formula One of the first formulas in which intersection products appeared was the genus formula , a straightforward rearrangement of the adjunction formula that describes the genus of a smooth curve on a smooth projective surface (we will generalize it to some C  S is a smooth curve of genus g on a smooth singular curves in Section 2.4.6). If surface, then K D .K I C C/ j C C S C is 2g 2 , this yields since the degree of the canonical class of 2  C C K C S g C 1: (2.2) D 2 Example 2.17 (Plane curves) . By way of examples, consider first a smooth curve 2 2 1 C d . If we let  2 A  . P of degree / be the class of a line, we have ŒCç D d and P D 3 , so the genus of C is K 2 P 2 3d C d 2/ 1/.d .d D D C 1 g : 2 2 Thus we recover, for example, the well-known fact that lines and smooth conics have genus 0 while smooth cubics have genus 1. 3 . Example 2.18 Q  P (Curves on a quadric) is a smooth quadric Now suppose that surface, and that C  Q is a smooth curve of bidegree .d;e/ — that is, a curve linearly equivalent to d e times a line of the other (equivalently, in times a line of one ruling plus 1 1 Q , the zero locus of a bihomogeneous polynomial  P P terms of the isomorphism Š 1 2 ̨ and ˇ of bidegree A .d;e/ .Q/ be the classes of the lines of the two rulings ). Let of Q , as in the discussion in Section 2.1.5 above, and let  D ̨ C ˇ be the class of a 3 plane section of . Applying adjunction to Q  P , we have Q K D .K 2ˇ: 2 ̨ C Q/ j D D 2 3 Q Q P

84 70 Chapter 2 First examples Thus, by the genus formula, 2 eˇ/ .d ̨ C C ˇ/.d ̨ C eˇ/ 2. ̨ D C 1 g 2 2de 2d 2e D C 1 2 .d 1/: D 1/.e 2.4.2 The self-intersection of a curve on a surface We can sometimes use the genus formula to determine the self-intersection of a 3 curve on a surface. For example, suppose that  P is a smooth surface of degree d S 1  2 is a line. Letting  L A and .S/ denote the plane class and applying adjunction S 3 L P S , we have K is 0, D .d 4/ , so that L  K  to d 4 ; since the genus of D S S the genus formula yields 2 L C d 4 0 D C 1; 2 or 2 L D 2 d: The cases d D 1 (a line on a plane) and d D 2 are probably familiar already; in the case 3 d a smooth surface S  P  of 3 , the formula implies the qualitative statement that . (See Exercise 2.60 below for a degree 3 or more can contain only finitely many lines 2 d 2 sketch of a proof, and Exercise 2.59 for an alternative derivation of .) D L We note in passing that we could similarly ask for the degree of the self-intersection 2 5 P of a 2-plane  X on a smooth hypersurface X  P ƒ . This is far harder (as the Š reader may verify, neither of the techniques suggested in this chapter for calculating the 3 self-intersection of a line on a smooth surface  P will work); the answer is given in S Exercise 13.22. 3 P 2.4.3 Linked curves in Another application of the genus formula yields a classical relation between what 3 linked curves in P are called . 3 Let P S;T be smooth surfaces of degrees s and t , and suppose that the scheme-  S \ T consists of the union of two smooth curves C and D with theoretic intersection no common components. Let the degrees of C and D be c and d , and let their genera be ́ g h respectively. By B ezout’s theorem, we have and c C d D st; so that the degree of C determines the degree of D . What is much less obvious is that

85 Curves on surfaces Section 2.4 71 C determine the degree and genus of . Here is one way to the degree and genus of D derive the formula. (2.2) on to determine the self-intersection of To start, we use the genus formula C .s 4/ , we have S : Since D K S 2 2 C C  C K C 4/c C .s S g D 1 D C C 1; 2 2 and hence 2 2 C .s 2g 4/c D (generalizing our formula in Section 2.4.2 for the self-intersection of a line). Next, since 1 ŒDç D t 2 A on .S/ , we can write the intersection number of C and D ŒCç C as S C D D C.t C/ D tc .2g 2 .s 4/c/ D .s C t 4/c .2g 2/:  D S : on This in turn allows us to determine the self-intersection of 2 D .2g C/ D td ..s C t 4/c D 2//: D.t D Applying the genus formula to , we obtain 2  K C D D S h D C 1 2 4/d td ..s C t 4/c .2g 2// C .s D 1: C 2 Simplifying, we get s C t 4 c/ h D .d g I (2.3) 2 in English, the difference in the genera of C and D is proportional to the difference in .s C t their degrees, with ratio . 4/=2 3  P The answer to Keynote Question (f) is a special case of this: If is a line, and L S and general surfaces of degrees s and S containing L , then, writing T \ T D L [ C , t we see that C is a curve of degree st 1 and genus t 4/.st 2/ C .s h : D 2 As is often the case with enumerative formulas, this is just the beginning of a much larger picture. The theory of liaison describes the relationship between the geometry C and D above. The theory in general is far more broadly of linked curves such as C D need only be Cohen–Macaulay, and we need no hy- and applicable (the curves S and T ), and ultimately provides a complete answer potheses at all on the surfaces 3 C;D  P to the question of when two given curves can be connected by a series of D curves D C linked as above. We will see ;C C ;:::;C and C with ;C D C i n i n 1 1 C 1 0 a typical application of the notion of linkage in Exercise 2.62 below; for the general theory, see Peskine and Szpiro [1974].

86 72 Chapter 2 First examples 2.4.4 The blow-up of a surface The blow-up of a point on a surface plays an important role in the theory of surfaces, ́ and we will now explain a little of this theory. Locally (in the analytic or etale topology), 2 at a point, which was treated in Section 2.1.9. such blow-ups look like the blow-up of P To fix notation, we let 2 p S be a point in a smooth projective surface and write 1 z z S  p . We write E D  W at .p/  S S for the preimage of p , ! S for the blow-up of 1 z 2 S/ e . , and exceptional divisor for its class. We will use the following called the A definitions and facts: z z  E is birational, and if q 2   S S is any point of the exceptional divisor, ! S W for the maximal ideal of O then there are generators x;y z;w and generators z S;q    y x D zw;  for the maximal ideal of such that D w , and E is defined O S;p z D 0 . In particular, locally by the equation S is smooth and E is a Cartier divisor. w z  p , then the proper transform If C of C , which is by C is a smooth curve through 1 z S  .C nf p g / , meets E transversely in one point. definition the closure in of z  ordinary m -fold point at p , then has an C meets E transversely C More generally, if m distinct points. Here we say that C has an ordinary m -fold point at p if the in C at has the form p completion of the local ring of   m Y ı y .x Š k ŒŒx;yçç y/  O i C;p D i 1  ;:::; for some distinct 2 k ; geometrically, this says that, near p , C consists 1 m of the union of m . smooth branches meeting pairwise transversely at p z in terms of A.S/ : A. S/ We can completely describe z be a smooth projective surface and  W ! S Proposition 2.19. S Let S the blow-up of 1 z e 2 A at a point p ; let S/ be the class of the exceptional divisor. S . z ̊ S/ D A.S/ (a) Z e as abelian groups. A.   1  for any ˇ D   . ̨ˇ/ ̨ ̨;ˇ 2 A  .S/ .  (b)  1  (c) ̨ D 0 for any ̨ 2 A e .S/ .  2 2 q D Œqç for any point (d) 2 E ( in particular, deg .e e / D 1 ) . 2  and  are inverse isomorphisms between A  .S/ and We first show that Proof:  2 z S/ is any class, we can write . ̨ 2 A A .S/ . By the moving lemma, if ̨ D ŒAç for some 0  z Z is .S/ with support disjoint from p ; thus  S/  A ̨ D ̨ . Likewise, if ̨ 2 A 2 . 0 0  any class, we can write ̨ D ŒAç for some A 2 Z ; thus .S/ with support disjoint from E 0    ̨ D ̨ . 

87 Curves on surfaces 73 Section 2.4 1 1 2 A We next turn to .S/ is any class, we can write ̨ D ŒAç for some A . If ̨  Z ̨ p ; thus   with support disjoint from 2 D ̨ . On the other hand, the A .S/ 1  z ! . Z S/  Z W .S/ is just the subgroup generated kernel of the pushforward map 1  1 e , the class of E . Thus we have an exact sequence by 1 1 z i! . 0 S/ ! A !h .S/ ! 0; A e  1 1 z .S/  A A . ! S/ splitting the sequence. W with 1 z is not torsion in A . S/ . This follows from the e It remains to show that the class 2 formula deg D 1 , which we will prove independently below. e   is a ring homomorphism. Part (b) of the proposition simply recalls the fact that For part (c) we use the push-pull formula:  .e    ̨/ D  0: e  ̨ D   C  S For part (d), let p , so that the proper transform be any curve smooth at z z  C S of C will intersect E transversely at one point q . We have then  z  ŒCç Cç C e; Œ D yields e and intersecting both sides with the class 2 C e D ; Œqç 0 2 is deg e D 1 . so the self-intersection number of e 2.4.5 Canonical class of a blow-up z S S We can express the canonical class of in terms of the canonical class of as follows: Proposition 2.20. With notation as above,  e: D  K K C S z S We must show that if ! is a rational 2-form on S , regular and nonzero at p , then Proof:  z S ! the pullback E . Let q 2 E   vanishes simply along , and let .z;w/ be generators of the maximal ideal of O such that there are generators .x;y/ for the maximal ideal z S;q of with O S;p   D zw and  x y D w:  It follows that   dx D z dw C wdz and   dy D dw: Thus   .dx ^ dy/ D w.dz ^ dw/:

88 74 Chapter 2 First examples  E is w D 0 , this shows that  Since the local equation of dx vanishes simply along at q E , as required. 2.4.6 The genus formula with singularities It will be useful in a number of situations to have a version of the genus formula C  S (2.2) geometric that gives the geometric genus of a possibly singular curve . (The genus of a reduced curve is the genus of its normalization.) To start with the simplest  S is a curve smooth away from a point p 2 case, suppose that of multiplicity m . C C p m -fold point, so that in particular the proper Assume moreover that is an ordinary z z is smooth. We can invoke the genus formula on to give a formula for the transform S C z of C genus in terms of intersection numbers on S . g As divisors,  z  C C mE; D C so that  z me: ŒCç Œ  D Cç From Proposition 2.20, we have   K K C e; D S z S z z  S and, putting this together with the genus formula for and Proposition 2.19, C we have 2 z z K C  C C z S D C 1 g 2 2    . C . C me/ C e/. me/ C K S D C 1 2   2 C K  C C m S C : 1 D 2 2 C  S More generally, if p , ;:::;p ;:::;m of multiplicity m has singular points 1 1 ı ı z at the points and the proper transform C in the blow-up Bl is p of S of C i ;:::;p f p g 1 ı p ), are ordinary smooth (as will, in particular, be the case if the C -fold points of m i i we have   2 X K  C C C m S i C : 1 g D 2 2 One can extend this further, to general singular curves C  S , by using iterated blow- ups, or by generalizing the adjunction formula, using the fact that any curve on a smooth surface has a canonical bundle (see for example Hartshorne [1977, Theorem III.7.11]).

89 Intersections on singular varieties Section 2.4 75 p 0 L L 1 Figure 2.10 The degree of intersection of two lines on a quadric cone is . 2 2.5 Intersections on singular varieties In this section we discuss the problems of defining intersection products on singular varieties. To begin with, the moving lemma may fail if X is even mildly singular: 3 3 2  X C  P Let be a smooth conic and . D p;C  P (Figure 2.10) P Example 2.21 2 X ... . Let L  P be a line (which necessarily contains p ). We p the cone with vertex claim that every cycle on that is rationally equivalent to L has support containing p , X and thus the conclusion of part (a) of the moving lemma does not hold for . X To show this, we first remark that the degrees of any two rationally equivalent curves on deg W A X are the same; that is, there is a function ! Z taking each irreducible .X/ 1 3 W ! i P curve to its degree. For, if is the inclusion, then for any curve D on X X we have deg D deg . D i .ŒDç//;   3  is the class of a hyperplane in P is odd. . In particular deg L D 1 where Now let D X be any curve not containing p . We claim that the degree of D   D ! C is a finite map W must be even. To see this, observe that the projection map p with the lines of X ; it follows that a general line whose fibers are the intersections of D 3 will intersect D transversely in deg . be a general in points. Now let H  P X / p 0 L;L H intersects X plane through . p  X , and so in the union of two general lines meets D transversely in 2 deg . is even. It follows that any cycle of / points, so deg D p dimension 1 on , effective or not, whose support does not contain p has even degree, X L and hence cannot be rationally equivalent to . Retaining the notation of Example 2.21, one might hope to define an intersection product on A.X/ even without the moving lemma. It seems natural to think that since 0 L;L p  X through two distinct lines meet in the reduced point p , we would have 0 H ç D Œpç . However, if  is the class of a general plane section ŒLçŒL \ X of X through p ,

90 76 Chapter 2 First examples p Ä ƒ t t N M t t Q Figure 2.11 The intersection product of and the class of a line cannot be defined. Œƒ ç t transversely in one point) we might also L then (since such a hyperplane meets each D Œpç . But  is rationally equivalent to the union of two lines through p expect ŒLç . Thus, if both expectations were satisfied, we would have 0 0 ç D 2ŒLçŒL ŒL ç D 2Œpç: Œpç D 1 D 2 . Applying the degree map, we would get the contradiction A.X/ ̋ There is a way around the difficulty, if we work in the ring : We can take Q the product of the classes of two lines to be one-half the class of the point , and our p contradiction is resolved. As Mumford has pointed out, something similar can be done for all normal surfaces (see Example 8.3.11 of Fulton [1984]). But in higher dimensions there are more difficult problems, as the following example shows: 4 3 P Let  Example 2.22. Q be a smooth quadric surface, and let X D p;Q be  P 4 3 with vertex p ... P the cone in . The quadric Q contains two families of lines f M P g t and f N g , and the cone X is correspondingly swept out by the two families of 2-planes t f ƒ ; see Figure 2.11. D p;M g g and f Ä p;N D t t t t  L not passing through the vertex p maps, under projection Now, any line X p , to a line of Q ; that is, it must lie either in a plane ƒ or in a plane from Ä X ; lines on t t 0 lie on one plane of each type. Note that since lines M p that do pass through  Q ;M t t 0 0 ¤ t and , while lines M of opposite rulings meet of the same ruling are disjoint for N t t t 0 0 M lying in a plane ƒ and is disjoint from in a point, a general line  ƒ for t ¤ t X t t meets each plane Ä transversely in a point. Thus, if there were any intersection product s A.X/ satisfying the fundamental condition (  ) of Theorem 1.5, we would have on ŒM猃 D Œqç D 0 and ŒMçŒÄ ç ç t t

91 Intersections on singular varieties Section 2.5 77 q 2 . Likewise, for a general line N  X lying in a plane Ä for some point , the X t opposite would be true; that is, we would have D Œrç and ŒNçŒÄ ç ç D 0: ŒN猃 t t and N — indeed, any two lines on X — are rationally equivalent! But the lines M Since any two lines in M is rationally equivalent to ƒ are rationally equivalent, the line t \ ƒ . Since any two lines in Ä are rationally equivalent, the the line of intersection Ä t s s M ) is rationally equivalent to an arbitrary line in Ä line of intersection (and thus also . s Since a point cannot be rationally equivalent to 0 on , we have a contradiction. Thus X ŒM猃 ç cannot be defined in A.X/ . products such as t   Despite this trouble, one can still define Œƒ f ç and f using methods of ç Œƒ t t M N  Fulton [1984]. In fact, one can define the pullback for an inclusion morphism f W B ! X that is a “regular embedding” (which means that f is locally a complete B , X ), or for the composition of such a morphism with a flat map. intersection in f Example 2.22 also shows that, even though is well-defined, pullbacks cannot M  be defined, at least in a way that makes the push-pull formula valid. If X were smooth,  ŒM猃 would be equal to f ç , where .ŒMçf then by the push-pull formula ç/ Œƒ t t  M M  ç the product Œƒ . This product is well- ŒMçf should be interpreted as being in A.M/ t M defined, as are the pullback and pushforward. But they do not allow us to compute the ŒM猃 , we would arrive at the contradiction ç I since product D ŒNç in A.X/ ŒMç t   f Œrç: D .f .f ç/ Œƒ Œƒ ç/ D ŒM猃 0 ç D ŒN猃 D ç D f t t t t N  M  M N There are, however, certain cycles (such as those represented by Chern classes of bundles) with which one can intersect, and this leads to a notion of “Chow cohomology”  groups .X/ A , which play a role relative to the Chow groups analogous to the role of co- homology relative to homology in the topological context: we have intersection products c d c C d A .X/ ! A .X/ ̋ .X/vspace A 5pt and c .X/ ̋ A .X/ .X/ A A ! k c k analogous to cup and cap products in topology. In the present volume we will avoid all of this by sticking for the most part to the case of intersections on smooth varieties, c where we can simply equate .X/ D A A .X/ ; for the full treatment, see Fulton c dim X [1984, Chapters 6, 8 and 17], and, for a visionary account of what might be possible, Srinivas [2010].

92 78 Chapter 2 First examples 2.6 Exercises 2 5 2 be the quadratic Veronese map. If Let ! P W  C  P P is a D  Exercise 2.23. 2;2 .C/ has degree 2d . (In particular, this plane curve of degree d , show that the image 5 P contains only curves of even degree!) More  S means that the Veronese surface n N n W P generally, if ! P  is the degree- d D   P Veronese map and is a X n;d k e , show that the image .X/ has degree d variety of dimension e . and degree k C r s .r 1/.s C 1/ 1  ! P Exercise 2.24. D W P  Let P be the Segre map, and let  r;s r s r s k X be a subvariety of codimension k  ŒXç 2 A P . P . Let the class  P  / be P given by k k k 1 ˇ ̨ ̨ ŒXç C c ˇ CC c c D 0 1 k r s 1  ̨;ˇ 2 A P . / are the pullbacks of the hyperplane classes, and we take (where P D 0 if i > s or k c i > r ). i (a) Show that all  0 . c i 1 .r C 1/.s C 1/ P . .X/ (b) Calculate the degree of the image  1 C 1/.s C 1/ .r (c)  † Using (a) and (b), show that any linear space  P ƒ contained r;s r s r Š P in the Segre variety lies in a fiber of either the map  P † ! P or the r;s s . corresponding map to P 2 2 Exercise 2.25. ! P ' be the rational map given by P W Let   1 1 1 - / ;x ' ; ; W ;x ; .x 1 0 2 x x x 2 0 1 or, equivalently, W .x ;x ;x /; ' x / 7! .x ;x x x ;x 1 0 1 0 2 2 1 0 2 2 2  P and let  P Ä be the graph of ' . Find the class ' 2 2 2 A ŒÄ . P ç  P 2 /: ' 2 2 8 W P be the Segre map. Find the class of the graph of  P Let ! P  Exercise 2.26. 2 2 8  P P  P  / A. in . 2 2 2  2 2 - Exercise 2.27. W P P  be the rational map sending .p;q/ 2 P Let  P P s 2 2 2  in A. P to the line  P p;q  P . Find the class of the graph of s / . n be a hypersurface of degree  P Exercise 2.28. X d . Suppose that X has an Let ordinary double point (that is, a point p 2 X such that the projective tangent cone T X is a smooth quadric), and is otherwise smooth. What is the degree of the dual C p  n  ? P  hypersurface X

93 Exercises Section 2.6 79 n p  P Exercise 2.29. be a variety of degree d and dimension k , and suppose X 2 Let X is a point of multiplicity m (see Section 1.3.8 for the definition). Assuming 2 p that n 1 X P W that the projection map  is birational onto its image, what is the degree ! p  ? .X/ of p Use Proposition 2.14. Hint: r r 1 k Exercise 2.30. P  is Show that the Chow ring of a product of projective spaces P O r r r 1 k i / D A. A. P  P P / C r 1 r C 1 k 1 ç=. ̨ /; ;:::; ̨ Z ;:::; ̨ D Œ ̨ 1 k 1 k where ̨ are the pullbacks of the hyperplane classes from the factors. Use this ;:::; ̨ 1 k to calculate the degree of the image of the Segre embedding 1 r 1/ C .r r  1/ C .r 1 1 k k P  , P W ! P  .  V V ! V ̋ ̋ V corresponding to the multilinear map 1 1 k k r r t ¤ 0 , let A be the automorphism W P Exercise 2.31. ! P For t r 2 ;X ŒX ;:::;X ç: ç 7! ŒX X ;tX ;X ;t ;:::;t X 2 r 1 0 2 r 0 1 1 r r  P  A P ˆ be the closure of the locus Let  ı Df .t;p;q/ j t ¤ 0 and q D A ˆ .p/ g : t Describe the fiber of over the point t D 0 , and deduce once again the formula of ˆ r r  P P . Section 2.1.6 for the class of the diagonal in In the simplest case, this construction is a rational equivalence between a smooth 1 1 3 P  plane section of a quadric P Q  P Š (the diagonal, in terms of suitable 1 identifications of the factors with P ) and a singular one (the sum of a line from each ruling), as in Figure 2.12. Let Exercise 2.32. n n n n P  P ‰  Df .p;q;r/ j p;q and r are collinear in P 2 g : P (Note that this includes all diagonals.) n n n 1 in P (a) Show that this is a closed subvariety of codimension  P n  P . (b) Use the method of undetermined coefficients to find the class n n n 1 n Œ‰ç A . P  P 2  P D /: (We will see a way to calculate the class Œ ç using Porteous’ formula in Exercise 12.9.)

94 80 Chapter 2 First examples 1 1 Figure 2.12 The diagonal in P  is equivalent to a sum of fibers. P and ;:::;F -tuples / Suppose that .F 1/ ;:::;G C / are general .r .G Exercise 2.33. r 0 r 0 C 1 variables, of degrees d and e respectively, so that of homogeneous polynomials in r r r r r P W and g W P ! ! P P sending x to .F in particular the maps .x/;:::;F f .x// r 0 r are regular. For how many points .x/;:::;G do .x// and x D .x .G ;:::;x P / 2 r 0 0 r D we have g.x/ f.x/ ? 2 P 2 p The next two exercises set up Exercise 2.36, which considers when a point will be collinear with its images under several maps: 2 4 . P Consider the locus / ˆ of 4-tuples of collinear points. Find the Exercise 2.34.  2 2 4 2 A class .. P ' / D / of ˆ by the method of undetermined coefficients, that is, by Œˆç intersecting with cycles of complementary dimension. 2 4 . P  ˆ With as in the preceding problem, calculate the class ' D Œˆç / Exercise 2.35. 2 3  . P / by using the result of Exercise 2.32 on the locus ‰ of triples of collinear points and considering the intersection of the loci ‰ and ‰ / of 4-tuples .p ;p ;p ;p 2 4 1;2;4 3 1;2;3 1 .p ;p with ;p each collinear. / and .p / ;p ;p 2 4 1 3 2 1 2 2 Exercise 2.36. C W Let A;B ! P and be three general automorphisms. For how P 2 2 P many points are the points p;A.p/;B.p/ and C.p/ collinear? p n at a point be the blow-up of P Exercise 2.37. B p , with exceptional divisor E as Let in Section 2.1.9. With notation as in that section, show that there is an affine stratification \ with closed strata for k D 1;:::;n and E e WD Ä . Let 1 E for k D 0;:::;n Ä k k k k be the class of E in terms . Show that e to describe the classes D  1 1 n n n 1 k k of  form a basis and e e and vice versa. Conclude that the classes and D ŒÄ ç k k k k k for the Chow group A.B/ .

95 Exercises Section 2.6 81 3 P Exercises 2.38–2.40 deal with the blow-up of along a line. To fix notation, let 3 3 3 3 1  along a line L  P be the blow-up of , that is, the graph X P P P  P X  ! W 1 3 1 -  P W P L . Let ̨ W X ! P of the rational map be given by projection from L projection on the second factor. 3 z H H be a plane containing L Exercise 2.38. P   X its proper transform. Let and 3 z P J be a plane transverse to L , Let J  X its proper transform (which is equal to  its preimage in ) and M  J a line not meeting L . Show that the subvarieties X z z z z z \ J; H; M; M \ H; H J X; , with open strata isomorphic to affine X are the closed strata of an affine stratification of z spaces. In particular, since only one (the subvariety M \ H ) is a point, deduce that 3 A Š Z . .X/ 1 2 z z Exercise 2.39. Hç , j D Œ D Jç 2 A h .X/ and m D ŒMç 2 A Œ .X/ be the classes Let of the corresponding strata. Show that 2 2 D 0; j h D m and deg .jm/ D deg .hm/ D 1: Conclude that 2 3 2 ;j Œh;jç=.h hj A.X/ /: D Z E be the exceptional divisor of the blow-up, and X Now let e D Exercise 2.40.  2 1 .X/ its class. What is the class e 2 ? ŒEç A 5 2 . be the space of conic curves in P Exercise 2.41. Let P ¤ (a) Find the dimension and degree of the locus of double lines (in characteristic 2 ). 5 Å P of singular conics (that is, line Find the dimension and degree of the locus  (b) pairs and double lines). 9 of plane cubics Exercises 2.42–2.54 deal with some of the loci in the space P described in Section 2.2. 9 9 Exercise 2.42. be the space of plane cubics and Ä  P Let the locus of reducible P 2 L;C  P be a line and a smooth conic intersecting transversely at two cubics. Let 2 p;q 2 P points ; let L C C be the corresponding point of Ä . Show that Ä is smooth at L C , with tangent space C j homogeneous cubic polynomials : Ä D P f T F g F.p/ D F.q/ D 0 C C L 2 Using the preceding exercise, show that, if p 2 ;:::;p P are general Exercise 2.43. 7 1 9  P points and is the hyperplane of cubics containing p H , then the hyperplanes i i H ;:::;H intersect Ä transversely — that is, the degree of Ä is the number of reducible 7 1 cubics through p ;:::;p . 1 7

96 82 Chapter 2 First examples Calculate the number of reducible plane cubics passing through seven Exercise 2.44. 2 general points 2 P , and hence, by the preceding exercise, the degree of Ä . ;:::;p p 7 1 9  P of triangles † We can also calculate the degree of the locus Exercise 2.45. (that is, totally reducible cubics) directly, as in Exercises 2.42–2.44. To start, show D L C that if L C C L is a triangle with three distinct vertices — that is, points 3 1 2 of pairwise intersection — then D with tangent space \ L p L † is smooth at C i i;j j g : T † D P f homogeneous cubic polynomials F j F.p i;j / D 0 for all i;j C L C Using the preceding exercise, Exercise 2.46. 2 is the number ;:::;p p 2 P show that if are general points, then the degree of † (a) 1 6 of triangles containing p ;:::;p ; and 1 6 (b) calculate this number directly. Consider a general asterisk — that is, the sum C D L L C L C Exercise 2.47. 2 3 1 9 . Show that the variety †  P of three distinct lines all passing through a point of p C p . Deduce triangles is smooth at , with tangent space the space of cubics double at 9  P that the space of asterisks is also smooth at C A . 2 p be general points. Show that any asterisk containing ;:::;p Let 2 P Exercise 2.48. 1 5 f p D ;:::;p L g consists, possibly after relabeling the points, of the sum of the line 5 1 1 ;.L ;p , the line L . D p / ;p L and the line L \ D p p 2 5 3 4 2 3 2 1 1 2 p ;:::;p 2 P are Using the preceding two exercises, show that, if Exercise 2.49. 1 5 9 of asterisks intersect the locus A  P H general points, then the hyperplanes p i transversely, and calculate the degree of A accordingly. 9 ¤ 3 ) the locus Z  P Exercise 2.50. of triple lines is a Show that (in characteristic cubic Veronese surface, and deduce that its degree is 9. 9 Exercise 2.51. X  P Let be the locus of cubics of the form 2L C M for L and M 2 . lines in P 2 2 is the image of P Show that  P (a) under a regular map such that the pullback of X 9 P is a hypersurface of bidegree .2;1/ . a general hyperplane in (b) Use this to find the degree of X . Exercise 2.52. If you try to find the degree of the locus X of the preceding problem by ;:::;H intersecting H X , where with hyperplanes p p 4 1 9 ; Df C 2 P H j p 2 C g p you get the wrong answer (according to the preceding problem). Why? Can you account for the discrepancy?

97 Exercises Section 2.6 83 5 2 P Let the space of plane P Exercise 2.53. denote the space of lines in the plane and 2 5   P be the closure of the locus of pairs ˆ conics. Let P j C is smooth and L is tangent to C g : f .L;C/ is a hypersurface, and, assuming characteristic 0, find its class Show that 2 ˆ Œˆç 2 5 1 . P /  . A P 9 Y  P be the closure of the locus of reducible cubics consisting Exercise 2.54. Let of a smooth conic and a tangent line. Use the result of Exercise 2.53 to determine the Y . degree of 2 14 14 Let Exercise 2.55. , and let †  P be the space of quartic curves in be the P P † , closure of the space of reducible quartics. What are the irreducible components of and what are their dimensions and degrees? 14  P Find the dimension and degree of the locus  Exercise 2.56. of totally reducible quartics (that is, quartic polynomials that factor as a product of four linear forms). 14 14 Exercise 2.57. be the space of plane quartic curves, and let ‚  P Again let be P the locus of sums of four concurrent lines. Using the result of Exercise 2.34, find the degree of . ‚ 14 A  P Exercise 2.58. of the preceding problem, this Find the degree of the locus time by calculating the number of sums of four concurrent lines containing six general 2 p , assuming transversality. ;:::;p points 2 P 6 1 A natural generalization of the locus of asterisks, or of sums of four concurrent lines, n N , of in P would be the locus, in the space of hypersurfaces of degree cones . We d P will indeed be able to calculate the degree of this locus in general, but it will require more advanced techniques than we have at our disposal here; see Section 7.3.4 for the answer. 3 S  P L be a smooth surface of degree d and Exercise 2.59. Let S a line. Calculate  1 D ŒLç 2 A the degree of the self-intersection of the class .S/ by considering the  3 containing L . P with a general plane intersection of H  S Exercise 2.60. Let S be a smooth surface. Show that if C  S is any irreducible curve such that the corresponding point in the Hilbert scheme H S (see of curves on H Section 6.3) lies on a positive-dimensional irreducible component of , then the degree 2 1 deg of the self-intersection of the class . ŒCç 2 A / .S/ is nonnegative. Using D this and the preceding exercise, prove the statement made in Section 2.4.2 that a smooth 3 surface  P of degree 3 or more can contain only finitely many lines . S 3 Let C  P Exercise 2.61. be a smooth quintic curve. Show that (a) if C has genus 2, it must lie on a quadric surface; (b) if C has genus 1, it cannot lie on a quadric surface; and

98 84 Chapter 2 First examples if has genus 0, it may or may not lie on a quadric surface (that is, some rational (c) C quintic curves do lie on quadrics and some do not). 3 C lies on P Let be a smooth quintic curve of genus 2. Show that C  Exercise 2.62. a quadric surface Q and a cubic surface S with intersection Q \ S consisting of the union of C and a line. Exercise 2.63. Use the result of Exercise 2.62 — showing that a smooth quintic curve of genus 2 is linked to a line in the complete intersection of a quadric and a cubic — to find the dimension of the subset of the Hilbert scheme corresponding to smooth curves of degree 5 and genus 2.

99 Chapter 3 Introduction to Grassmannians and lines 3 P in Keynote Questions 3 3 will meet all ;:::;L  (a) P Given four general lines , how many lines L  P L 1 4 four? (Answer on page 110.) 3 Given four curves C , how many lines will ;:::;C ;:::;d  P d of degrees (b) 1 1 4 4 meet general translates of all four? (Answer on page 112.) 3 0 C;C  P If are two general twisted cubic curves, how many chords do they have (c) in common? That is, how many lines will meet each twice? (Answer on page 115.) 3 If Q are four general quadric surfaces, how many lines are tangent ;:::;Q (d)  P 4 1 to all four? (Answer on page 125.) 3.1 Enumerative formulas In this chapter we introduce Grassmannian varieties through enumerative problems, of which the keynote questions above are examples. To clarify this context we begin by discussing enumerative problems in general and their relation to the intersection theory described in the preceding chapters. In Section 3.2 we lay out the basic facts about Grassmannians in general. (Sections 3.2.5 and 3.2.6 may be omitted on the first reading, but will be important in later chapters.)

100 3 86 P Chapter 3 Introduction to Grassmannians and lines in 3 P Starting in Section 3.3 we focus on the Grassmannian of lines in . We calculate the Chow ring and then, in Sections 3.4 and 3.6, use this to solve some enumerative 3 P . In Section 3.5 we introduce the key problems involving lines, curves and surfaces in technique of specialization , using it to re-derive some of these formulas. 3.1.1 What are enumerative problems, and how do we solve them? of objects Enumerative problems in algebraic geometry ask us to describe the set ˆ of a certain type satisfying a number of conditions — for example, the set of lines 3 P in meeting each of four given lines, as in Keynote Question (a), or meeting each 3 C P , as in Keynote Question (b). In the most common of four given curves  i ˆ to be finite and we ask for its cardinality, whence the name situation, we expect enumerative geometry. Enumerative problems are interesting in their own right, but — as van der Waerden is quoted as saying in the introduction — they are also a wonderful way to learn some of the more advanced ideas and techniques of algebraic geometry, which is why they play such a central role in this text. There are a number of steps common to most enumerative problems, all of which will be illustrated in the examples of this chapter. If we are asked to describe the set ˆ of objects of a certain type that satisfy a number of conditions, we typically carry out the following five steps:  Find or construct a suitable parameter space H . Suitable, for us, for the objects we seek will mean that H should be projective and smooth, so that we can carry out calculations A. in the Chow ring / . Most importantly, though, for each condition imposed, the locus H Z  H of objects satisfying that condition should be a closed subscheme (which means i T in turn that the set D Z of solutions to our geometric problem will likewise have ˆ i the structure of a subscheme of H ). In our examples, the natural choice of parameter space H G D is the Grassmannian 3 .1;3/ P , which we will construct and describe in Sections 3.2.1 parametrizing lines in G and 3.2.2 below; as we will see, it is indeed smooth and projective of dimension 4. As 3  G of lines ƒ † P we will see in Sections 3.3.1 and 3.4.2, moreover, the locus  C 3 C meeting a given curve P will indeed be a closed subscheme of codimension 1.   Describe the Chow ring H / of H . This is what we will undertake in Section 3.3 A. below; in the case of the Grassmannian G .1;3/ , we will be able to give a complete description of its Chow ring. (In some circumstances, we may have to work with the cohomology ring rather than the Chow ring, as in Appendix D, or with a subring of A. H / including the classes of the subschemes Z , as in Chapter 8.) i  Find the classes ŒZ . ç 2 A. H / of the loci of objects satisfying the conditions imposed i A.G/ of Thus, in the case of Keynote Question (b), we have to determine the class in the locus Z ; the answer is given in Section 3.4.2.  G of lines meeting the curve C i i

101 Enumerative formulas Section 3.1 87  Calculate the product of the classes found in the preceding step . If we have done everything correctly up to this point, this should be a straightforward combination of the two preceding steps. At this point, we have what is known as an : It describes the enumerative formula H / , of the scheme ˆ  H of solutions to our geometric problem, under the class, in A. assumption that this locus has the expected dimension and is generically reduced — that is, the cycles  H intersect generically transversely. (If the cycles Z are all locally Z i i Cohen–Macaulay, then by Section 1.3.7 the enumerative formula describes the class of ˆ  H the subscheme ˆ has the expected dimension; under the weaker hypothesis that that is, the cycles Z are dimensionally transverse.) i  H , indeed has the expected Verify that the set of solutions, viewed as a subscheme of dimension, and investigate its geometry . We will discuss, in the following section, what exactly we have proven if we simply stop at the conclusion of the last step. But ideally Z H  we would like to complete the analysis and say when the cycles do in fact meet i generically transversely or dimensionally transversely. In particular, if the geometric problem posed depends on choices — the number of lines meeting each of four curves — we would like to be able to say that for general , for example, depends on the C C i i choices the corresponding scheme ˆ is indeed generically reduced. Thus, for example, in the case of Keynote Question (b), the analysis described above and carried out in Section 3.4.2 will tell us that if the subscheme ˆ  G of lines Q 3 meeting each of four curves P C is zero-dimensional then it has degree 2  deg .C . / i i not tell us that the actual number of lines meeting each of the four curves But it does Q is in fact deg .C , or for that matter for any. That is addressed in 2 for general C / i i Section 3.4.2 in characteristic 0; we will also see another approach to this question in Exercises 3.30–3.33 that also works in positive characteristic. One reason this last step is sometimes given short shrift is that it is often the hardest. For example, it typically involves knowledge of the local geometry of the subschemes Z — their smoothness or singularity, and their tangent spaces or  H i tangent cones accordingly — and this is usually finer information than their dimensions and classes. But it is necessary, if the result of the first four steps is to give a description of the actual set of solutions, and it is also a great occasion to learn some of the relevant geometry. 3.1.2 The content of an enumerative formula Because the last step in the process described above is sometimes beyond our reach, it is worth saying exactly what has been proved when we carry out just the first four steps in the process. Q ̨ D In general, the computation of the product of the classes of ŒZ ç 2 A. H / i some effective cycles Z in a space H tells us the following: i

102 3 88 P Chapter 3 Introduction to Grassmannians and lines in deg ̨ ̨ 2 A 0 . H / and (for example, if . ̨/ ¤ 0 ), we can conclude that the ¤ (a) If 0 T Z is nonempty intersection . This is the source of many applications of enumerative i geometry; for example, it is the basis of the Kempf/Kleiman–Laksov proof of the existence half of the Brill–Noether theorem, described in Appendix D. is a positive intersect in the expected dimension, then the class ̨ (b) If the cycles Z i T Z . In particu- linear combination of the classes of the components of the intersection i T 2 A . lar, if H / has dimension 0, then the number of points of ̨ Z . ̨/ is at most deg . i 0 This in turn implies: T ̨ 2 A is . H / and deg . ̨/ < 0 , we may conclude that the intersection (i) If Z i 0 rather than finite. More generally, if is not the class of an effective cycle, ̨ infinite T we can conclude that has dimension greater than the expected dimension . Z i T ̨ 2 A must either be empty or . H / and deg . ̨/ D 0 , then the intersection Z (ii) If i 0 T T we can conclude that either ̨ Z infinite. (In general, if D ¿ or 0 Z has D i i dimension greater than the expected dimension.) So, suppose we have carried out the first four steps in the process of the preceding section in the case of Keynote Question (a): We have described the Grassmannian D G .1;3/ and its Chow ring, found the class G of lines D ŒZç of the cycle Z  1 3 4 P L , and calculated that deg . meeting a given line  / D 2 . What does this tell us? 1 4  really only tells us Without a verification of transversality, the formula deg D 2 1 that the number of intersections is either infinite or 1 or 2. Beyond this, it says that if 3 P that meet the four the number of “solutions to the problem” — in this case, lines in given lines — is finite, then there are two counted with multiplicity — that is, either two solutions with multiplicity 1, or one solution with multiplicity 2. In order to say more, we T need to be able to say when the intersection Z has the expected dimension; we need i to be able to detect transversality and, ideally, to calculate the multiplicity of a given solution. (The third of these is often the hardest. For example, in the calculation of the 3 number of lines meeting four given curves C  P , we see in Exercises 3.30–3.33 how i to check the condition of transversality, but there is no simple formula for the multiplicity when the intersection is not transverse.) A common aspect of enumerative problems is that they themselves may vary with parameters: If we ask how many lines meet each of four curves C , the problem varies i with the choice of curves C . In these situations, a good benchmark of our understanding i is whether we can count the actual number of solutions for a general such problem: for example, whether we can prove that if ;:::;C are general conics, then there are C 1 4 exactly 32 lines meeting all four. Thus, in most of the examples of enumerative geometry we will encounter in this book, there are two aspects to the problem. The first is to find the “expected” number of solutions by carrying out the first four steps of the preceding section to arrive at an enumerative formula. The second is to verify transversality — in other words, that the actual cardinality of the set of solutions is indeed this expected number — when the problem is suitably general.

103 Introduction to Grassmannians Section 3.1 89 3.2 Introduction to Grassmannians A Grassmannian , is a projective variety whose closed points Grassmann variety , or correspond to the vector subspaces of a certain dimension in a given vector space. Projective spaces, which parametrize one-dimensional subspaces, are the most familiar examples. In this chapter we will begin the study of Grassmannians in general, and then 3 focus on the geometry and Chow ring of the Grassmannian of lines in , the first and P most intuitively accessible example beyond projective spaces. Our goal in doing this is to introduce the reader to some ideas that will be developed in much greater generality (and complexity) in later chapters: the Grassmannian (as an undetermined coefficients and special- example of parameter spaces), the methods of ization for computing intersection products more complicated than those mentioned in Chapter 2, and questions of transversality, treated via the tangent spaces to parameter spaces. For more information about Grassmannians, the reader may consult the books of Harris [1995] for basic geometry of the Grassmannian, Griffiths and Harris [1994] for the basics of the Schubert calculus and Fulton [1997] for combinatorial formulas, as well as the classic treatment in the second volume of Hodge and Pedoe [1952]. -dimensional G.k;V / As a set, we take the Grassmannian k G D to be the set of V . We give this set the structure of a projective vector subspaces of the vector space Pl , ucker embedding variety by giving an inclusion in a projective space, called the ̈ and showing that the image is the zero locus of a certain collection of homogeneous polynomials. is the same n -dimensional vector space V -dimensional vector subspace of an A k 1 n -dimensional linear subspace of P V Š P , so the Grassmannian as a .k 1/ G.k;V / could also be thought of as parametrizing 1/ -dimensional subspaces of P V . We .k will write the Grassmannian as G .k 1; P V / when we wish to think of it G.k;V / this way. When there is no need to specify the vector space V but only its dimension, say n G.k;n/ or G .k 1;n 1/ . Note also that there is a natural , we will write simply identification  G.k;V / G.n k;V / D ?  ƒ  V sending a ƒ k  V -dimensional subspace . to its annihilator There are two points of potential confusion in the notation. First, if ƒ  V is a k -dimensional vector subspace of an n -dimensional vector space V , we will often use the same symbol ƒ G D G.k;V / . When we need to denote the corresponding point in Œƒç 2 G for the point corresponding to make the distinction explicit, we will write to the plane ƒ  V . Second, when we consider the Grassmannian G D G .k; P V / we will sometimes need to work with the corresponding vector subspaces of . In V z ƒ  P V is a k -plane, we will write these circumstances, if ƒ for the corresponding .k C 1/ -dimensional vector subspace of V .

104 3 90 P Chapter 3 Introduction to Grassmannians and lines in ucker embedding 3.2.1 The Pl ̈ -dimensional vector subspaces of a given vector space To embed the set of V k in a ƒ  the one-dimensional k projective space, we associate to a -dimensional subspace V subspace V V k k  I ƒ V V k ;:::;v that is, if , we associate to it the point of P . ƒ v V / corresponding has basis 1 k . This gives us a map of sets ^^ v to the line spanned by v 1 k n V k 1 . / k ; G.k;V / V / Š P . ! P . To see that this map is one-to-one, observe that if Pl called the ucker embedding ̈ ;:::;v in are a basis of ƒ  V , then a vector v annihilates  D v v ^^ v 1 1 k k v is in the span ƒ of the exterior algebra if and only if . ;:::;v ƒ ; thus  determines v 1 k n V Concretely, if we choose a basis g for V , and so identify ;:::;e with k f , we e 1 n as the row space of a k  n matrix may represent ƒ 0 1 a a  a 1;n 1;2 1;1 B C a  a a 2;1 2;2 2;n B C A D : B C : : : : : : : : @ A : : : :  a a a k;n k;1 k;2 V k V In these terms, a basis for is given by the set of products f ; ^^ e g e i n i 1  i  < 

105 Introduction to Grassmannians Section 3.2 91 e 2 we could replace it by any nonzero element is nothing special about the vector V i V k 2 V . Repeating this idea, we see that a nonzero element V can be written in  v 2 if and only if ^^ v the form for some (necessarily independent) v v ;:::;v V 2 1 1 k k the kernel of the multiplication map  ^ V C k 1 V ! V ̈ has dimension at least ucker embedding is k . That is, the image of the Pl ̊ ^  V V k k C 1 n  ! 2 G D V V /  j rank .V ; k V k k C 1 on and this is the zero locus of the homogeneous polynomials of degree V that n V k C 1 ! ^  W V k .n C 1/ -st-order minors of the map V written out as a matrix. are the G is an algebraic set, it follows that G is a variety: Its ideal is Once we know that the kernel of the map of polynomial rings k ç Œp ç Œx ! k  i ;:::;i < 

106 3 92 P Chapter 3 Introduction to Grassmannians and lines in 4  matrix with One way to obtain this relation is to note that the determinant of the 4 repeated rows 1 0 a a a a 1;1 1;2 1;3 1;4 C B a a a a C B 2;3 2;1 2;4 2;2 C B a a a a A @ 1;3 1;1 1;4 1;2 a a a a 2;2 2;4 2;3 2;1 must be 0. Expanding this determinant as a sum of products of minors of the first two ̈ rows and of the last two rows, all of which are Pl ucker coordinates, we obtain (3.1) p 0: p D p p C p p 1;4 2;3 1;3 3;4 1;2 2;4 dim G.2;4/ 4 ), it generates the homogeneous As this is an irreducible polynomial (and D 5 , which is thus a smooth quadric.  G.2;4/ ideal of P n G.2;n/ is cut out by In fact, for any , the ideal of the Grassmannian of 2-planes ̈ ucker coordinates similar to the polynomial (3.1) quadratic polynomials in the Pl above. P V 2 e is a basis of V and  D ;:::;e p More precisely, if e , then the ^ e 2 V 1 n a b a;b polynomials ̊ p WD p n p  p a < b < c < d p  C p 1 g j D 0 a;c b;c a;d c;d a;b a;b;c;d b;d Pl minimally generate the ideal of the Grassmannian. These are the ucker relations in ̈ ̈ . We will describe the Pl ucker relations in G.2;n/ the special case of the Grassmannian general following Proposition 3.2. g Another way to characterize the collection of polynomials g defining f a;b;c;d .1;n/ , in characteristic not equal to 2, is that they are the coefficients of the element G V V 4 2 2 2 0 — in other words, an element D 2   is decomposable if and only if V V ^  D 0 . These coefficients may be characterized (up to a factor of 2) as the Pfaffians  of a skew-symmetric matrix. Exercises 3.17–3.22 describe a number of aspects of the projective geometry of the ̈ Grassmannian in the Pl ucker embedding. 3.2.2 Covering by affine spaces; local coordinates Like a projective space, a Grassmannian D G.k;V / can be covered by Zariski G .n k/ -dimensional subspace open subsets isomorphic to affine space. To see this, fix an Ä  V , and let U : be the subset of k -planes that do not meet Ä Ä U Df ƒ 2 G j ƒ \ Ä D 0 : g Ä G : In fact, if we take w This is a Zariski open subset of to be any basis for ;:::;w 1 n k Ä and set  D w , then we have ^^ w 1 n k V k ¤  ^ ! j V / ; g 0 P  G 2 Œ!ç Df U . Ä

107 Introduction to Grassmannians Section 3.2 93 U corre- is the complement of the hyperplane section of from which we see that G Ä ̈ sponding to the vanishing of a Pl ucker coordinate (though not all hyperplane sections of ̈ G ucker embedding have this form). in the Pl k/ k.n A G.k;n/  . We claim now that the open set is isomorphic to affine space U Ä U To see this, we first choose an arbitrary point that will play the role of the Œç 2 Ä -plane   V complementary to Ä , so that we have a direct-sum k origin; that is, fix a D decomposition ̊ Ä . Any k -dimensional subspace ƒ  V complementary V  Ä Ä modulo  — call this map  to — and projects isomorphically to  projects to Ä Ä  modulo . Thus ƒ is the graph of the linear map — call this map  1    Ä W '  V D  ̊ Ä ! ƒ  Ä: ! Ä W Ä is a subspace ƒ   ̊ ! D V  Conversely, the graph of any map ' . These two correspondences establish a bijection Ä complementary to k/ k.n Š : .;Ä/ Hom Š U A Ä V e ;:::;e To make this explicit, suppose we choose a basis for consisting of a basis 1 k followed by a basis e for ;:::;e  for Ä . If ƒ 2 U -plane then the preimages is a k n Ä C 1 k 1 1 is the row space of the matrix ;:::; form a basis for  ƒ e . Thus 2 ƒ e ƒ 1 k   0 1 ::: a 1 0 ::: 0 a a 1;2 1;1 1;n k B C ::: a 0 1 ::: 0 a a 2;1 2;2 2;n k B C D B ; B C : : : : : : : : : : : : : : @ A : : : 0 : : : : 0 0 ::: 1 a ::: a a k;n k;2 k;1 k A D .a in the / is the matrix representing the linear transformation ' W  ! Ä where i;j given bases. Since there is a unique vector in projecting (mod Ä ) to each e ƒ 2  , this i matrix representation is unique. The bijection defined above sends 2 U to the linear ƒ Ä D  transformation given by the transpose of the matrix A ! .a . / Ä i;j If we start with any representation of ƒ as the span of the rows of a k  n matrix 0 ̈ V , then the Pl B ucker coordinate p with respect to the given basis of , which 1;2;:::;k 0 is the determinant of the submatrix consisting of the first B , is nonzero. columns of k 0 on the left by the inverse of this submatrix gives us back the matrix B B Multiplying 0  k minors of B above, and thus the k  k minors of B k multiplied by the are the 0 inverse of the determinant of the first k  k minor of B . On the other hand, we can realize the entry a minor of of A , up to sign, as a k  k i;j k  k submatrix in which we take all the first k : It is (up to sign) the determinant of the B i -th, and put in instead the .k C j/ -th column. Thus we may write columns except for the p .ƒ/ O 1;:::;i C 1; j i;i C 1;:::;k;k ; D a ̇ i;j p .ƒ/ 1;:::;k

108 3 94 Chapter 3 Introduction to Grassmannians and lines in P and this expression shows that is a regular function on . Thus the bijection a U i;j Ä k.n k/ Š is a biregular isomorphism. U A Ä ̈ .ƒ/ .ƒ/=p ucker of Pl More generally, it turns out that the ratios p ;:::;a a 1;2;:::;k 1 k . coordinates are, up to sign, precisely the determinants of submatrices (of all sizes) of A J are sets of indices. Write A I To express the result, suppose that for the minor of J and I 0 I and columns with indices in J . Write I A the matrix involving rows with indices in f g ) of I , and, if J Df j for the complement (in the set of row indices ;:::;j 1;:::;k g , t 1 k;:::;j k for the “translated” set of indices f j write C J . With this notation, C k g C t 1 J  t minor A the t of A is equal, up to sign, to the k  k minor of B involving the I 0 columns . To see this as a regular function on U I we need only divide by the J [ Ä : 1;:::;k minor involving columns D f i With notation as above, suppose that ;:::;i Proposition 3.2. I are row g 1 t k Df j indices and ;:::;j J g are column indices with each j . We have > k i t 1 0 p .ƒ/ I [ .J C k/ J A ̇ det : D I p .ƒ/ 1;:::;k For example, the 3  3 minor of the matrix 6 1 7 2 3 4 5 0 1 1 0 0 a a a a 1;1 1;2 1;3 1;4 @ A B D 0 1 0 a a a a 2;4 2;1 ; 3 2 2 2 ; a 0 0 1 a a ; a 3;4 3;1 3 ; ; 3 2 3 2  2 minor involving columns 1, 5 and 6, is, up to sign, the   a a 2;2 2;3 det a a 3;3 3;2 .a / of the matrix . i;j ̈ Proof: ucker coordinates on the right is independent of the matrix The expression in Pl ̈ , so we may compute the two Pl representation chosen for ucker coordinates in terms ƒ 0 B in the form given above, so p is the .ƒ/ D 1 and p .ƒ/ of the matrix [ .J k/ I 1;:::;k C 0 B I . Expanding this minor in terms of the minor of involving the columns k/ [ .J C J 0  .k t/ .k I , we see that all but the term ̇ 1  A t/ minors involving the rows of I are zero. Having established Proposition 3.2, it is easy to describe the Pl ucker relations , the ̈ ̈ quadratic polynomials in the Pl ucker coordinates that generate the homogeneous ideal of V k   . V / . With notation as above, consider the expansion of any G.k;V / P t minor t of A along one of its rows or columns. Replacing each factor of each term that appears ̈ p ucker coordinates, with denominator by the ratio of two Pl , and multiplying .ƒ/ 1;:::;k 2 p satisfied .ƒ/ , we get a homogeneous quadratic polynomial in the p through by I 1;:::;k identically in U and hence in all of G.k;V / . For more information we refer the reader Ä to De Concini et al. [1980, Section 2] or Fulton [1997, Section 9.1].

109 Introduction to Grassmannians Section 3.2 95 3.2.3 Universal sub and quotient bundles In this section and the following, we will introduce the universal bundles on the and show how to describe the tangent bundle to Grassmannian G.k;n/ G.k;n/ in terms of them. These constructions are of fundamental importance in understanding the geometry of Grassmannians. G.k;V / -dimensional vector space, Let D be an the Grassmannian of k - V n G , and let V WD G  V be the trivial vector bundle of rank n on G planes in V whose fiber V at every point is the vector space (here we are thinking of a vector bundle as a variety, for the rank- subbundle of V whose S k rather than as a locally free sheaf). We write 2 G is the subspace ƒ fiber at a point Œƒç itself; that is, D ƒ  V D V S : Œƒç Œƒç S is called the on G ; the quotient Q D V = S is called the universal universal subbundle n 1 Š G D P V D P . In the case 1 — that is, — the universal quotient bundle k  is the line bundle O ) subbundle . S ; similarly, in the case k D n 1 (so G D P V 1/ V P  Q is the line bundle O the universal quotient bundle .1/ . V P We have said “the rank- k subbundle of V whose fiber at a point Œƒç 2 G is the subspace ƒ at most one bundle, since we have itself,” and this certainly describes V D  V . Who would doubt that it is an algebraic unambiguously defined a subset of G ? To prove this, however, something more is necessary. Most primitively, V subbundle of we must check that it is trivial on an affine open cover, and that the transition functions are regular on the overlap of any two open sets of the cover. Alternatively, and equivalently, we may show that the subset S is an algebraic subset, and that over an open cover it is isomorphic, as an algebraic variety, to a trivial bundle. Here is a proof: S Proposition 3.3. V whose fiber over a point Œƒç 2 G D G.k;V / is the The subset of ƒ subspace V is a vector bundle over G .  Of course, it follows that D V = S is also a vector bundle. Q Let be the incidence correspondence Proof: S Df .ƒ;v/ 2 G S V j v 2 ƒ g :  The set S is an algebraic subset of G  V , since if we represent ƒ by a vector  2 V V V k k C k 1 V ^ D ƒ 2  , it is given by the equation v  V . Explicitly, if ƒ is the row 0 space of the matrix A , as in Section 3.2.2, then the condition v 2 ƒ is equivalent to the 0 vanishing of the C 1/ -st-order minors of the matrix obtained from A by adjoining v .k as the .k C 1/ -st row. These minors can be expressed (by expanding along the new row 0 ̈ and the Pl ) as bilinear functions in the coordinates of v A of ucker coordinates, proving that S is an algebraic subset.

110 3 96 P Chapter 3 Introduction to Grassmannians and lines in Ä  of dimension n k and consider the preimage Now pick a subspace V as before, we can identify U . Choosing a complement  to Ä G U  with of Ä Ä . Moreover, if ƒ 2 U V then the projection ˇ Hom W .;Ä/ !  with kernel Ä ;Ä Ä takes D ƒ  V isomorphically to  . In other words, this projection gives an S Œƒç . This proves that isomorphism to the trivial bundle   U S is actually a vector S U Ä Ä bundle, which we identify as S . S The following result is the reason that we refer to as the universal subbundle. A proof may be found in Eisenbud and Harris [2000]. If X ' W X ! G are in a one-to-one Theorem 3.4. is any scheme then the morphisms corresponds to the subbundles  V k O correspondence with rank- such that ' F ̋ X  bundle F D ' S . S . Viewing There is also a projective analog of the vector bundle as G .k 1; P V / G (that is, as parametrizing 1/ -planes in P V ), we set .k j Df ˆ G  P V .ƒ;p/ p 2 ƒ g : 2 The space ˆ can also be realized as the projectivization of the universal subbundle S , where by the projectivization E on a scheme X we mean P E WD of a vector bundle  whose fiber over a point . / — a locally trivial fiber bundle over X E p 2 X is Sym Proj . E in Section 4.8.1, where we will discuss flag / . (We will see more of the space ˆ P p manifolds in general and in particular; we will deal with projective bundles in general ˆ ˆ D P S is called the universal k -plane over G in Chapter 9.) . Theorem 3.4 may be interpreted as saying that the Grassmannian represents the k V , in the sense that the contravariant functor of families of -dimensional subspaces of functor from schemes to sets given on objects by 7! Mor .X;G.k;V // is naturally X isomorphic to the functor given by X 7! f rank- k subbundles of V ̋ O . Again, in g X the language that we will develop in Section 6.3, this says that the Grassmannian V .k 1; P V / is the Hilbert scheme of .k 1/ -planes in G . See Eisenbud and Harris P [2000, Chapter 6] for an introduction to these ideas and a proof of this statement. 3.2.4 The tangent bundle of the Grassmannian Knowledge of the tangent bundle of the Grassmannian is the key to its geometry. It turns out that the tangent bundle can be expressed in terms of the universal bundles S and Q : Theorem 3.5. The tangent bundle T is isomorphic to the Grassmannian G D G.k;V / G to om are the universal sub and quotient bundles. . S ; H / , where S and Q Q G Proof: Consider the open affine set j U Df ƒ 2 G 0 ƒ \ Ä D g Ä

111 Introduction to Grassmannians Section 3.2 97 Ä is a subspace of of dimension n k . Fixing a described in Section 3.2.2, where V Ä 2 and decomposing V as  ̊ point , we get an identification of U Œç with the U Ä Ä .;Ä/ under which the point Œç goes to the linear transformation 0. vector space Hom restricted to U is the trivial bundle and the fiber In particular, the tangent bundle T Ä G is Hom . Œç over .;Ä/ j The bundle by the composite map S   U is isomorphic to the trivial bundle Ä U Ä j ; U ! V  U V=Ä ! S  U  D  Ä Ä Ä U Ä and the bundle Q j is isomorphic to the trivial bundle Ä  U via the tautological Ä U Ä ̋ O V projection Q . This gives an identification of fibers, depending on Ä : ! G T /: / Q . Hom .;Ä/ D Hom . S ; D   G  . T Š H To prove that these identifications extend to an isomorphism , / S ; Q om G G we must check that the gluing map for T on an intersection and that for H om / . S ; Q G G 0 Œç U U D containing the point \ agree on the fiber over Œç (and thus agree U Ä Ä  U as maps of bundles). We may regard D Hom .;Ä/ as the set of linear U Ä 0 Ä , and this representation is related to the transformations whose graphs do not meet 0 0 U . The gluing ! by the isomorphisms Ä representation of U V=  Ä Ä ̨ ˇ  W T j j / j / j T . d' ! . G U U G U U 0 0 Ä Ä Ä Ä along this set is by the differential of the composite linear transformation 1 ˇ ̨ 0 ' .;V=/ .;Ä/ W ! ! Hom .;Ä Hom / Hom induced by these isomorphisms. Of course, the differential of a linear transformation is the same linear transformation. The same isomorphisms give the gluing of the bundle om H . S ; Q / . G From the identification of tangent vectors to G D G.k;V / at ƒ with the space Hom .ƒ;V=ƒ/ , we can see that not all tangent vectors at a given point are alike: We can associate to any tangent vector its , and this will be preserved under automorphisms rank of G (see Exercise 3.24 and, for a nice application, Exercise 3.23). In particular, this means that when 1 < k < dim V 1 the automorphism group of G.k;V / does not act transitively on nonzero tangent vectors, and hence Kleiman’s theorem (Theorem 1.7) does not apply in positive characteristic. Nevertheless, the conclusions it gives for intersections of Schubert cycles are correct in all characteristics (and may be proven by a different method). n The Euler sequence on P The isomorphism of Theorem 3.5 is already useful in the case of projective space n . D G .0;n/ . In this setting Theorem 3.5 gives rise to the Euler sequence P

112 3 98 Introduction to Grassmannians and lines in Chapter 3 P 1 P v v 3 v V 2 v 1 2 1 v to A ;v all map to the tangent vector to P v . Figure 3.1 The tangent vectors ;v 1 3 2 n D C 1/ -dimensional vector space and P V .n P V its projectivization. Let be an We consider the quotient map n U q nf 0 g! P W D V n 2 to the corresponding point p D Œvç v P 2 . The tangent V sending a nonzero vector at v is the same as the tangent space to V at v , which is to say the vector U space to V itself, and the kernel of the differential space n T W U ! T dq P p v v Q p D h v i  V spanned by v . Thus dq is the one-dimensional subspace induces an v isomorphism n  ; ! V= T Q P p p as illustrated in Figure 3.1. This isomorphism does not, however, give a natural identification of the vector n spaces and T V= P p . Even though both these vector spaces depend only on the point Q p n 2 , the isomorphism dq p between them depends on the choice of the vector v . P v Indeed, if is any nonzero scalar, the differential dq . is equal to dq  divided by  v v is any linear functional, then the map Wh v i! But, by the same token, if l l.v/  dq k v is independent of the choice of v , and so we have a natural identification  n  ̋ V= h v i! i T : P h v Œvç This is the identification n n n T Š / Q .1/ ̋ Q D H om . O ; O . 1/;Q/ D H om . S P P P asserted (more generally) in Theorem 3.5.

113 Introduction to Grassmannians Section 3.2 99 x To put it another way, in terms of coordinates ;:::;x on V , a constant vector 0 n field V does not give rise to a vector field on P V , but the vector field @=@x on i @ x w.x/ D j @x i does. This gives us a map on V n ̋ V ! T ; O .1/ P V P whose kernel is the Euler vector field X @ e.x/ x D : i @x i The resulting exact sequence 7! 1 e n ! 0 O O ! ! ̋ V ! T 0 .1/ P V V P P is called the Euler sequence . To relate this to the identification of the tangent bundle above, start with the universal sequence on P V : ! S ! O 0 ! ̋ V ! Q 0: V P   S O , we arrive at the O .1/ ; since Now tensor with the line bundle ̋ S D Š S V P V P sequence  n S O ! .1/ ̋ V ! O 0 ̋ Q ! 0: ! P V P n By Theorem 3.5 the term on the right is , and we obtain the Euler sequence again. T P 3.2.5 The differential of a morphism to the Grass- mannian W X Suppose that the morphism G.k;n/ corresponds via the universal property ' ! n  O to a subbundle E , so that E is the pullback of the universal subbundle S on G.k;n/ . X n Q O Set = E , so that F is the pullback of the universal quotient bundle F on G.k;n/ . D ' is by definition a homomorphism of vector bundles The differential of   T /: ! ' d' T F D ' W H om ; . S ; Q / D H om E . X X G.k;n/ G.k;n/ The local description of the Grassmannian above makes it easy to identify this homo- morphism locally. A global section of the is called a vector field . Recall that a vector field may be T X identified with a derivation @ W O (Eisenbud [1995, Chapter 16]). This works ! O X X even if X is singular: In that case we define T to be the dual of the sheaf of differential X forms; of course then T is a coherent sheaf, not necessarily a vector bundle. (A famous X question posed by Zariski (see Lipman [1965]) asks whether, with this definition — and always assuming that the characteristic is 0 — T is a vector bundle if and only if X is X smooth. See Hochster [1977] for some partial results.)

114 3 100 Chapter 3 P Introduction to Grassmannians and lines in be a variety and W X ! G.k;n/ the morphism corresponding Proposition 3.6. Let ' X n n n = E ! F W O E to a subbundle O ; set and let  i O W be the projection. Let ! D F X X X k Š E is trivial and W O U  X be an open subset over which E a trivialization of E U U F @ is a vector field on U , then in U the homomorphism .d'/.@/ 2 H om over . E ; U / . If U of the @. ̨/ , where @. ̨/ is the derivative with respect to @ ı  is the composition composite map i k n W O ̨ D i ! E I ı ! O ' U U U @ that is, is the map obtained by applying to each entry of a matrix representing .d'/.@/ n ! F . O  and composing the result with the projection ̨ W U U .d'/.@/ E , and Note that the map described above depends only on the subbundle k k k E ! O Š W O not on the trivialization O is an invertible matrix over chosen: If ˇ X X X , then O X .@ ̨/ˇ C ̨.@ˇ/; D @. ̨ˇ/ . and the second term vanishes when we project to F Proof: The desired result follows at once from the description of the affine spaces n ̨ covering the Grassmannian: We can change bases in so that O is given by the U k n matrix  0 1 ̨  1 0  0 ̨ ̨ 1;1 1;2 k 1;n B C  ̨  ̨ 0 ̨ 0 1 2;1 2;2 k 2;n B C ; B C : : : : : : : : : : : : : : : : @ A : : : : : : : : 0 0 0 1 ̨  ̨ ̨ k k;2 k;n k;1 ̨ where the U , and give the morphism ' in local coordinates. The are functions on i;j derivative of ' @ , is then by definition obtained by applying @ to each of the , applied to ̨ . coordinate functions i;j 3.2.6 Tangent spaces via the universal property There is another way to approach the tangent space, which depends on a pretty and well-known bit of algebra. Let O be a local ring with maximal ideal m , and suppose for simplicity that O k . contains a copy of its residue field There are natural one-to-one correspondences between Proposition–Definition 3.7. the following sets: 2 -vector spaces). . m = (a) Hom ; k / (homomorphisms of k m k (b) Der k . O ; k / ( k -linear derivations; that is, k -vector space homomorphisms d W O ! k d.fg/ D fd.g/ C gd.f / ). that satisfy Leibniz’ rule 2 Hom -algebras). k . O ; k Œç=. // (c) (homomorphisms of -algebras k 2 Mor .D; Spec O / (morphisms of schemes). D D Spec k Œç=. (d) / , where

115 Introduction to Grassmannians Section 3.2 101 k The first two of these are naturally -vector spaces, and the correspondence preserves this structure; we regard the other two as equipped with this structure as well. Any of Zariski tangent space of these spaces, with its vector space structure, is called the O at its closed point. Spec O , we think of a variety (or scheme) at a closed point x When O is the local ring X;x as the Zariski tangent space of at x , and of course X of the Zariski tangent space of O -schemes carrying the (unique) k the set in (d) is the same as the set of morphisms of — which we will call 0 — to x . point of D ' ' j is as in (c), then The sets in (c) and (d) are the same by definition. If Proof: m 2 2 2 W m annihilates m m ! ./=. = / Š k and induces a vector space homomorphism ' 2 as in (b) induces a k -linear map d as in (a). Similarly, a derivation . W m = m d ! k j m We leave to the reader the construction of the inverse correspondences. Consider the case G.k;V / . The tangent space at x D Œƒç is, by the argument D X ! G.k;V / sending 0 to Œƒç . By Theorem 3.4, giving a D above, the collection of maps D ! G.k;V / is the same as the giving a rank- k subbundle W of map  D ; the map V takes 2 D to Œƒç 2 G .k;V / if and only if the fiber 0 . is equal to ƒ W 0 We can understand the identification of the tangent space T G.k;V / to the Grass- ƒ mannian with the space Hom .ƒ;V=ƒ/ using this description together with the universal property of the Grassmannian described in Theorem 3.4. Since D is the same as a locally free module D is affine, a vector bundle over 2 Œç=. / . Since this ring is local, Nakayama’s lemma shows that such a module k over is free (see for example Eisenbud [1995, Exercise 4.11]). Thus only the inclusion D ƒ V  D varies.  ! Putting this together, we get a new way to look at the identification of the tangent spaces to the Grassmannian: Let ƒ  V be a k -dimensional subspace, and let ' W ƒ ! V=ƒ be a Proposition 3.8. G.k;V / at the homomorphism. As an element of the tangent space to the Grassmannian Œƒç , ' corresponds to the free submodule point 2 2 0 Œç=. ̋ / ! V ̋ k Œç=. ƒ /; v ̋ 1 7! v ̋ 1 C ' k .v/ ̋ ; 0 ! ƒ ! V is any map that when composed with the projection V W V=ƒ where ' ' . gives Any map ƒ  D ! V  D that reduces to the inclusion modulo  has the Proof: 0 0 ̋ 1 7! v ̋ 1 C ' form .v/ ̋  for some ' v . If we work in the affine coordinates V corresponding to a subspace complementary to ƒ and use the splitting Ä D ƒ ̊  ,

116 3 102 P Chapter 3 Introduction to Grassmannians and lines in ƒ V  then the point corresponds to the matrix 0 1 0  1 0  0 0 B C 0 1  0 0  0 B C : : B D : : B C : : : : : 0  0 0 : @ A  0 0 0  1 0 0 is represented by the last In this matrix representation k columns of ' ' , and taking a n 0 corresponds to making a different choice of the first k  k block of ' different lifting of . '  k  k block, adding We can do row operations to clear all the terms from the first a multiple of  times certain rows to other rows. This corresponds to composing with ƒ  D , and thus does not change the image of an automorphism of  D ! V  D . ƒ Since we add after multiplying by  , this does not change the block representing ' . Thus 0 k  k block of ' is 0; equivalently, the first we may assume that the first k  k block corresponding to the map  ! V  D is the identity. ƒ D .1;3/ G 3.3 The Chow ring of Before launching into the geometry of general Grassmannians in the next chapter, .1;3/ , the we will spend the remainder of this chapter studying the geometry of G 3 . This is the simplest case beyond the projective spaces. The P Grassmannian of lines in general results are in many ways similar, but more combinatorics is involved, and in the 3 P case of lines in it is possible to visualize more of what is going on. Once the reader .1;3/ the general results will seem more natural. has absorbed the case of G 3.3.1 Schubert cycles in .1;3/ G 3 3 on P To start, we fix a ; that is, a choice of a point p 2 P complete flag , a line V 3 3 P p containing L , and a plane H  P  containing L (Figure 3.2). We can give a stratification of G by considering the loci of lines ƒ 2 G .1;3/ .1;3/ p having specified dimension of intersection with each of the subspaces L and H . , These are called Schubert cells and their closures, which are irreducible subvarieties, are called Schubert cycles (or sometimes Schubert varieties); the classes of these cycles are the . As we shall see, the Schubert cells form an affine stratification Schubert classes of G .1;3/ , and it will follow from Proposition 1.17 that the Schubert classes generate the Chow group A. G .1;3// . Using intersection theory, we will be able to show that in fact A. G .1;3// is a free Z -module having the Schubert classes as free generators. In the next chapter, we will see that the same situation is repeated for all Grassmannians.

117 The Chow ring of G Section 3.3 103 .1;3/ 3 P L p H 3 P H in  Figure 3.2 A complete flag . L  p More formally, we begin not with the Schubert cells but with the Schubert cycles: † G D .1;3/; 0;0 Df ƒ j ƒ \ † ¤ ¿ g ; L 1;0 ; Df ƒ j p 2 ƒ † g 2;0 ƒ j ƒ  H g ; † Df 1;1 ; Df ƒ j p 2 ƒ  † g H 2;1 g L ƒ j ƒ D † Df : 2;2 The four nontrivial ones are illustrated in Figure 3.3. In each case we take the reduced scheme structure. To show, for example, that † is an irreducible variety, we note first 1;0 that † is the image of the incidence correspondence 1;0 0 0 Ä ;p/ 2 G .1;3/  L j p 2 L .L g : Df Ä under projection to L , the set of lines through a given point p 2 L , is The fiber of 2 P ; since all fibers are irreducible and of the same dimension and the isomorphic to Ä projection is proper, it follows that † , are irreducible. The proof that , and with it 1;0 the other Schubert cycles are irreducible follows in exactly the same way. Thus † in a denotes the set of lines meeting the .2 a/ -dimensional plane of V a;b point and the b/ -dimensional plane of V .3 in a line. This system of indexing may seem peculiar at first; the reasons for it will be clearer when we discuss Schubert cycles in general in the following chapter. For now, we will mention that the codimension of † a;b a C b , as will be clear in examples and as we will prove in general in Theorem 4.1. is We often drop the second index when it is 0, writing for example † instead of 1 † . When the choice of flag is relevant, we will sometimes indicate the dependence by 1;0 writing †. V / , or simply note the dependence on the relevant flag elements by writing, . † L .L/ for the cycle of lines ƒ meeting for example, 1

118 3 104 P Chapter 3 Introduction to Grassmannians and lines in L H † .L/ † .H/ 1 1;1 H p p .p/ † † .p;H/ 2 2;1 Figure 3.3 Schubert cycles in G . .1;3/ It is easy to see that there are inclusions † 2 L gD † .1;3/: † † f G 1 2;2 2;1 † 1;1 ı † For each index we define the Schubert cell to be the complement in .a;b/ of † a;b a;b the union of all the other Schubert cycles properly contained in † . To show that the a;b ı † † form an affine stratification, it suffices to show that each is isomorphic to an a;b a;b affine space. We will do the most complicated case, leaving the others for the reader (the G.k;n/ is done in Theorem 4.1). general case of a Schubert cell in Example 3.9. We will show that the set ı p g D † H n .† 6 [ † ƒ / Df ƒ j ƒ \ L ¤ ¿ but † ... ƒ and 1;1 2 1 1 3 0 A . Let be a general plane containing the point p but not containing is isomorphic to H L . Any line meeting L but not passing through p and not contained in H meets the line 0 0 0 H/ n .H H \ in a unique point contained in (Figure 3.4). Thus we have maps H 2 ı 1 ı 0 0 p g / Š A † and † ! A ! .H .L n .H \ H nf // Š 1 1 0 ƒ ƒ \ L sending to \ H ƒ respectively. The product of these maps gives us and an isomorphism 3 1 ı 2  † Š A A D A : 1

119 The Chow ring of G Section 3.3 105 .1;3/ ƒ 0 H H p L ı 0 L;ƒ \ H Figure 3.4 The map / defines an isomorphism 7! .ƒ ƒ ! \ † 1 3 2 1 0 . .H H  H / / Š A g  A p Š A nf n .L \ b C a A 2 ç . G .1;3// does not depend on the choice By Theorem 1.7, the class Œ† a;b GL ; we will denote the class of flag, since any two flags differ by a transformation in 4 of by † a;b b a C 2 A D ç  . G .1;3//: Œ† a;b a;b 0 . G .1;3// is isomorphic to Z and is generated by the A By Proposition 1.8, the group 4 is also D Œ G .1;3/ç fundamental class A  . G .1;3// ; by Theorem 1.7, the group 0;0 Z  of a point in G .1;3/ . (In particular isomorphic to and is generated by the class 2;2 G .1;3/ any two points in are rationally equivalent.) 3.3.2 Ring structure We can now determine the structure of the Chow ring of G .1;3/ completely: a C b Theorem 3.10. 2 A 0 The six Schubert classes  . G .1;3// , , freely  b  a  2 a;b A. .1;3// as a graded abelian group, and satisfy the multiplicative relations generate G 2 1 1 2 A .A C D   ! A  / I  2 1;1 1 3 2 1  D   D  I .A / A A  !  1 1 1;1 2;1 2 4 3 1 ! D .A   A  I A /  1 2;2 2;1 2 2 2 2 4 0 .A D  ;  A  ! D    A D /: 2 1;1 2;2 2 1;1 3 4 2 D 2 From these formulas we deduce that ;   D 2 D , and  2;2 1;1 2;1 1 1 1 2    D .1;3// , such . Since dim . G > 4 D 4 , any product that would have degree 2;2 2 1 as   , is 0. 2 2;1

120 3 106 Chapter 3 Introduction to Grassmannians and lines in P As we said, we know by Proposition 1.17 that the Schubert Proof of Theorem 3.10: 4 A. G .1;3// . That they are free generators follows for A classes  G .1;3// generate . a;b from Proposition 1.21, and will follow for the remaining Chow groups from the intersec- tions products above: For example, the formulas show that the matrix of the intersection 2 and  pairing on is nonsingular, so A  . G .1;3// is freely generated by these 1;1 2 two classes. It remains to prove the formulas. We will consider the intersections of pairs of cycles, 0 ; . To simplify notation we will taking these with respect to generically situated flags V V 0 0 † henceforth write , respectively. † † for . V / and † / . V and a;b a;b a;b a;b We begin with the case of cycles of complementary dimension, starting with the  intersection number of with itself. By Kleiman transversality we have 2 0 2  #  \ † D ; / .†  2;2 2 2 2 and since the intersection 0 0 † † g Df ƒ j p 2 ƒ and p \ 2 ƒ 2 2 0 0 and through ƒ p;p p D ), consists of one point (corresponding to the unique line p we conclude that 2 D  :  2;2 2 Similarly, 2 0 I D  .†  \ † #  / 1;1 2;2 1;1 1;1 since 0 0 H † Df ƒ j ƒ  H and †  \ ƒ g 1;1 1;1 0 H \ ƒ consists of the unique line , we conclude that D H 2  D  2;2 1;1 0 0 Df j p 2 ƒ g and † as well. On the other hand, ƒ Df ƒ j ƒ  H † g are disjoint, 2 1;1 0 p ... H since , so that   D 0: 1;1 2 Finally, 0 0 0 † H g Df ƒ j ƒ \ L ¤ ¿ and p † 2 ƒ  : \ 1 2;1 0 will intersect H in one point Since q , and any line ƒ satisfying all the above conditions L 0 p can only be the line ;q (Figure 3.5), this intersection is again a single point. Thus   D  1 2;1 2;2 as well.

121 The Chow ring of G Section 3.3 107 .1;3/ L 0 H q 0 p 0 0 0 0 Df .p / ;H g .L/ \ p † ;q † . Figure 3.5 1 2;1 We now turn to the intersections of cycles whose codimensions sum to less than 4. 0 \ First, the intersection † † is the locus of lines ƒ meeting L and containing the 1 2 0 , which is to say the Schubert cycle † point with respect to a flag containing the p 2;1 0 0 and the plane ;L , so we have point p p :   D  2;1 1 2 0 and In a similar fashion, the intersection of † with is a cycle of the form † † 2;1 1 1;1 0 respect to a certain flag; specifically, it is the locus of lines containing the point L H \ 0 H , so that and lying in   D  : 1 1;1 2;1 2  The last and most interesting computation to be made is the product . (This is 1 such a crucial case that we will prove it twice: here and in Section 3.5.1!) The difference 0 between this case and the preceding ones is that the locus \ † † of lines meeting each 1 1 0 L and L is not a Schubert cycle. of the two general lines We will use the method of undetermined coefficients, first introduced in Section 2.1.6. 2 We have by now established that A G .1;3// is freely generated by the classes  and . 1;1  , so that we may write 2 2 D (3.2)  ̨ ˇ C 2 1;1 1 for some (unique) 2 Z . We can then determine the coefficients ̨ ˇ ˇ by and ̨ and taking the product of both sides of (3.2) with classes of complementary dimension. G .1;3// and the previous A. One way to do this is by invoking the associativity of calculations: We have 2 C ˇ  / D . ̨  ; D    D   . D  / 2 2 1 1 2 2;1 1 1;1 2;2 2 1 2  D . Similarly, from D  1 and  D  ̨ and since 0 we get 2 2;2 1;1 2 2 ˇ / D    . ̨ D  .  / D   C D  1;1 1;1 2 1;1 1;1 1 2;1 1 2;2 1 1 2 and . In sum, we have D   we see that ˇ D 1 2;2 1;1 2  D  C  ; 2 1;1 1 and this completes our description of the Chow ring A. G .1;3// .

122 3 108 Chapter 3 Introduction to Grassmannians and lines in P M M 0 0 M M 00 H 00 p L L 0 00 00 0 / \ Figure 3.6 .M/ \ † † .M / / Df L g ; † . .H .p † \ † g .M/ \ † L .M Df / 2 1 1 1;1 1 1 2 2   and It is instructive to compute   geometrically, without invoking asso- 1;1 2 1 1 ciativity as in the proof above. To determine , we used ̨ 2   : D . ̨ ̨ C  D /   ˇ 2 1;1 2 2;2 2 1 By Kleiman transversality, we have 8 9 ƒ ; ¿ ¤ L \ ˆ > < ˇ = ˇ 0 ̨ D # ƒ L ƒ ¤ ¿ and \ ˇ ˆ > : ; 00 2 p ƒ 3 0 00 L for general lines and p L a general point in P and . Any line ƒ satisfying the 00 00 0 p three conditions must lie in each of the planes ;L and , and so must be their p ;L D 1 . intersection; thus ̨ Similarly, to determine ˇ we used 2 :   ˇ D . ̨ D C ˇ  /   1;1 1;1 2;2 1;1 2 1 Again, by generic transversality, we get: 8 9 ¿ ¤ L ; ƒ \ ˆ > ˇ = < ˇ 0 ƒ # D ˇ \ L ƒ ¤ ¿ and ˇ ˆ > ; : 00 ƒ H  3 0 00 general lines and for H L a general plane in P and . The only line ƒ satisfying L 00 0 00 H these conditions is the line joining the points and L L \ H \ , so again ˇ D 1 (Figure 3.6). Tangent spaces to Schubert cycles 0 0 , guaranteed by Kleiman’s and † The generic transversality of the cycles † ;b a a;b theorem in characteristic 0, played an essential role in the computation above. By

123 The Chow ring of .1;3/ 109 G Section 3.3 describing the tangent spaces to the Schubert cycles, we can prove this transversality p directly and hence extend the results to characteristic . 0 † † . Tangent spaces to other We will carry this out here for the intersection \ 2 2 are described in Exercises 3.26 and 3.27; they will be Schubert cycles in G .1;3/ treated in general in Theorem 4.1. The key identification is given in the following result: 3 D † Proposition 3.11. .p/ Let P † D P V that be the Schubert cycle of lines in 2 z L 2 † contain .p/ p , and suppose that for the two-dimensional . Writing  V L 2 z z L;V= T with Hom . , and identifying .1;3/ L L/ , we have G subspace corresponding to L † Df ' j '. T p/ D 0 g : Q L 3 0 the cycles p;p 2 Given Proposition 3.11, it follows immediately that for general P 0 0 0 † of .p and / meet transversely: If p ¤ p .p/ , then at the unique point L D p;p † 2 2 0 0 † D .p/ and † † intersection of the Schubert cycles † , we have .p D / 2 2 0 0 † \ T g † Q Df ' j '. T p/ D '. Q p ; / D 0 gDf 0 ŒLç ŒLç 0 z Q p p span Q L . since and z L Ä  V complementary to and We choose a subspace Proof of Proposition 3.11: identify the open subset g Df ƒ 2 G U j ƒ \ Ä D 0 .1;3/ Ä z as the graph of Hom L;Ä/ by thinking of a 2-plane ƒ 2 U with the vector space . Ä z a linear map from L to Ä , just as in the beginning of the proof of Theorem 3.5. It z U L;Ä/ \ † Hom is the linear space in is immediate from the identification that . 2 Ä . Thus its tangent space at a point such that p  Ker ' Q ŒLç has the consisting of maps .'/ same description. As a consequence of Theorem 3.10, we have the following description of the Chow G .1;3/ : ring of ; Z Œ ç 1 2 D G A. .1;3// Corollary 3.12. : 3 2 2 2 /  . ;    2 2 1 2 1 1 We will generalize this to the Chow ring of any Grassmannian, and prove it by applying the theory of Chern classes, in Chapter 5. A point to note is that the given presentation of the Chow ring has the same number of generators as relations — that is, given that the Chow ring A. G .1;3// has Krull dimension 0, it is a complete intersection . The analogous statement is true for all Grassmannians.

124 3 110 Chapter 3 Introduction to Grassmannians and lines in P x L 1 M x L 2 L 3 L , the lines and x Figure 3.7 An apparent double point: when viewed from appear L 3 2 x , and therefore there is a unique line through to cross at a point in the direction M x L and L meeting . 2 3 3 3.4 Lines and curves in P In this section and the next we present several applications of the computations above. 3.4.1 How many lines meet four general lines? This is Keynote Question (a). Since is the class of the locus † .L/ of lines  1 1 L , and generic translates of meeting a given line .L/ are generically transverse, the † 1 number is 4 deg D 2:  1 We can see the geometry behind this computation — and answer more refined L ;:::;L questions about the situation — as follows. Suppose that the lines are , and 4 1 if consider first the lines that meet just the first three. To begin with, we claim that 3 L is any point, there is a unique line and meeting x  P 2 passing through x M 1 x x L L to a , as in Figure 3.7. To see this, note that if we project L and and L from 3 2 3 2 H , we get two general lines in plane , and these lines meet in a unique point y . The H 3 M . WD x;y is then the unique line in P L containing x and meeting L line and 2 3 x L L and (Informally: If we look at , we see an “apparent , sighting from the point x 3 2 0 M then the — see Figure 3.7.) Moreover, if x ¤ x crossing” in the direction of the line x 0 M ;M are disjoint: If they had a common point, they would lie in a plane, and lines x x would be coplanar, contradicting our hypothesis of generality. L ;L ;L all three of 2 1 3

125 3 Lines and curves in Section 3.4 111 P Q L 4 p L 1 q L 2 L 3 ;:::;L Figure 3.8 Two lines that meet each of L . 4 1 M The union of the lines is a surface that we can easily identify. There is a three- x 1 P . Each restriction map dimensional family of quadratic polynomials on each Š L i   0 0 O ! H H O . Since there is  is linear, so its kernel has codimension 3 .2/ .2/ 3 L i P 3 a -dimensional vector space of quadratic polynomials on P , there is thus at least one 10 ́ containing L . By B ;L quadric surface and L Q ezout’s theorem, any line meeting 3 1 2 at least three times, must be contained in L ;L . ;L Q each of Q , and thus meeting 1 3 2 Since the union of the lines M is a nondegenerate surface (the lines are pairwise disjoint, x and so cannot lie in a plane), it follows that Q is unique, and is equal to the disjoint union a M D Q: x L 2 x 1 Since the degree of L Q is general, L is 2, and meets Q in two distinct points p 4 4 q ; the two lines M are the unique lines meeting and and q passing through p and M q p L (Figure 3.8). Thus we see again that the answer to our question is 2. ;:::;L all of 4 1 For which sets of four lines are there more or fewer than two distinct lines meeting all four? The geometric construction above will enable the reader to answer this question; see Exercise 3.29. 3.4.2 Lines meeting a curve of degree d We do not know a geometric argument such as the above one for four lines that would enable us to answer the corresponding question for four curves, Keynote Question (b). In this case, intersection theory is essential. The basic computation is the following: 3 Let C  Proposition 3.13. be a curve of degree d . If P Ä g WDf L 2 G .1;3/ j L \ C ¤ ¿ C C , then the class of Ä is is the locus of lines meeting C 1 G ç .1;3//: d   . 2 A ŒÄ D 1 C

126 3 112 P Chapter 3 Introduction to Grassmannians and lines in To see that Ä is a divisor, consider the incidence correspondence Proof: C Df C  G .1;3/ j p 2 L g ; .p;L/ 2 † under the projection to is whose image in . The fibers of † Ä C are all projective .1;3/ G C has pure dimension 3 . On the other hand, the projection to planes, so † is generically Ä C one-to-one, so also has pure dimension 3. (See Exercise 3.20 for a generalization.) Ä C 1 .1;3// Ä in A be the class of Now let G , and write . C C D ̨   1 C ̨ 2 Z for some ̨ , we intersect both sides with the class  and get . To determine 2;1   deg D ̨ deg . ̨:   D / 2;1 1 2;1 C 3 3 p 2 P containing and a plane H  P If .p;H/ is a general pair consisting of a point , p the Schubert cycle L .p;H/ L j p 2 Df  H g † 2;1 Ä will intersect the cycle transversely. (This follows from Kleiman’s theorem in C characteristic 0, and can be proven in all characteristics by using the description of † .p;H/ the tangent spaces to Ä in in Exercise 3.27 and of the tangent spaces to 1 C Exercise 3.30.) Therefore ̨ D # .Ä : \ † g .p;H// D # f L j p 2 L  H and L \ C ¤ ¿ 2;1 C H (being general) will intersect transversely in d To evaluate this number, note that C 2 q ;:::;q points f ; since p will be collinear H is general, no two of the points q g i 1 d p . Thus the intersection Ä \ with † , as in .p;H/ will consist of the d lines p;q i 2;1 C ̨ d , so Figure 3.9. It follows that D d D   : 1 C We will revisit Proposition 3.13 in Section 3.5.3, where we will see how to calculate . by the method of specialization C 3 ;:::;C C  P Proposition 3.13 makes it easy to answer Keynote Question (b): If 4 1 Ä are general translates of curves of degrees are generically , then the cycles d ;:::;d 4 1 C i transverse by Kleiman transversality, so the number of lines meeting all four is 4 4 4 Y Y Y : /  .d D ŒÄ d 2 deg deg D ç i 1 i C i 1 i 1 i D D 1 D i One can verify the necessary transversality by using our description of the tangent spaces, too; as a bonus, we can see exactly when transversality fails. This is the content of Exercises 3.30–3.33.

127 3 Lines and curves in 113 P Section 3.4 C q 3 H p q 2 q 1 with Figure 3.9 The intersection of † . Ä .p;H/ 2;1 C 3.4.3 Chords to a space curve n C  P of degree d and Consider now a smooth, nondegenerate space curve g . We define the locus ‰ , or .C/  G .1;n/ of chords genus secant lines to C , to be 2 q of the locus of lines of the form p;q with p and .1;n/ distinct points the closure in G - G .1;n/ W .C/ is the image of the rational map  of C  C C . Inasmuch as ‰ 2 ‰ to p;q , we see that .p;q/ sending .C/ will have dimension 2. 2 n ‰ such .C/ as the locus of lines L  P Note that we could also characterize 2 that the scheme-theoretic intersection \ C has degree at least 2. As we will see in L Exercise 3.38, this characterization differs from the definition given when we consider singular curves, or (as we will see in Exercise 3.39) higher-dimensional secant planes to n P we will show in Exercise 3.37 they agree, and we curves; but for smooth curves in can adopt either one. (For much more about secant planes to curves in general, see the discussion in Section 10.3.) n D 3 of smooth, nondegenerate space Let us now restrict ourselves to the case 3 2 G C , and ask: What is the class, in A P . curves .1;3// , of the locus ‰ of .C/  2 secant lines to C ? We can answer this question by intersecting with Schubert cycles of complementary codimension (in this case, codimension 2). We know that .C/ç D ̨ C ˇ Œ‰ 2 1;1 2 3 ̨ and ˇ . To find the coefficient ˇ we take a general plane H  P for some integers , and consider the Schubert cycle : g .H/ Df L 2 † .1;3/ j L  H G 1;1

128 3 114 Introduction to Grassmannians and lines in Chapter 3 P C H  deg C lines. Figure 3.10 .H/ \ ‰ † .C/ consists of 1;1 2 2 By our calculation of A. G .1;3// and Kleiman transversality, we have  .H/   Œ‰ D .C/ç deg D # .† .C//: ‰ \ ˇ 1;1 2 1;1 2 The cardinality of this intersection is easy to determine: The plane H will intersect C d points p , no three of which will be collinear (Arbarello et al. [1985, Sec- ;:::;p in 1 d  d lines p tion 3.1]), so that there will be exactly ;p joining these points pairwise; thus j i 2   d ˇ D 2 (see Figure 3.10). 3 p Similarly, to find P ̨ be a general point and we let 2 .p/ † Df L 2 G .1;3/ j p 2 L gI 2 we have as before  .C//:  D  Œ‰ ̨ deg D # .† .p/ \ ‰ .C/ç 2 2 2 2 To count this intersection — that is, the number of chords to C — through the point p 2 2 C ! P  . This map is birational onto its image W  P C , consider the projection p which will be a curve having only nodes as singularities (see Exercise 3.34), and these C through p . (These chords were classically nodes correspond exactly to the chords to apparent nodes C (Figure 3.11): If you were looking at C with your eye of called the p , and had no depth perception, they are the nodes you would see.) By the at the point genus formula for singular curves (Section 2.4.6), this number is   d 1 g: ̨ D 2 Thus we have proven: 3 C  P g If is a smooth nondegenerate curve of degree d and genus Proposition 3.14. , C then the class of the locus of chords to C is      d 1 d 2 D  . C .C/ç .1;3//: g Œ‰  A 2 G 1;1 2 2 2 2

129 Specialization Section 3.4 115 p q C r .C/  p Figure 3.11 Another apparent node. We can use this to answer the third of the keynote questions of this chapter: If C 0 are general twisted cubic curves, by Kleiman’s theorem the cycles C D ‰ .C/ and S 2 0 0 D ‰ , we have and S .C intersect transversely; since the class of each is  3 C / 2 2 1;1 0 2 C D . deg 3 10: / / D S \ .S # 1;1 2 Exercises 3.40 and 3.41 explain how to use the tangent space to the Grassmannian to prove generic transversality, and thus verify this result, in all characteristics. 3.5 Specialization There is a another powerful approach to evaluating the intersection products of inter- esting subvarieties: . In this section we will discuss some of its variations. specialization 3.5.1 Schubert calculus by static specialization 2 2 As a first illustration we show how to compute the class  A. G .1;3// by 1 specialization. The reader will find a far-reaching generalization to the Chow rings of Grassmannians and even to more general flag varieties in the algorithms of Vakil [2006a] and Cos ̧kun [2009]. 0 .L † and † / .L/ The idea is that instead of intersecting two general cycles 1 1 0 0 . The goal is to choose , we choose a special pair of lines L;L representing  L and L 1 0 † is readily identifiable, .L/ \ special enough that the class of the intersection / .L † 1 1 but at the same time not so special that the intersection fails to be generically transverse.

130 3 116 Introduction to Grassmannians and lines in Chapter 3 P 0 and to be distinct but incident. The intersection L L We do this by choosing 0 0 † .L \ / is easy to describe: If p D L \ L .L/ is the point of intersection of the † 1 1 0 0 and ƒ meeting L D L L;L either passes lines and the plane they span, then a line H 0 H (since it then meets L and through p in distinct points). Thus, as sets, or lies in L we have 0 0 † † .L .L/ / Df ƒ j ƒ \ L ¤ ¿ and ƒ g L \ ¤ ¿ \ 1 1 ƒ j 2 ƒ Df p  H g ƒ or † [ .p/ D † .H/: 1;1 2 If we now show that the intersection is generically transverse, we get the desired 2 D   C . To check this transversality, we can use the description of formula  1;1 2 1 0 .L/ and † is .L the tangent spaces to / † given in Exercise 3.26. First, suppose ƒ 1 1 0 .p/ † .L/ \ a general point of the component of .L † / , that is, a general line † 1 2 1 0 0 K D ƒ;L and K through D ƒ;L p be the planes spanned by ƒ together ; we will let 0 G L . Viewing the tangent space T and . L .1;3// as the vector space of linear with ƒ z z ! ƒ maps V= ' ƒ , we have W ̊ ̊ 0 0 z z z z Q p/  : K= .† ƒ K and T ƒ .† .L// .L D // D T ' j '. Q p/  ' j '. = 1 1 ƒ ƒ 0 ƒ are distinct, they intersect in and , so that the intersection is K Since K 0 g .L// \ T T .† 0 .L .† // Df ' j '. Q p/ D 1 1 ƒ ƒ 0 Œƒç .L/ \ † . .L Since this is two-dimensional, the intersection / is transverse at † 1 1 0 † Similarly, if .H/ of † , .L/ ƒ † / .L is a general point of the component \ 1 1 1;1 0 0 L and L so that in distinct points q and q ƒ , we have meets ̊ ̊ 0 0 Q z z z z q/  ; H= .† ƒ D and T ƒ .† .L// .L D // T ' ' j '. j q '. /  Q H= 1 1 ƒ ƒ so ̊ 0 z z ƒ/ T .† H .L .† // D : ' j .L// \ T  '. 1 1 ƒ ƒ 0 .L .L/ \ Again this is two-dimensional and we conclude that . Œƒç † / is transverse at † 1 1 2  Before going on, we mention that the computation of given here is an example 1 static specialization of the simplest kind of specialization argument, what we may call : We are able to find cycles representing the two given classes that are special enough that the class of the intersection is readily identifiable, but general enough that they still intersect properly. In general, we may not be able to find such cycles. Such situations call for a more dynamic specialization . There, we powerful and broadly applicable technique, called .A consider a one-parameter family of pairs of cycles ;B specializing from a “general” / t t pair to a special pair .A ;B / , which may not intersect dimensionally transversely 0 0 at all! The key idea is to ask not for the intersection \ B of the limiting cycles, A 0 0 but rather for the limit lim of their intersections. For an example of / B .A \ t 0 t ! t

131 Specialization Section 3.5 117 dynamic specialization, see Section 4.4 of the following chapter, where we consider 4 of lines in P in the Grassmannian the self-intersection of the cycle of lines G .1;4/ 4 meeting a given line in . P 3.5.2 Dynamic projection Problems situated in projective space tend to be especially amenable to specialization n P techniques: We can use the large automorphism group of to morph the objects we are dealing with into potentially simpler, more tractable ones. One fundamental example , which we will describe here and use in dynamic projection of this is the technique of the following section to re-derive the formulas for the class of the locus of lines incident to a curve. n (the “repellor”) that span P . Fix two disjoint planes A (the “attractor”) and R n A Choose coordinates ;:::;y on P ;:::;x so that the equations of ;y are f x D 0 g x a r 0 i 0 R f y and the equations of D are g , and consider the action ‰ of the multiplicative 0 i n given by P G group on m .tx ;:::;tx ;y ;:::;y / 7! ;:::;x .x ;y ;:::;y / W t r a 0 r a 0 0 0 1 1 ;t D .x y /: ;:::;x ;:::;t y 0 a 0 r .x ;y ;:::;y ;:::;x / to .x;y/ ; for example, we (In what follows, we will abbreviate 0 0 r a remain fixed under will write D .tx;y/ .) It is clear that the points of A and R .x;y/ t the action of G . On the other hand, we can say intuitively that a point not in A or R m will “flow toward A ” as t approaches zero, and will “flow toward R ” as t approaches 1 . More precisely, note that any point ... A [ R lies on a unique line that meets both p R and p;A , being an .a C 1/ -plane, must meet . (The span . Since A and R are A R p;A can meet R only in a point q 2 R ; the line p;q is then the unique line disjoint, p and meeting A and R .) This line is the closure of the orbit of p under the containing n G . In particular, any point in P R n given action of has a well-defined limit in A as m t approaches zero. n X  Now suppose is any variety. We consider the images of X under the P automorphisms . In other words, we set , and in particular their flat limit as t ! 0 t 1 n ı 2 A ¤  P Df j t .t;p/ 0 and p 2 Z .X/ gI t n 1 ı A  P Z be the closure of Z we let , and look at the fiber X  of Z over t D 0 . 0 (Note that even if X is a variety, X as the may well be nonreduced.) We think of X 0 0 limit of the varieties approaches 0 (see Figure 3.12 for an illustration; see .X/ as t t also Eisenbud and Harris [2000, Chapter 2] for a discussion of flat limits in general). The following properties of the limit X make it easy to analyze some interesting 0 cases:

132 3 118 Chapter 3 Introduction to Grassmannians and lines in P C R y a x A D R;y D R;z R;x 3 P to a line A . from a line Figure 3.12 Dynamic projection of a conic in R With notation as above: Proposition 3.15. n  P (a) is stable under the action of X G . m 0 X R D X \ R . \ (b) 0 (in case / X (c) \ R with base A .X X \ R D ¿ , is contained in the cone over red 0 0 0 .X we take this to mean /  A ). 0 red In addition, we know (as for any rational equivalence) that is equidimensional and X 0 X dim D dim X , and, by Eisenbud [1995, Exercise 6.11], that the Hilbert polynomial 0 X of is the same as the Hilbert polynomial of X . 0 1 n  on the product A G For the first part, consider the action of P Proof: given as the m 1 n ; that is, on A product of the standard action of and the action ‰ above of G G on P m m W ! .ts; .p//: ' .s;p/ t t n to itself, so it carries Z g P to itself and the fiber f X This carries to itself. But it 0 0 n 0 g P above; thus via the action ‰ acts on the fiber f . is invariant under ‰ X 0 The second point is more subtle. (In particular, it is asymmetric: The same statement, with replaced by A , would be false.) It is not, however, intuitively unreasonable: Since R n flow away from n R points in P as t ! 0 , the only way a point p 2 R can be a limit R of points . .p R / is if it is there all along, that is, if p 2 X \ t t R In any case, note first that one inclusion is immediate: Since is fixed pointwise by the automorphisms , we have t 1  .X \ R/  Z P and hence X \ R  X . To see the other inclusion, we want to show that the ideal \ R 0 2 \ R/ is contained in I.X . We can then write \ R/ . Let f.x/ I.X I.X \ R/ 0 f.x/ D g.x;0/ for some g.x;y/ 2 I.X/:

133 Specialization Section 3.5 119 3 q q c p p H H 2 p 2 p 1 1 p p 3 3 Figure 3.13 A space curve specializes to a union of lines. C Now observe that f.x;ty/ j f 2 I.X/ g ; I.Z/ f , we see that D 2 I.Z/ . Setting t D 0 g.x;ty/ g.x;0/ 2 I.X h.t;x;y/ / , and hence so 0 2 I.X . \ f R/ 0 G To prove the third assertion, note that the -orbit of any point not contained in m A is stable under R to a point of A . Since X R [ is a straight line joining a point of 0 it contains. A [ R , it is the union of such lines, together with any points of G m dynamic projection : We are realizing the We sometimes call this construction projection map n  P n R ! A; W R 7! .0;y/; .x;y/ n P as the limit of a family of automorphisms . As we will see, though, considering of t .X/ yields more information than simply taking the projection the limit of the images t  . See Figures 3.12 and 3.13 for examples. .X/ R n  P Example 3.16. be a subvariety of dimension m and degree d X Let . We will ex- X whose limit is a d -fold m -plane (that is, a scheme whose hibit a dynamic projection of m -plane and that has multiplicity d at the general point of that plane), and support is an d another whose limit is the generically reduced union of -planes containing a distinct m .m 1/ -plane. fixed n  P To make the first construction, let be any m -dimensional subspace, and A n R to be an .n m 1/ -plane R  P choose disjoint from X and from A . Since  X D ¿ , we see that X \ \ R D ¿ as well, and it follows that .X . Since / A R red 0 0 X , and computing D dim X dim dim A , we see that the support of X is exactly A D 0 0 the degree we have h X iD d h A i , as claimed. 0

134 3 120 Chapter 3 Introduction to Grassmannians and lines in P to be a general To make the second construction, choose the repellor subspace R m , so that R \ X consists of deg X distinct points, and take A to n plane of dimension / 1/ R . We see that .X .m -plane disjoint from is contained in the union of the be an red 0 deg X points of X \ R D X m -planes that are the cones over the R . Also, X must \ 0 0 X X , it is equidimensional and has degree equal to contain all these points. Since deg 0 X follows that is the generically reduced union of these distinct planes, as required. 0 Note that while the multiplicities of X are determined in both cases, the actual 0 scheme structure of will depend very much on the geometry of X . X 0 3.5.3 Lines meeting a curve by specialization As an example of how dynamic projection can be used in specialization arguments, of the locus we revisit the computation of the class  G .1;3/ of lines meeting a Ä C C 3 C curve from Proposition 3.13.  P 3 3  be a curve of degree d . Choose a plane H  P intersecting C C Let P 3 P transversely in points , and q 2 ;:::;p p any point not lying on H . Consider the 1 d f A one-parameter group g PGL ; that is, choose with repellor plane H and attractor q 4 t 3 D ;:::;Z , ç on P ŒZ such that q coordinates Œ1;0;0;0ç and H is given by Z 0 D 0 3 0 3 t ¤ 0 the automorphisms of P and consider for given by 0 1 1 0 0 0 B C 0 t 0 0 B C D A : B C t 0 0 t 0 @ A 0 0 0 t 3 1  D .C/ , and let ˆ  A A Let P C be the closure of the locus t t ı 1 3 2 ˆ A  Df .t;p/ j t ¤ 0 and p 2 C : g P t As we saw in the preceding section, the limit of the curves C as t ! 0 (that is, the fiber t of ˆ over t D 0 ) is supported on the union of the d lines p , and has multiplicity 1 at ;q i a general point of each, as shown in Figure 3.13. Ä and We can use this construction to give a rational equivalence between the cycle C 1 .1;3/ . p G ;q/ in G the sum of the Schubert cycles . Explicitly, take ‰  A †  .1;3/ i 1 to be the closure of the locus 1 .t;ƒ/ 2 A 0  G .1;3/ j t ¤ f and ƒ \ C : ¤ ¿ g t As we will verify in Exercises 3.35 and 3.36, the fiber is supported on of ‰ ‰ t D 0 over 0 the union of the Schubert cycles † and has multiplicity 1 along each, establishing . p ;q/ i 1 the rational equivalence  . d  D 1 C

135 Specialization Section 3.5 121 over ˆ 0 (that is, the flat limit lim t The fiber of D C not of the curves C ) is 0 t ! t t p necessarily equal to the union of the ;q : it may have an embedded point at the lines d i ‰ , being a divisor in G .1;3/ , will not have embedded point . Nonetheless, the fiber q 0 components. 3.5.4 Chords via specialization: multiplicity problems One of the main difficulties in using specialization is the appearance of multiplicities. We will now illustrate this problem by trying to compute, via specialization, the class of 3 P the chords to a smooth curve in . C WD A Consider again the family of curves .C/ described in the previous section. t t What is the limit as 0 of the cycles ‰ G .C ? To interpret /  ! .1;3/ of chords to C t t t 2 1 .1;3/   this question, let A be the closure of the locus ... G ı 1 .t;ƒ/ 2 ... A  G .1;3/ j t ¤ 0 and ƒ 2 ‰ Df .C : / g t 2 What is the fiber of this family? ... 0 is easy to identify. It is contained in the locus of lines whose ... The support of 0 D C lim intersection with the flat limit contains a scheme of degree at least 2, C t 0 t 0 ! which is to say the union of the Schubert cycles . p of lines lying in a plane ;p ;q/ † j i 1;1 p spanned by a pair of the lines , and the Schubert cycle † .q/ of lines containing the ;q 2 i ;p q . Moreover one can show that the Schubert cycles . p all appear with † ;q/ point j i 1;1 ... multiplicity 1 in the limiting cycle , from which we can deduce that the coefficient of 0  d  in the class of ‰ .C/ is . 1;1 2 2 The hard part is determining the multiplicity with which the cycle † .q/ appears 2 in : This will depend in part on the multiplicity of the embedded point of C ... at q , 0 0 which will in turn depend on the genus of C (see for example Exercises 3.43 and 3.44). g Note the contrast with the calculation in Section 3.5.3 of the class of the locus Ä of C incident lines via specialization: There, the embedded component of the limit scheme also depended on the genus of lim Ä , but did not affect the limiting cycle. C ! 0 t C t An alternative approach to this problem would be to use a different specialization to  capture the coefficient of : Specifically, we could take the one-parameter subgroup 2 3 q  P . The limiting H with repellor a general point and attractor a general plane  d 1 C scheme C will be a plane curve of degree d with ı D nodes D lim g 0 ! t 0 t 2 r , with a spatial embedded point of multiplicity 1 at each node. The limit of ;:::;r 1 ı ‰ .C /  G .1;3/ will correspondingly be supported on the the corresponding cycles t 2 † . In this case .H/ and the union of the Schubert cycle Schubert cycles † / .r ı i 2 1;1 the coefficient of the Schubert cycle † .H/ is the mysterious one (though calculable: 1;1 given that a general line ƒ  H meets C points, we can show that it is the limit of in d 0  d chords to C as t ! 0 ). On the other hand, one can show that the Schubert cycles t 2 † , from which we .r / / all appear with multiplicity 1 in the limit of the cycles ‰ .C 2 i 2 t can read off the coefficient ı of . .C/ in the class of ‰  2 2

136 3 122 P Chapter 3 Introduction to Grassmannians and lines in We will fill in some of the details involved in this calculation in Exercise 3.45. 3.5.5 Common chords to twisted cubics via special- ization To illustrate the artfulness possible in specialization arguments, we give a different specialization approach to counting the common chords of two twisted cubics: We will not degenerate the twisted cubics; we will just specialize them to a general pair of twisted 0 lying on the same smooth quadric surface .1;2/ and cubic curves , of types C;C Q .2;1/ respectively. 0 and C Q (the The point is, no line of either ruling of C will be a chord of both 0 C , and vice versa for lines of the other C lines of one ruling are chords of but not of 0 0 C ruling). But since Q , any line meeting C [ C [ in three or more distinct C  0 . It follows that the only common chords to C and C will be the points must lie in Q 0 \ C lines joining the points of intersection pairwise C ; since the number of such points  5 0 0 D deg .ŒCçŒC \ ç/ D 5 , the number of common chords will be C is # .C / D 10 . 2 Of course, to deduce the general formula from this analysis, we have to check that the 0 .C/ \ ‰ ‰ .C / is transverse; we will leave this as Exercise 3.46. intersection 2 2 0 C C to twisted cubics lying on What would happen if we specialized Q , both and  4 0 6 D , giving rise to ? Now there would only be four points of C having type C .1;2/ \ 2 common chords. But now the lines of one ruling of Q would all be common chords to 0 both. Thus .C/ \ ‰ would have a positive-dimensional component: Explicitly, .C ‰ / 2 2 1 0 ‰ ‰ .C / .C/ would consist of six isolated points and one copy of P \ . It might seem 2 2 that in these circumstances we could not deduce anything about the intersection number 0 from the actual intersection, but in fact the excess intersection .C/ç  Œ‰ /ç/ deg .Œ‰ .C 2 2 0 /ç/ ; .C/ç  Œ‰ .Œ‰ formula of Chapter 13 can be used in this case to determine deg .C 2 2 see Exercise 13.35. 3 3.6 Lines and surfaces in P 3.6.1 Lines lying on a quadric 3 Q  P F be a smooth quadric surface and Let D F the locus of .Q/  G .1;3/ 1 lines contained in ŒFç 2 A. G .1;3// . ( . In this section we will determine the class is Q F Fano scheme , a construction that will be treated extensively in Chapter 6.) an example of a 1 1 P Via the isomorphism  P Q , the lines on Q are fibers of the two projections Š 1 3 ! P maps ; in particular, we see that dim F D 1 . Since A Q . G .1;3// is generated by  , we must have 2;1 ŒFç D ̨   2;1

137 3 Lines and surfaces in Section 3.6 123 P 3 (1,3) C 1 Q “ D ” ”=” C 2 3 Figure 3.14 The rulings of a quadric surface correspond to conic curves Q  P 5  .1;3/  P C ; thus ŒF . .Q/ç D 4 G 2;1 1 i 3 . If L  P .L/ is a general line and † the Schubert cycle for some integer  G ̨ .1;3/ 1 L , then by Kleiman transversality we have of lines meeting deg   / ̨ .ŒFç D 1 .† D .L/ \ F/ # 1 # f M D G .1;3/ j M  Q and M \ L ¤ ¿ g : 2 Now L , being general, will intersect Q in two points, and through each of these points there will be two lines contained in Q ; thus we have ̨ D 4 and ŒFç D 4 : 2;1 We will see how to calculate the class of the locus of linear spaces on a quadric hyper- surface more generally in Section 4.6. is actually the union [ C of two disjoint curves in the Grass- The variety C F 1 2 ; each of these curves has class 2 mannian, corresponding to the two rulings of , Q 2;1 5 ̈ and thus has degree 2 as a curve in the Pl P ucker embedding in (see Figure 3.14). For details see Eisenbud and Harris [2000]. 3.6.2 Tangent lines to a surface 3 Next, let  P be any smooth surface of degree d , and consider the locus S .S/ T  G .1;3/ of lines tangent to S . Let ˆ be the incidence correspondence 1 ˆ Df .q;L/ 2 S  G .1;3/ j q 2 L  T ; S g q q where S denotes the projective plane tangent to S at T . The projection ˆ ! S on the q 1 ˆ as a P first factor expresses -bundle over S , from which we deduce that ˆ , and hence its image T .S/ in G .1;3/ , is irreducible of dimension 3 . 1

138 3 124 P Chapter 3 Introduction to Grassmannians and lines in p H S \ S C H D . \ T .† .S// D 2 .p;H/ Figure 3.15 deg 2;1 1 T .S/ , we write To find the class of 1 ; .S/ç D ̨   ŒT 1 1 3 2 H and a general point p P H . By Kleiman and choose a general plane  transversality, D ŒT .S/ç ̨   2;1 1 # .† D \ T .S// .p;H/ 2;1 1 D # f M 2 G .1;3/ j q 2 M  T : S for some q 2 S and p 2 M  H g q 2 S in a smooth plane curve C  H Š , being general, will intersect Now of H P 2   being general in H , the line p degree  P d , and, dual to p will intersect the dual p 2    P C curve transversely in deg .C  / points. By Proposition 2.9, we have  deg / D d.d 1/ .C and hence ŒT 1/ .S/ç D d.d 1 1 (see Figure 3.15).

139 Exercises Section 3.6 125 This gives the answer to the last keynote question of this chapter: How many lines are tangent to each of four general quadric surfaces ? Once more, Kleiman’s theorem Q i T assures us that the cycles / intersect transversely, a fact we can verify in all .Q 1 i characteristics by explicit calculation. The answer is thus Y 4 .2 deg D deg .Q 32: / ŒT D /ç 1 i 1 3.7 Exercises ƒ;Ä 2 G be two points in the Grassmannian G D G.k;V / . Show Let Exercise 3.17. V k . ƒ;Ä P V / is contained in  if and only if the intersection ƒ \ Ä  V that the line G V has dimension k 1 . of the corresponding subspaces of Exercise 3.18. Using the fact that the Grassmannian V k D . G.k;V / P G  V / Œƒç G is the point corresponding to a is cut out by quadratic equations, show that if 2 V k T k G -plane P . ƒ then the tangent plane V / intersects G in the locus  Œƒç G T gI G Df Ä j dim .Ä \ ƒ/  k 1 \ Œƒç k ƒ in codimension 1. that is, the locus of -planes meeting Exercise 3.19. Let be an .n C 1/ -dimensional vector space, and consider the universal V k -plane over G D G .k; P V / introduced in Section 3.2.3: ˆ Df 2 G  P V j p 2 ƒ g : .ƒ;p/ C G V of dimension k C .k P 1/.n k/ , and  Show that this is a closed subvariety of V k C 1  P . that it is cut out on G V by bilinear forms on P  . V V / P n Use the preceding exercise to show that, if  X P is any subvariety of Exercise 3.20. l < n k , then the locus dimension Ä g Df ƒ 2 G .k;n/ j X \ ƒ ¤ ¿ X of -planes meeting X is a closed subvariety of G .k;n/ of codimension n k l k . Exercise 3.21. l < k < n , and consider the locus of nested pairs of linear subspaces Let n P of of dimensions l and k : F .l;k I n/ Df .Ä;ƒ/ 2 G .l;n/  G .k;n/ j Ä  ƒ g : .k;n/ Show that this is a closed subvariety of .l;n/  G G , and calculate its dimension. (These are examples of a further generalization of Grassmannians called flag manifolds , which we will explore further in Section 4.8.1.)

140 3 126 P Chapter 3 Introduction to Grassmannians and lines in Again let Exercise 3.22. m  l consider the locus of pairs of l < k < n , and for any n l and k intersecting in dimension at least m : P of dimensions linear subspaces of I m I n/ Df .Ä;ƒ/ 2 G .l;n/  G .k;n/ j dim .Ä F ƒ/  m g : .l;k \ G  G .k;n/ and calculate its dimension. Show that this is a closed subvariety of .l;n/ n B Let G .1;n/ be a curve in the Grassmannian of lines in P , with Exercise 3.23.  n have rank 1. Show that the lines in B the property that all nonzero tangent vectors to P B either parametrized by (a) all lie in a fixed 2-plane; (b) all pass through a fixed point; or n P (c) are all tangent to a fixed curve .  C (Note that the last possibility actually subsumes the first.) Show that an automorphism of carries tangent vectors to tangent G.k;n/ Exercise 3.24. vectors of the same rank (in the sense of Section 3.2.4), and hence for 1 < k < n the cannot act transitively on nonzero tangent vectors. group of automorphisms of G.k;n/ does act transitively Show, on the other hand, that the group of automorphisms of G.k;n/ on tangent vectors of a given rank. ı Exercise 3.25. In Example 3.9, we demonstrated that the open Schubert cell † D 1 3 n .† [ † / is isomorphic to the affine space A . For each of the remaining † 1 2 1;1 ı a;b † Schubert indices , show that the Schubert cell  G .1;3/ is isomorphic to the a;b 4 a b . affine space of dimension Exercise 3.26. Consider the Schubert cycle † : Df ƒ 2 G .1;3/ j ƒ \ L ¤ ¿ g 1 Suppose that 2 † ƒ and ƒ ¤ L , so that ƒ \ L is a point q and the span ƒ;L a 1 plane ƒ is a smooth point of † K , and that its tangent space is . Show that 1 ̊ z z z z ' 2 Hom . .† ƒ;V= : ƒ/ j '. Q q/  / K= D ƒ T 1 ƒ Exercise 3.27. Consider the Schubert cycle † : D † g .p;H/ Df ƒ 2 G .1;3/ j p 2 ƒ  H 2;1 2;1 † is smooth, and that its tangent space at a point ƒ is Show that 2;1 ̊ z z z z ƒ  .'/ : 0 H= ƒ;V= and Im ƒ/ j '. Q p/ D .† . ' D / T 2 Hom 2:1 ƒ Exercise 3.28. Use the preceding two exercises to show in arbitrary characteristic that general Schubert cycles † intersect transversely, and deduce the ;† .1;3/  G 2;1 1 equality deg . 1  . D /  2;1 1

141 Exercises Section 3.7 127 C L preserving incidence with a curve Figure 3.16 Deformation of a line C L . 3 3 Exercise 3.29.  P L be four pairwise skew lines and ƒ ;:::;L P  a line Let 4 1 meeting all four; set D ƒ \ L p and H : D ƒ;L i i i i Show that Œƒç 2 G fails to be a transverse point of intersection of the Schubert cycles equals the .L † / exactly when the cross-ratio of the four points p ƒ ;:::;p 2 4 i 1 1 H cross-ratio of the four planes ;:::;H . in the pencil of planes containing ƒ 4 1 3 ;:::;C C  P Exercises 3.30–3.33 deal with a question raised in Section 3.4.2: If 4 1 3 , do the corresponding cycles Ä are general translates of four curves in P  G .1;3/ C i C of lines meeting the intersect transversely? i To start with, we have to identify the smooth locus of the cycle Ä  G .1;3/ of C lines meeting a given curve , and its tangent spaces at these points; this is the content C of the next exercise, which is a direct generalization of Exercise 3.26 above. 3 3 P  Let be any curve, and L  P Exercise 3.30. a line meeting C at one smooth C point p of C and not tangent to C . Show that the cycle Ä of lines meeting  G .1;3/ C is smooth at the point , and that its tangent space at ŒLç is the space of linear C ŒLç 4 z z z = k L carrying the one-dimensional subspace Q !  L L to the one-dimensional maps p 4 z z z z . C C L of k T = L/= L (see Figure 3.16). subspace p Next, we have to verify that, for general translates C of any four curves, the i Ä corresponding cycles are smooth at each of the points of their intersection. A key C i fact will be the irreducibility of the relevant incidence correspondence: 3 B ;:::;B 2 Exercise 3.31. P Let be four irreducible curves and let ' ;:::;'  1 1 4 4 3 ' P PGL ; let C . Show that the incidence D be four general automorphisms of / .B i i i 4 correspondence 4 Df .' g ;:::;' i ;L/ 2 . PGL for all / ˆ  G .1;3/ j L \ ' ¿ .B ¤ / i i 4 4 1 is irreducible.

142 3 128 Introduction to Grassmannians and lines in Chapter 3 P Using this, we can prove the following exercise — asserting that for general trans- C L meeting all four, the cycles Ä lates of four given curves and any line are smooth at i C i ;L/ ;:::;' ŒLç — simply by exhibiting a single collection satisfying the conditions .' 4 1 in question: 3  ;:::;B Exercise 3.32.  P Let be four curves and ' PGL ;:::;' B four 4 4 4 1 1 3 3 C general automorphisms of D ' ; let .B P / . Show that the set of lines L  P i i i C , C , , C meeting and C L is finite, and that, for any such 4 2 1 3 L meets each C ; at only one point p (a) i i ; and is a smooth point of C p (b) i i C for any (c) i . L is not tangent to i 3 3 Exercise 3.33. Let C P  be any four curves, and L  P ;:::;C a line meeting all 4 1 four and satisfying the conclusions of Exercise 3.32. Use the result of Exercise 3.30 to give a necessary and sufficient condition for the four cycles G .1;3/ to intersect Ä  C i ŒLç , and show directly that this condition is satisfied for all lines meeting transversely at C when the C are general translates of given curves. ;:::;C 1 i 4 3 3 Exercise 3.34. 2 be a smooth curve and p  P Let a general point. Show that C P p does not lie on any tangent line to C ; (a) p C ; and (b) does not lie on any trisecant line to does not lie on any stationary secant such C (that is, a secant line q;r to C to p (c) that the tangent lines T and T C C meet). q r 2 Deduce from these facts that the projection C ! P W is birational onto a plane  p 2 curve  P having only nodes as singularities. (Note that as a consequence the same C 0 n 2 P is true for the projection of a smooth curve from a general .n 3/ -plane to P C .)  Exercises 3.35 and 3.36 deal with the approach, described in Section 3.5.3, to calculating the class of the variety  G .1;3/ of lines incident to a space curve † C 3 P C by specialization. Recall from that section that we choose a general plane  3 3 P be the meeting C at d points p H and a general point q 2 P  , and let f A g t i .C/ PGL with attractor q and repellor H ; we let C one-parameter subgroup of D A t t 4 1 G A ‰ and take  .1;3/ to be the closure of the locus  ı 1 Df .t;ƒ/ 2 A ‰  G .1;3/ j t ¤ 0 and ƒ \ C : ¤ ¿ g t Show that the support of the fiber ‰ is exactly the union of the Schubert Exercise 3.35. 0 cycles † . . p ;q/ i 1 Show that ‰ has multiplicity 1 at a general point of each Schubert Exercise 3.36. 0 . † ;q/ . p cycle i 1

143 Exercises Section 3.7 129 r C Let be a smooth curve. Show that the rational map ' W  Exercise 3.37. P 2 - G .1;r/ to the line p;q when sending ¤ q actually extends to a C .p;q/ p 2 sending .p;p/ to the projective tangent line regular map on all of C C . Use this T p r L  P ' such that the to show that the image of coincides with the locus of lines \ has degree at least 2. L scheme-theoretic intersection C Show by example that the conclusion of the preceding exercise is false Exercise 3.38. r  C to be smooth. Is it still true if we allow C to in general if we do not assume P have mild singularities, such as nodes? Similarly, show by example that the conclusion of Exercise 3.37 is Exercise 3.39. false if we consider higher-dimensional secant planes: For example, the image of the rational map 3 - C .2;r/; ' G W p;q;r; .p;q;r/ 7! r P whose scheme-theoretic intersection need not coincide with the locus of 2-planes ƒ  has degree at least 3. with C S D ‰ Exercise 3.40. .C/ contains the locus of lines Show that the smooth locus of 2 3 P L such that the scheme-theoretic intersection L \ C consists of two reduced  T L identify the tangent plane S as a subspace of T . G points, and for such a line L L C a smooth point of ‰ (When is a tangent line to .C/ ?) 2 3 0 C;C Exercise 3.41.  P Use the result of the preceding exercise to show that if are 0 of chords .C/;‰ .C two general twisted cubic curves, then the varieties /  G .1;3/ ‰ 2 2 0 to C and C intersect transversely. 3 C  P be a smooth, nondegenerate curve of degree Exercise 3.42. d and genus g , Let 3 L;M P be general lines.  and let Find the number of chords to C meeting both L and M by applying Proposition 3.14. (a) (b) Verify this count by considering the product morphism 1 1    W C ! P P  M L 1 M ; ) and comparing the W C ! P  are the projections from L and (where M L arithmetic and geometric genera of the image curve. 3 C  P Let be a smooth, irreducible nondegenerate curve of degree d , Exercise 3.43. 1 3  A and let  P ˆ be the family of curves specializing C to a scheme supported 3 on the union of lines joining a point 2 P to the points of a plane section of C , as p constructed in Section 3.5.3. Show that C , and that may have an embedded point at p 0 the multiplicity of this embedded point may depend on the genus of the curve C , by considering the examples of curves of degrees 4 and 5.

144 3 130 Chapter 3 Introduction to Grassmannians and lines in P ‰ Exercise 3.44. In the situation of the preceding problem, let /  G .1;3/ be .C 2 t C t ¤ the locus of chords to . Suppose that the degree of C is 4. Show that the 0 for t .p/ will be in the flat limit with multiplicity depending on the genus of C . component † 2 3  P Exercise 3.45. Again, suppose d ; choose a general C is any curve of degree 3 3 H and point p 2 P P , and consider the one-parameter group f A plane g  PGL 4 t ;:::;Z and attractor plane — that is, choose coordinates ŒZ with repellor point H p ç 3 0 3 Œ0;0;0;1ç such that p D on and H is given by Z P D 0 , and consider for t ¤ 0 the 3 3 automorphisms of P given by 0 1 1 0 0 0 B C 0 1 0 0 B C : D A C B t 0 0 1 0 A @ 0 0 0 t C ¤ D A . .C/ , and for t Let 0 let ‰ C .C be the locus of chords to /  G .1;3/ t t t t 2  d appears as a component of multiplicity † in .H/ Show that the Schubert cycle 1;1 2 . ‰ .C / the limiting scheme lim 2 ! t 0 t 1 Hint:  A Let  G .1;3/ be the closure of the family ‰ ı g j t ¤ 0 and L 2 ‰ Df .C ; / .t;L/ ‰ t 2  H is a general line then in a neighborhood of the point .0;L/ and show that if 2 L  d 1 A consists of the union of  G smooth sheets, each intersecting the the family ‰ 2 f . g G .1;3/ transversely in the Schubert cycle † 0 .H/ fiber 1;1 3 0 Exercise 3.46.  Let  P C;C be general twisted cubic curves lying on a smooth Q quadric surface Q , of types .1;2/ and .2;1/ respectively. Show that the intersection 0 ‰ \ ‰ .C / of the corresponding cycles of chords is transverse. .C/ 2 2 3 d  P Let be a smooth curve of degree Exercise 3.47. and genus g , and let C 3  G .1;3/ be the locus of its tangent lines. Find the class ŒT.C/ç 2 A . T.C/ G .1;3// of in the Grassmannian G .1;3/ . T.C/ 3 3 C  P and genus be a smooth curve of degree d Let g , and let S  P Exercise 3.48. be a general surface of degree e . How many tangent lines to C are tangent to S ?

145 Chapter 4 Grassmannians in general Keynote Questions 1 n 2n C 1 2n C P P ;:::;V V Š are four general n -planes, how many lines L P If (a)   4 1 meet all four? (Answer on page 150.) 5 P G .1;3/ Let C  be a twisted cubic curve contained in the Grassmannian (b)  5 3 P G of lines in P .1;3/ , and let  [ 3 P D ƒ  S Œƒç C 2 be the surface swept out by the lines corresponding to points of C . What is the S ? How can we describe the geometry of S ? (Answer on page 145.) degree of 4 S ;Q  P If are general quadric hypersurfaces and Q D Q (c) \ Q their surface 2 1 2 1 of intersection, how many lines does S contain? More generally, if and Q Q are 2 1 2n Q general quadric hypersurfaces in D Q \ P and , how many .n 1/ -planes X 1 2 X does contain? (Answer on page 157.) n G .1;n/ of lines in P What is the degree of the Grassmannian , embedded in (d) ̈ projective space via the Pl ucker embedding? (Answer on page 150.) In this chapter, we will extend the ideas developed in Chapter 3 by introducing Schubert cycles and classes on , the Grassmannian of k -dimensional subspaces G.k;n/ n -dimensional vector space in an , and analyzing their intersections, a subject that V goes by the name of the Schubert calculus . Of course we may also consider G.k;n/ in its projective guise as G .k 1;n 1/ , the Grassmannian of projective .k 1/ -planes in n 1 P , and in places where projective geometry is more natural (such as Sections 4.2.3 and 4.4) we will switch to the projective notation.

146 132 Chapter 4 Grassmannians in general 4.1 Schubert cells and Schubert cycles Let G.k;V / be the Grassmannian of k -dimensional subspaces of an n - G D . Generalizing the example of G D G.2;4/ , the center dimensional vector space V .1;3/ Schubert varieties called or of our study will be a collection of subvarieties of G.k;n/ V Schubert cycles V , that is, a nested , defined in terms of a chosen complete flag in sequence of subspaces V 0  V  V D  V n n 1 1 i . V D with dim i The Schubert cycles are indexed, in a way that will be motivated below, by sequences a D ;:::;a of integers with / .a 1 k k  a 0:  a  n a  1 2 k Young diagrams (Such sequences are often described by — see Section 4.5.) For such a  Schubert cycle , we define the . a / † G to be the closed subset sequence V a i  g ƒ/ \ : i for all Df dim j G 2 ƒ .V V . † / a C i n a k i Œ† . V /ç 2 A.G/ does not depend on the choice of Theorem 1.7 shows that the class a flag, since any two flags differ by an element of . In general, when dealing with a GL n property independent of the choice of V , we will shorten the name to † , and we define a  I WD Œ† A.G/ ç 2 a a Schubert classes these, naturally, are called is . We shall see (in Corollary 4.7) that A.G/  form a basis. a free abelian group and that the classes a To simplify notation, we generally suppress trailing zeros in the indices, writing r † to denote . Also, we use the shorthand in place of † † , † ;:::;a ;0;:::;0 a a p p;:::;p ;:::;a s s 1 1 r indices equal to p . with To elucidate the rather awkward-looking definition of . V / , suppose that ƒ  V † a is a k -plane. If ƒ is general, then V i \ ƒ D 0 for i  n k , while dim V D ƒ \ i C n i k dim . Thus we may describe † for i > n ƒ such that k V i \ ƒ  as the set of j a j that is a occurs for a value of steps sooner than expected. i Equivalently, we may consider the sequence of subspaces of ƒ  .V \ ƒ/  .V (4.1) \ 0  .V \ ƒ/  .V \ ƒ/ D ƒ: ƒ/ n 1 1 2 n Each subspace in this sequence is either equal to the one before it or of dimension one k times. The Schubert cycle † . V / is greater, and the latter phenomenon occurs exactly a the locus of planes ƒ for which “the i -th jump in the sequence (4.1) occurs at least a i steps early.” Here are two common special cases to bear in mind:

147 Schubert cells and Schubert cycles Section 4.1 133  k ƒ meeting a given space of dimension l nontrivially is The cycle of -subspaces the Schubert cycle † : g 0 ¤ V . V / Df ƒ j ƒ \ l 1 C k n l -dimensional subspaces meeting a given .n k/ - In particular, the Schubert cycle of k dimensional subspace nontrivially is \ / Df ƒ j ƒ † V . V : ¤ 0 g 1 k n ̈ in the Pl This is a hyperplane section of ucker embedding. (But not every hyper- G G plane section of is of this form. This follows by a dimension count: the family of k/ k/ k;n/ — has dimension k.n G.n , whereas .n -planes — the Grassmannian  n ̈ 1 .) ucker coordinates has dimension the space of linear forms in the Pl k  k -subspaces contained in a given l -subspace is the The sub-Grassmannian of Schubert cycle † ƒ : g . V / Df ƒ j V  k l .n l/ Similarly, the sub-Grassmannian of planes containing a given r -plane is the Schubert cycle r : g ƒ † . V / Df ƒ j V  r k/ .n , defined for , and the cycles  i  n k † † 0 , , defined for 0  i  k The cycles i i 1 special Schubert cycles. As we shall see in Section 5.8, their classes are are called G , and each of the intimately connected with the tautological sub and quotient bundles on corresponding sequences of classes forms a minimal generating set for the algebra A.G/ . Our indexing of the Schubert cycles is by no means the only one in use, but it has several good properties:  It reflects the partial order of the Schubert cycles defined with respect to a given V by inclusion: if we order the indices termwise, that is, .a flag ;:::;a  / 1 k 0 0 0 0 0 (writing / if and only if a a  a and  for 1  i  k ;:::;a a < a .a when a i 1 i k 0 0 ; that is, a ¤ < a a a for some i ), then i i  † † b: ” a  a b This follows immediately from the definition.  It makes the codimension of a Schubert cycle apparent: By Theorem 4.1 below, X ; codim G/ D .† a  i a P . a jWD so that a j is the degree of  A.G/ in a i  It is preserved under pullback via the natural inclusions i W 1/ ! G.k C 1;n C G.k;n/ ,

148 134 Chapter 4 Grassmannians in general C 1/ V (whose image is the set of ) and .k -subspaces containing 1 G.k;n/ , G.k;n C W 1/ j ! -subspaces contained in ); that is, k (whose image is the set of V n   / D  : and j / .  . i D a a a a > n k , or when a Here we adopt the convention that when a > 0 , we 1 1 k C take  0 as a class in A.G.k;n// . (This convention is consistent with the D a restriction to sub-Grassmannians. For example, †  G.k;n C 1/ is the k C 1 n k subset of the -planes containing a fixed general one-dimensional subspace, and G.k;n/ † thus the intersection of of subspaces contained in a fixed with the k 1 C n  j codimension- 1 subspace is empty, so that .) It follows A.G.k;n// 2 0 D  1 C k n that if we establish a formula X  D   c a b I a;b c 0 0 G.k in the Chow ring of ;n G.k;n/ / with , the same formula holds true in all 0 0  0  n k k or  n k . Whenever it happens that i  k j and is an isomorphism jCj b j a j , A.G.k;n C 1// or A.G.k C 1;n C 1// on , the formula will also hold in A respectively. Conditions for this are given in Exercise 4.32.  G.n k;V G.k;V / / obtained by associating Š There is a natural isomorphism ?  ƒ  V the .n k/ -dimensional subspace ƒ to a  V k -dimensional subspace V ƒ . This duality carries consisting of all those linear functionals on that annihilate each Schubert cycle to another Schubert cycle. For example, one checks immediately † ƒ .W / , which is the set of k -planes that meeting a fixed .n k C 1 i/ -plane i 0 nontrivially, is carried into the Schubert cycle † such that W of .n k/ -planes ƒ i 1 0 ? 0 ? .ƒ /  i , that is, such that dim \ C W W ̈ V . See Section 4.5 for the general case. ƒ 4.1.1 Schubert classes and Chern classes Schubert classes provide fundamental invariants of vector bundles. Recall from n E is a vector bundle of rank generated by a space W Š k Theorem 3.4 that, if of r global sections on a variety , then there is a map X ! G.n r;W / sending a point X x 2 X to the subspace in W consisting of the sections vanishing at x . The pullbacks Chern of the Schubert classes  give a fundamental set of invariants of E called the a classes of E — see Section 5.6.2. We will see that every Schubert class is a polynomial in the special Schubert classes (see Section 4.7).

149 Schubert cells and Schubert cycles Section 4.1 135 4.1.2 The affine stratification by Schubert cells As in the case of D G.2;4/ , the Grassmannian G.k;n/ has an affine G .1;3/ stratification. To see this, set   [  ı † : D † † a b a b>a ı The are called † . Schubert cells a ı The locally closed subset †  G is isomorphic to the affine space Theorem 4.1. a a k/ j j k.n ı † is smooth and irreducible, and the Schubert variety † is ; in particular A a a ı a j in G.k;n/ . The tangent space to † at a point is irreducible and of codimension j Œƒç a T the subspace of Hom .ƒ;V=ƒ/ consisting of those elements ' that send D G Œƒç \ ƒ  ƒ V a n k C i i into V C ƒ C i a n k i V=ƒ  ƒ i 1;:::;k . D for Choose a basis .e so that ;:::;e / Proof: for V n 1 Dh e V ;:::;e i : i i 1 Suppose Œƒç 2 † . By definition, the , and consider the sequence (4.1) of subspaces of ƒ a first nonzero subspace in the sequence will be V , the first of dimension 2 ƒ \ 1 a C k n 1 V / ƒ for will be ;:::;v .v , and so on. Thus we may choose a basis ƒ \ 1 2 k C k n a 2 v , and so on. In terms of this basis, and the basis 2 V V 2 v , with 1 2 C k n n a k C 2 a 1 2 1 .e has the form ;:::;e ƒ / for V , the matrix representative of n 1 1 0    0 0 0 0 0 0 B C      0 0 0 0 B C : B C       0 0 0 A @ 0         (This particular matrix corresponds to the case k D 4 , n D 9 and a D .3;2;2;1/ .) If ƒ , and we chose a basis for ƒ in this way, the corresponding matrix G were general in would look like 1 0       0 0 0 C B 0 0        C B : C B 0         A @          Thus the Schubert index a is the number of “extra zeros” in row i . i

150 136 Grassmannians in general Chapter 4 ı 0 † ƒ 2 † 2 but not in any of the smaller varieties † Now suppose that ; that is, ƒ a a a 0 for e ... V a . In this case v , so, for each i , the coefficient of > a i 1 k C n n k C i a a i i i in the expression of v as a linear combination of the e is nonzero, and this condition ̨ i ı † † characterizes elements of . (It follows in particular that † among elements of a a a ı † is the closure of .) Given that the coefficient of e in v is nonzero, we can i a k C n i a i v ƒ represented by multiply by a scalar to make the coefficient 1, obtaining a basis for i the rows of a matrix of the form 0 1   1 0 0 0 0 0 0 B C     1 0 0 0 0 B C ; B C 1 0 0 0      @ A 1 0        in the i -th row appears in the .n k C i a D / -th column for i where the 1;:::;k . 1 i v Finally, we can subtract a linear combination of to kill the ;:::;v from v 1 i 1 i for e in the expression of v e as a linear combination of the coefficients of i ̨ k C j a n j j < i given by the row vectors of the matrix , to arrive at a basis of ƒ 0 1   1 0 0 0 0 0 0 B C 0 1 0 0 0 0    B C A D : B C 0 1 0 0 0  0   @ A   0  0 0  1 0 Setting b n k C 1 a D f ;:::;n a , we may describe this by saying that the g 1 k A of A (that is, the submatrix involving columns from b ) is the identity -th submatrix b b has a unique basis of this form. Indeed, any other basis of ƒ matrix. We claim that ƒ k  k has a matrix obtained from this one by left multiplication by a unique invertible g , and thus has submatrix A D matrix g . b ı It follows that † is contained in the open subset U  G consisting of planes ƒ a complementary to the span of the k basis vectors whose indices are not in b . By the n ı D † same argument, any element of n U k  has a unique basis given by the rows of a 0 matrix with submatrix A D I , that is, of the form b 1 0  0  0 0    1 B C 0  1 0     0 B C : D A C B 0    0  0 1  A @ 1   0 0  0   k/ k.n ı defined by the vanishing of A is a coordinate subspace of † Thus Š j a j U a ı coordinates, and it follows that † is smooth and irreducible, and of codimension j a j , a ı † j is the closure of † as claimed. Since a , it is also irreducible and of codimension j a a ̈ in G.k;n/ (but it may be singular; for example, one sees from the Pl ucker relation 3 4 ). over a smooth quadric in P is the cone in P G.2;4/  † (Example 3.1) that 1

151 Schubert cells and Schubert cycles Section 4.1 137 The statement about tangent spaces follows from the explicit coordinate description k.n k/ A of U Š with the set of k  n matrices † above. We identify the open set a . Since the tangent space to an affine space having an identity matrix in positions from b may be identified with the corresponding vector space, the tangent space Hom .ƒ;V=ƒ/ 0 ƒ at b complementary is given by the set of matrices in the positions from to G.k;n/ , or more properly by the transposes of these matrices. Given such a tangent b to those in  n matrix with submatrix A vector, we may complete it uniquely to a D I , and k b ƒ V D V=ƒ ̊ ƒ inducing ! this (or rather its transpose) corresponds to the lifting ! ƒ . Thus the set of tangent directions at ƒ to the affine subspace the identity map ƒ ı is identified with the set of matrices in that subspace, and this corresponds precisely † a to the set of maps in Hom .ƒ;V=ƒ/ whose lifting as above sends V into k C i a n i V , as claimed. ƒ C i a C k n i is at least generated as an abelian group A.G/ From Proposition 1.17 we see that k/ k.n W deg , and the existence of the degree homomorphism ! Z by the classes  A a k.n k/ .G/ that counts points shows that is actually free on the class of a point, which A i  .G/ is the generator . In Corollary 4.7 we will prove that all the A are free, by k k/ .n intersection theory and results on transversality. The description of the tangent spaces in Theorem 4.1 can be used to prove this transversality. Here is an example: Corollary 4.2. Let G D G.k;n/ . Then k k/ k k.n n D . D .G/ I /  . A 2 / k k k n k/ 1 .n k k n are both equal to the class of a point in the Chow ring and . / . / that is, k k n 1 of G . Proof: We know that .G/ is generated by the class  of a point, so it suffices A k 0 k/ .n k/ k .n / . and . to show that both / are of degree 1. k k n 1 G as the variety of k -dimensional subspaces ƒ of the We regard -dimensional n vector space V . If H  V is a codimension-1 subspace, then † ; g .H/ Df ƒ  V j ƒ  H k 1 † .H/ at the point corresponding to ƒ is and the tangent space to k 1 : .† g .H// T Df ' 2 Hom .ƒ;V=ƒ/ gj '.ƒ/  H k Œƒç 1 If H ƒ ;:::;H -plane k are general codimension-1 subspaces, then there is a unique 1 k n T T k k † H .H / in . Further, the tangent spaces ƒ D , namely, the intersection k i i 1 1 i D 1 D i intersect only in the zero homomorphism, so the intersection is transverse. This proves .n k/ that . / is the class of a point. k 1

152 138 Chapter 4 Grassmannians in general k . / , we can make an analogous To prove the corresponding statement for n k Š G.n k;n/ argument, or we can simply use duality: the isomorphism G.k;n/  introduced above carries , as we have already remarked, and preserves the  to k k 1 degree of 0-cycles. 4.1.3 Equations of the Schubert cycles N ̈ D G.k;n/ , ! It is a remarkable fact that under the Pl P every ucker embedding G ;:::;e † V /  G defined relative to the standard flag V Schubert cycle Dh e is the . i i a i 1 N with a coordinate subspace of G , that is, a subspace defined by the intersection of P ̈ vanishing of an easily described subset of the Pl ucker coordinates. This is true even at the level of homogeneous ideals: N Theorem 4.3. G.k;n/  P † be a Schubert cycle, and let b be the strictly Let  a k -tuple b D .n k C 1 a . The homogeneous ;:::;n k C 2 a increasing ;:::/ 1 2 N † in P is generated by the homogeneous ideal of the Grassmannian ideal of the a 0 ucker relations, page 94) together with those Pl p ucker coordinates (the Pl such that ̈ ̈ b 0 6 b b in the termwise partial order. The equations of the † were studied in Hodge [1943], and this work led to the a notions of a straightening law (Doubilet et al. [1974]) and Hodge algebra (De Concini et al. [1982]). A proof of Theorem 4.3 in terms of Hodge algebras may be found in the latter publication, along with a proof that the homogeneous coordinate ring of is † a Cohen–Macaulay. The ideas have also been extended to homogeneous varieties for other reductive groups by Lakshmibai, Musili, Seshadri and their coauthors (see for example Seshadri [2007]). Avoiding this theory, we will prove Theorem 4.3 only in the easy case 5 .1;3/  G.2;4/ . In Exercise 4.17 we invite the reader to give the easier proof G D P of the set-theoretic version of Theorem 4.3. consists of : G.2;4/ , the Schubert cycle † Proof of Theorem 4.3 for In G.2;4/ a;b those two-dimensional subspaces that meet V nontrivially and are contained in a 3 V . We must show that the homogeneous ideal of 4 b 5   P † G.2;4/ a;b ̈ is generated by the Pl g WD p p p p C p p together with ucker relation 1;2 2;3 1;4 1;3 3;4 2;4 ̈ the Pl ucker coordinates p j .i;j/ f .3 g a;4 b/ — i;j (note that the condition .i;j/ — .3 a;4 b/ means i > 3 a or j > 4 b ). Specifically:  0 D p ; that is, it is the cone over the nonsingular † is the hyperplane section  G 3;4 1 3 . g D quadric p P C p p p in 2;3 2;4 1;3 14

153 Intersection products Section 4.1 139  † p is the plane D p . D p 0 D 3;4 2;4 2 2;3  is the plane p p p D D 0 . † D 1;4 3;4 1;1 2;4  † D p D p p D p D 0 . is the line 2;3 2;1 1;4 3;4 2;4  . is the point p 0 D p D D p D D p † p 2;4 3;4 1;4 1;3 2;2 2;3 L 2 A subspace has a basis whose first vector is in V † , and therefore has its a 3 a;b C 1 coordinates equal to 0, and whose second vector is in V last a , and thus has its b 4 last b coordinates equal to 0. If B is the matrix whose rows are the coordinates of these p involving is (up to sign) the determinant of the submatrix of two vectors, then B i;j i p j . It follows that if i > 3 a or j > 4 b , then the columns and .L/ D 0 , so the i;j ̈ given subsets of Pl ucker coordinates do vanish on the Schubert cycles as claimed. To show that the ideals of the Schubert cycles are generated by the relation g and the given subsets, observe that each of the subsets is the ideal of the irreducible subvariety described above, and these have the same dimensions as the Schubert cycles. For example, we know that dim † , and the ideal D 2 1;1 .g;p ;p ç ;p ;:::;p / D .p Œp ;p k ;p  / 2;4 3;4 1;2 3;4 3;4 1;4 1;4 2;4 is the entire ideal of a plane. 4.2 Intersection products 4.2.1 Transverse flags G D G.k;V / be the Grassmannian of k -dimensional linear Throughout, we let n V . We start with one useful definition. As subspaces of an -dimensional vector space we said, Kleiman’s theorem assures us (in characteristic 0) that, for a general pair of /;† V W on V , the Schubert cycles † flags . V and intersect generically . W /  G a b transversely. In this case, we can actually say explicitly what “general” means: Definition 4.4. We say that a pair of flags V and W on V are transverse if any of the following equivalent conditions hold: (a) \ W V . i D 0 for all i i n \ W (b) dim / D .V .0;i C j n/ for all i;j . min i j e (c) There exists a basis ;:::;e for V in terms of which n 1 W V e ;:::;e i and Dh Dh e ;:::;e i : n i C 1 j i n j 1 Note that any two transverse pairs can be carried into one another by a linear automorphism of V . Moreover, transverse pairs form a dense open subset in the space of all pairs of flags, so any statement proved for a general pair of flags (such as the generic

154 140 Chapter 4 Grassmannians in general W † \ † transversality of the intersection . / /  G ) holds for any transverse pair, . V a b and vice versa. Here is a lemma that will prove useful in intersecting Schubert cycles, though we will not use its full strength until the proof of Pieri’s formula (Proposition 4.9):  . V /;† Let . W / † G be Schubert cycles defined relative to transverse Lemma 4.5. a b V flags V . If ƒ 2 † V . and / \ † W . W / is a general point of their intersection, on a b then: 0 does not lie in any strictly smaller Schubert cycle ƒ . V / ̈ † (a) . V / . † a a (that is, consisting of intersections with W on ƒ V ƒ with The flags induced by and (b) flag elements and W V ) are transverse. ̨ ˇ V W V Note that, by the first part, the flags on ƒ induced by ƒ and W are, and ƒ explicitly, V W D ƒ \ V D ; i W \ ƒ D ƒ ƒ and 1;:::;k: a i C k n n C i b k i i i i The first part of the statement is immediate for dimension rea- Proof of Lemma 4.5: 0 † † V and . V / \ being transverse, the intersection sons: the flags . W / will have W a b V . dimension strictly less than † \ † . . W / / a b V W and ƒ As for the second part, we have to show that the subspaces are ƒ i k i complementary, that is, that \ V 0: W ƒ \ D i a C k n i b n i k i We do this by a dimension count: consider the incidence correspondence ̊ 2 .† j . V / \ † . ˆ //  P .V D : \ W v .ƒ;Œvç/ ƒ 2 / W a k i b C i a b n n i k i † dim dim .† ! . We will show that / \ ˆ < ˆ . W // , and thus the projection V a b † cannot be dominant, proving the lemma. Note that by the first part of . V / \ † / . W a b ˆ by the preimage of the complement the lemma we can replace of U \ W P .V / and P .V W \ / n i b k C 1 i a n k C i a 1 n n i b k i i i k i .V in / . \ W P a k C n i b n i k i i V are transverse, we have Since the flags and W .V 1: \ W b dim P / D n k a i i i a n n k i C b k i k i (If a is C b ˆ is 0 and W  n k , then the intersection V \ i a k n i n i b C i k k i i 2  Œvç correspondingly empty, so we are done in that case.) Next, suppose that U ˆ P .V W , we consider the quotient Œvç over / . To describe the fiber of \ k n i b C i a n i k i 0 0 0 0 V= h v i , and the flags V space and V W on V D comprised of images of subspaces V i 0 ; that is, under the projection V W V and ! i  a i C .V ; Ch v i /= h v i if j < n k j i 0 V D j V a i = h v i ; j C 1  n k C if 1 C j i

155 Intersection products Section 4.2 141 and similarly  .W ; Ch v i /= h v i if j < n i b j k i 0 W D j 1 h v i if j C =  n i b W : j 1 C i k 0 0 Now we just observe that, if ƒ .ƒ;Œvç/ D ƒ= h v i V 2 belongs ˆ , then the plane to the Schubert cycles 0 0 0 1;V G.k  . V /: / and † / † W . ;:::; y a a ;:::;a 1 k i b ;:::;b ;:::; b b 1 k k i ˆ over P .V Thus the fibers of have dimension / W \ a i C n i b n k i i k X X k/ / .k /; b a j b j . 1/.n b a D .k 1/.n k/ . j a j j i j i k ¤ i j ¤ i j k and altogether we have  .n k a / b D 1/ C ˆ .k 1/.n k/ . j a j a dim / . j b j b i i i i k k D k.n k/ j a jj b j 1 dim .† . V / \ † < . W //; a b as desired. 4.2.2 Intersections in complementary dimension As in the case of the Grassmannian G .1;3/ , we start our description of the Chow by evaluating intersections of Schubert cycles in complementary codimension. ring of G † . / and † V . W / defined in Here as before we use the fact that Schubert cycles a b V ; W always intersect generically transversely; this follows from terms of general flags Kleiman’s theorem, or, in arbitrary characteristic, from Theorem 4.1. / and W are transverse flags in V and † V . V /;† are Schubert . W If Proposition 4.6. a b j a jCj b jD k.n k/ , then † cycles with . V / and † intersect transversely in a . W / a b unique point if a , and are disjoint otherwise. C b 1;:::;k D i for each k D n i i C k 1 Thus  if for all k D 1 , a C b i n i i C 1 k  deg D  a b otherwise : 0 As observed, since the two flags Proof: and W are transverse, the Schubert cycles V will meet generically transversely, and hence (since the intersection is zero-dimensional) transversely. Thus deg  //  W D # .† . . V / \ † a a b b   ˇ  i; ƒ/ \ dim .V n C i a k ˇ i D ƒ # i for all : ˇ .W dim \ ƒ/  i; k C i b n i

156 142 Grassmannians in general Chapter 4 To evaluate the cardinality of this set, consider the conditions in pairs; that is, for each , i consider the . V / : i † -th condition associated to the Schubert cycle a \ ƒ/  i .V dim k n i a C i i C in combination with the -st condition associated to † .k . W / : 1/ b .W 1: C i k dim  ƒ/ \ n b 1 C i k i C 1 If these conditions are both satisfied, then the subspaces \ ƒ and W V ƒ; \ b C i n i C 1 a n k C i 1 i k having greater than complementary dimension in ƒ , must have nonzero intersection; in particular, we must have V \ W 0; ¤ a n k n i C 1 b C i i i k C 1 and are general, this in turn says we must have and, since the flags V W k C i a 1 C n i C n b 1; C n  i 1 i k C or, in other words, a b  n k: C i k i C 1 V If equality holds in this last inequality, the subspaces and W C i a k n i C 1 n b k i C 1 i will meet in a one-dimensional vector space , necessarily contained in ƒ . (In the Ä i .) i notation of Definition 4.4, Dh e Ä i i a C n k i We have thus seen that . V / and † † . W / will be disjoint unless a  C b a i k C 1 i b k i . But from the equality n for all k X a b jD j jCj b .a k.n C k/; D / i i 1 k C D i 1 , then we must have a C b k n D b  we see that if k for all i C a n i i k i 1 1 C i k C for all i . Moreover, in this case any ƒ in the intersection † must contain . V / \ † / . W a b k each of the subspaces , so there is a unique such ƒ , equal to the span of these Ä i one-dimensional spaces, as required. Corollary 4.7. The Schubert classes form a free basis for A.G/ , and the intersection G m m dim A .G/ A forms .G/ ! Z have the Schubert classes as dual bases.  m m k/ k.n In view of the explicit duality between A and .G/ .G/ given by Propo- A sition 4.6, it makes sense to introduce one more bit of notation: for any Schubert  , we will define the .a D ;:::;a index / D dual index to be the Schubert index a a 1 k .n k a / ;:::;n k a / . In these terms, Proposition 4.6 says that deg . 1  D 1 a k b  if b D a and is 0 otherwise.

157 Intersection products Section 4.2 143 Corollary 4.7 suggests a general approach to determining the coefficients in the G Ä  expression of the class of a cycle as a linear combination of Schubert classes: If is any cycle of pure codimension m , we can write X ŒÄç  D : a a j m a jD  To find the coefficient , we intersect both sides with the Schubert cycle . V / D † a a † . V / for a general flag V ; we then have k a k a ;:::;n n 1 k   deg .ŒÄç   //: / D # .Ä \ † V D . a a a undetermined coeffi- We have used exactly this approach — called the method of — in calculating classes of various cycles in G .1;3/ cients in the preceding chapter; Proposition 4.6 and Corollary 4.7 say that it is more generally applicable in any Grass- mannian. Explicitly, we have: m 2 A Corollary 4.8. .G/ is any class, then If ̨ X  ̨ :  D deg . ̨ /  a a jD a j m  G.k;n/ and  , then 2 A.G/ are any Schubert classes on G D In particular, if a b   is equal to the product a b X ;  c I c a;b j c jDj a jCj b j where  deg . /: D   a c I c b a;b  , V and W the Schubert cycles † W . U /;† / . V / and † Since for general flags U . a c b   D deg . are generically transverse by Kleiman’s theorem, the coefficients  / a c b a;b c I Littlewood–Richardson coefficients , and they are nonnegative integers. They are called appear in many combinatorial and representation-theoretic contexts. If we adopt the j a j 0 2 A  convention that D .G.k;n// if a fails to satisfy the conditions a 0 n  a l > k;  a for all  k and a 0 D 1 l k then the Littlewood–Richardson coefficients , c and depend only on the indices a , b c a;b I and not on k and n . Corollary 4.8 shows that knowing the Littlewood–Richardson coefficients suffices to determine the products of all Schubert classes. In the case of the Grassmannians G.2;n/ they are either 0 or 1 (see Section 4.3), but a Littlewood–Richardson coefficient > 1 appears already in G.3;6/ (Exercise 4.35). a;b I c

158 144 Chapter 4 Grassmannians in general G .1;n/ n P C X n G swept out by a one-parameter family C Figure 4.1 The surface  .1;n/ of P X  lines. . We will give one in There exist beautiful algorithms for calculating the c a;b I kun [2009] Section 4.7, and much more effective methods are given for example in Co s ̧ 0 ¤ ?” and and Vakil [2006a]. But even simple questions such as, “when is a;b c I > 1 ?” do not seem to admit simple answers in general. “when is a;b I c We will return to Schubert calculus shortly, but we take a moment here to use what we have already learned to answer Keynote Question (b). 4.2.3 Varieties swept out by linear spaces n swept .k;n/ be an irreducible curve, and consider the variety X  P C G  Let C ; that is, out by the linear spaces corresponding to points of [ n X P D ƒ  C Œƒç 2 X C ; in particular, (See Figure 4.1). We would like to relate the geometry of to that of 5 when C Keynote Question (b) asks us to find the degree of G .1;3/  P X is a  twisted cubic curve. n To begin with, observe that P X : If is indeed a closed subvariety of n j G .k;n/  P Df 2 p 2 ƒ g ˆ .ƒ;p/ k -plane over G is the universal , as described in Section 3.2.3, and ̨ W ˆ ! G .k;n/ .k;n/ n and ˇ W ˆ ! P are the projections, then we can write 1 ˇ. ̨ .C//: X D x 2 X lies on a unique k -plane ƒ 2 C — that is, the Now, suppose that a general point n 1 ̨ X map .C/ ! ˇ  P W is birational, so that in particular dim .X/ D k C 1 . The .n degree of is the number of points of intersection of X with a general X k 1/ -plane

159 Intersection products Section 4.2 145 n  X , and so lies on a unique P Ä ; since each of these points is a general point of , the number is the number of Ä -planes ƒ that meet ƒ . In other words, we have k k -plane # .X \ Ä/ D deg .X/ .C \ † D .Ä// # 1 deg  D / (by Kleiman’s theorem)  .ŒCç 1 .C/; deg D ̈ where by the degree of ucker embedding of G .k;n/ . C we mean the degree under the Pl 3 These ideas allow us to answer Keynote Question (b): The surface P X swept  5  G .1;3/  P out by the lines corresponding to a twisted cubic , times the degree C ˇ of the map itself has degree 3 or 1. In defined above, is equal to 3. Thus the surface X would be contained in a Schubert cycle C , and as we have the latter case, the curve † 1;1 seen in the description on page 138, this Schubert cycle is contained in the 2-plane in 5 ̈ defined by the vanishing of three Pl P ucker coordinates. Since a twisted cubic is not contained in a 2-plane, this shows that the surface X has degree 3. More of the geometry X is described in Exercises 4.23-4.25. of n X G If is a variety of any dimension m , we can form the variety   P Z .k;n/ swept out by the planes of Z . Its degree — assuming it has the expected dimension k C dim Z and that a general point of X lies on only one plane ƒ 2 Z — is expressible in terms of the Schubert coefficients of the class ŒZç 2 . G .k;n// , though it is not in A m Z general equal to the degree of . This is the content of Exercise 4.22; we will return to this question in Section 10.2, where we will see how to express the answer in terms of Chern and Segre classes. 4.2.4 Pieri’s formula One situation in which we can give a simple formula for the product of Schubert D  . Such classes are  classes is when one of the classes has the special form b;0;:::;0 b . special Schubert classes called Proposition 4.9 . For any Schubert class  (Pieri’s formula) 2 A.G/ and any integer b , a X .   / D  c a b c jDj a jC b j  a c  a i 8 1 i i i 1 Proof: By Corollary 4.8, Pieri’s formula is equivalent to the assertion that, for any with j c jDj a jC b , Schubert index c  c for all i;  a a if 1  i i 1 i  / . deg  D  a c b 0 otherwise : 1 This proof was shown to us by Izzet Cos ̧kun

160 146 Chapter 4 Grassmannians in general † To prove this, we will look at the corresponding Schubert cycles /;† . . U / and V a b  / , defined with respect to general flags V , . and W ; we will show that their † U W c violates the condition a c  c intersection is empty if  a , and consists of i for any 1 i i i i a single point if these inequalities are all satisfied. By Kleiman’s theorem, the intersection multiplicity will be 1 in the latter case. By definition, . V / Df ƒ j dim .ƒ \ † V g i for all i  / a n C k i a i and  W / Df i j dim .ƒ \ W † . : g i for all /  ƒ c i C c i 1 C k Set W V A \ ; D i i n k C k C 1 i C c a i i A so that either 0 or dim A D D c -th condition in the a i C 1 . Combining the i i i i C 1 i/ -th condition in the second, we see that for any first definition and the .k  † 2 V / \ † ƒ . . W / we have a c \ A 0: ¤ ƒ i   / 0 for some i then A D D 0 , so that † If . V / \ †  , . W < a D ¿ , and deg   c a i i a i c c b c .  as required. Thus we may assume that i for every a i i We claim that the A . are linearly independent if and only if c i  a for all 1 i i i D e V as in Section 4.2.2, so that Dh e W ;:::;e and i To see this, choose a basis i j i i 1 ;:::;e e i . With this notation h n n C 1 j ; Dh e A i ;:::;e i a c C k n k C i n i i i and the condition c  a amounts to the condition that the two successive ranges of 1 i i n c k C i 1 indices a i ;:::;n k C i 1 a C k and n k C i c ;:::;n i 1 1 i i i do not overlap. In other words, if we let A Dh A i ;:::;A 1 k be the span of the spaces A , then we have i X k c dim a b; C 1 D A  C i i a with equality holding if and only if  for all i . Note that by Lemma 4.5 the c 1 i i ƒ is spanned by its intersections with the A plane ; that is, ƒ  A . i Now we introduce the conditions associated with the special Schubert cycle . U / † . b This is the set of k -planes that have nonzero intersection with a general linear sub- U D . For there to be any space  V of dimension n k C 1 b U 1 b n k C  , and hence, † 2 . V / \ † 0 ƒ ¤ U \ satisfying this condition requires that A / . W a c since U is general, that dim A  k , then we will have b . Thus, if c i > a for any C 1 i i

161 Grassmannians of lines Section 4.2 147  V \ † . . We can accordingly assume . W / \ † † . U / D ¿ , and c /  a i for all 1 i i a c b A D hence dim C b . k  V is a general subspace of codimension k C b 1 , it will meet A Finally, since U any nonzero vector in this intersection. Since v in a one-dimensional subspace. Choose L A D v uniquely as a sum A , we can write i v CC v v D with v : 2 A i i 1 k  2 † V . Suppose now that / \ † satisfies all the Schubert conditions . U / \ † / ƒ . W a c b  A and ƒ \ U ¤ 0 , ƒ must contain the vector above. Since , and, since ƒ is v ƒ , it follows that ƒ must contain the vectors v A spanned by its intersections with the i i  † . V / \ as well. Thus, we see that the intersection will consist of . U / \ † / † . W a c b ƒ D h v ;:::;v the single point corresponding to the plane i spanned by the v , and 1 i k we are done. As a corollary of the Pieri formula, we can prove a relation among the special Schubert classes that is an important special case of a theorem of Whitney used for computing Chern classes (Theorem 5.3): Corollary 4.10. In A.G.k;n// , we have k C  1: C  D CC  /  .1 /.1  1/ C   . CC k 1;1;1 1;1 1 1 2 n k 1 Proof: We can use Pieri to calculate the individual products appearing in this expression. To start, Pieri tells us that m m    C  : D 1 m 1 l l;1 1;1 l C in the product on the left, then, the sum d When we write out the terms of degree d > 0 , telescopes: For d X i C C   C . /   . D  / . 1/ i d 1;1 i d d d 1;1 d 2;1;1 d 1 0 D i d 1 d  / . C  . C 1/ . 1/ C d d d 2 1 1 2;1 0: D 4.3 Grassmannians of lines G D G.2;V / be the Grassmannian of two-dimensional subspaces of an .n C 1/ Let - n dimensional vector space V , or, equivalently, lines in the projective space P V Š P . The Schubert cycles on G with respect to a flag V are of the form † : g V  . V / Df ƒ j ƒ \ V ƒ and 0 ¤ C a n a ;a n a 1 2 2 1 1

162 148 Grassmannians in general Chapter 4 In this case, Pieri’s formula (Proposition 4.9) allows us to give a closed-form expression for the product of any two Schubert classes: Proposition 4.11. a Assuming that  b a b , 2 1 2 1 CC  D   C   ;a a a b C b C ;a a C b b C ;b a ;b C b 1 1;a C a b C 2 1 2 2 2 1 1 1 2 2 1 1 2 2 1 1 X D  : c ;c 2 1 j jDj b c jCj a j b a a C b  c C  2 1 1 1 1 Proof: We will start with the simplest cases, where the intersection of general Schubert b , then the Schubert cycle D b is D b cycles is again a Schubert cycle: If † / . W 2 1 b;b equal to ;  W ƒ j f g ƒ n b so that for any a ;a we have 2 1 8 9 0; ¤ V \ ƒ ˆ > a 1 n 1 ˆ ˇ > < = ˇ ˇ † D / W . . † \ / V ƒ ƒ  V ; ;a a n a b;b 2 1 2 ˇ ˆ > ˆ > : ; W ƒ  b n ( ) ˇ ƒ \ .V ¤ / W 0; \ a n 1 n b 1 ˇ D ƒ ˇ ƒ .V  \ W / a n n b 2 W /: W \ .V ;V \ † D 1 a n a n b n a b C C b b;a n 1 2 2 1 Thus by Kleiman’s theorem we have  (4.2)  : D  ;a a a b;a C b b;b C 2 1 2 1 Now, suppose we want to intersect an arbitrary pair of Schubert classes  and ;a a 2 1  . We can write b ;b 2 1  /   /.  D . ;a a a ;0 a a ;a b b ;b ;0 b b ;b 1 2 2 2 1 2 1 2 2 2 2 1   D  ; a ;0 a b C b ;a ;0 b a b C 2 1 2 2 1 2 2 2 and if we can evaluate the product of the first two terms in the last expression, we can use (4.2) to finish the calculation. a  b , Pieri says that But this is exactly what Pieri gives us: if   D  C  CC  ; a;0 a a C b 1;1 C b;0 a;b b;0 and the general statement follows.

163 Grassmannians of lines Section 4.3 149 .1;n/ G (and a little combinatorics) We can use this description of the Chow ring of G .1;n/ D to answer Keynote Question (d): What is the degree of the Grassmannian ̈ ucker embedding? We observe first that, since the hyperplane under the Pl 1/ C G.2;n V 2 n C 1 1 / class on pulls back to the class k P 2 A . .G.2;n C 1// , we have  1 2n 2 deg . .G.2;n D 1// C deg /: 1  To evaluate this product, we make a directed graph with the Schubert classes in a G.2;n as vertices and with the inclusions among the corresponding Schubert cycles C 1/ . V / indicated by arrows (the graph shown is the case n D † ): 5 a  0 6  1  6   1;1 2  6 6   3 2;1   6 6    3;1 2;2 4   6 6   4;1 3;2   6   3;3 4;2  6  4;3   4;4 In terms of this graph, the rule expressed in Proposition 4.11 for multiplication  by is simple: The product of any Schubert class  is the sum of all with  1 1 a;b   — that is, the Schubert classes in the row below immediate predecessors of a;b a;b 2 2n by an arrow. In particular, the degree deg .. that are connected to /  of / 1 a;b the Grassmannian is the number of paths upward through this diagram starting with “1”s 1  . If we designate such a path by a sequence of and ending with  n 0;0 1;n n 1 and 1 “2”s, corresponding to whether the first or second indices change (these are n the vertical and diagonal arrows in the graph shown) reading from left to right, there are never more “2”s than “1”s. Equivalently, if we associate to a “1” a left parenthesis and to a “2” a right parenthesis, this is the number of ways in which n 1 pairs of parentheses can appear in a grammatically correct sentence. This is called the 1/ -st Catalan .n ; a standard combinatorial argument (see, for example, Stanley [1999]) gives number 2/Š .2n c D : 1 n nŠ.n 1/Š

164 150 Chapter 4 Grassmannians in general In sum, we have: V 2 n C 1 Proposition 4.12. 1/ k  P . / C The degree of the Grassmannian G.2;n is .2n 2/Š C 1/ D G.2;n : deg nŠ.n 1/Š This number also represents the answer to the enumerative problem of how many n n 2/ 2 general .n P -planes V lines in ;:::;V meet each of  P 2n . 2 1 2n Pieri’s formula (Proposition 4.9) gives us the means to answer the generalization  of Keynote Question (d) to all Grassmannians: Since is the class of the hyperplane 1 ̈ ucker embedding, the degree of the Grassmannian section of the Grassmannian in its Pl k/ k.n in that embedding is the degree of  . This will be worked out (with the aid of the 1 hook formula from combinatorics) in Exercise 4.38; the answer is that k 1 Y iŠ .G.k;n// D deg : k//Š .k.n .n k C i/Š 0 D i We can also use the description of A. G .1;n// given in Proposition 4.11 to answer 1 2n C Keynote Question (a): If P  are four general n -planes, how many ;:::;V V 1 4 2n C 1  P meet all four? The answer is the cardinality of the intersection L lines T .V † ; given transversality — a consequence of Kleiman’s theorem /  G .1;2n C 1/ i n in characteristic 0, and checked directly in arbitrary characteristic via the description of tangent spaces to Schubert cycles in Theorem 4.1 — this is the degree of the product 4  2 A. G .1;2n C 1// . Applying Proposition 4.11, we have n 2  I D   C  C  CC 2n 1;1 C 1;n 1 n 2n n;n n since each term squares to the class of a point and all pairwise products are zero, we have 4 . D 1; / deg n C n and this is the answer to our question. We will see in Exercise 4.26 another way to arrive at this number, in a manner analogous to the alternative solution to the four-line problem given in Section 3.4.1; Exercise 4.27 gives a nice geometric consequence. 4.4 Dynamic specialization In Section 3.5.1, we started to discuss the method of specialization , and used it to determine the products of some Schubert classes. We can compute intersection numbers in other cases only by using a stronger and more broadly applicable version of this technique, called dynamic specialization .

165 Dynamic specialization Section 4.4 151 Recall that in Section 3.5.1 we described an alternative approach to establishing the 2 D  C  relation  in the Chow ring of the Grassmannian G .1;3/ . Instead of taking 11 2 1 — whose intersection was  G .1;3/ two general translates of the Schubert cycle .L/ † 1 necessarily generically transverse, but whose intersection class required additional work 3 0 0 † to calculate — we considered the intersection .L † / , where L;L were  P .L/ \ 1 1 not general, but incident lines. The benefit here is that now the intersection is visibly a 0 \ L D p L union of Schubert cycles: Specifically, if is their point of intersection and 0 their span, we have D L;L H 0 \ † † .L .L/ / D † .H/: .p/ [ † 1;1 2 1 1 The trade-off is that we cannot just invoke Kleiman to see that the intersection is indeed generically transverse; this can however be established directly by using the description of the tangent spaces to the two cycles given in Theorem 4.1. 4 D Suppose now we are dealing with the Grassmannian .1;4/ of lines in P G G 2 4  2 A and we try to use an analogous method to determine the product .G/ — that is, 2 the class of the locus of lines meeting each of two given lines. We would try to find a 4 L;M  P such that the two cycles pair of lines \ .L/ Df ƒ j ƒ \ L ¤ ¿ g and † ¿ .M/ Df ƒ j ƒ † M ¤ g 2 2  representing the class are special enough that the class of the intersection is clear, but 2 still sufficiently general that they intersect generically transversely. L However, there are no such pairs of lines. If the lines M are disjoint, they and are effectively a general pair, and the intersection is not a union of Schubert cycles. On L meets M at a point the other hand, if , then the locus of lines through p forms a p three-dimensional component of the intersection , so the intersection is .L/ \ † .M/ † 2 2 not even dimensionally transverse. 1 4 2 M f , parametrized by t P A We can nevertheless consider a family of lines , g in t M with for t ¤ 0 and with M L meeting L at a point p . This gives disjoint from t 0 / a family of intersection cycles \ † † .M . To make this precise, we consider .L/ t 2 2 the subvariety ı 1 .t;ƒ/ 2 A Df  G j t ¤ 0 and ƒ 2 † ˆ .L/ \ † g .M / t 2 2 1  A D  G . Since and its closure is disjoint from L for t ¤ 0 , the fiber ˆ ˆ M t t 2 ˆ \ † does .M .L/ / of ˆ over t ¤ 0 represents the class  † , and it follows that 2 0 2 t 2 ˆ we are looking not at the as well. The key point is that when we look at the fiber 0 † \ .L/ intersection † of the limiting cycles, but rather at the flat limit of the .M / 0 2 2 intersection cycles † , which is necessarily of the expected dimension. .L/ \ † / .M t 2 2 † ˆ is contained in the intersection † , but has smaller The fiber \ / .M .L/ 2 0 0 2 dimension. Thus a line ƒ arising as the limit of lines ƒ must meeting both L and M t t satisfy some additional condition beyond meeting both L and M , and to characterize 0 ˆ we need to say what that condition is. 0

166 152 Chapter 4 Grassmannians in general 4 3 . L and M For together span a hyperplane H 0 D L;M t Š P , the lines  P ¤ t t t . If goes to H Let as t H 0 be the hyperplane that is the limit of the f ƒ is a family g t t 0 ƒ must be meeting both L and of lines with ƒ for t ¤ 0 , then the limiting line M t t 0 H contained in . 0 \ D p ƒ L does not pass through the point M Of course, if , then it must be 0 0 P contained in the 2-plane L;M is redundant. In , so the new condition ƒ H  D 0 0 0 sum, we conclude that the support of ˆ must be contained in the union of the two 0 two-dimensional Schubert cycles ˆ /: f ƒ j ƒ  P g[f ƒ j p ;H 2 ƒ  H .p gD † † .P/ [ 2;2 3;1 0 0 0 0 0 .p .P/ † We will see in Exercise 4.28 that the support of ˆ [ † , is all of / ;H 0 0 3;1 2;2 0 ˆ and in Exercise 4.29 that is generically reduced. Thus, the cycle associated to 0 .p the scheme is exactly the sum † , and we can deduce the .P/ C † / ˆ ;H 0 0 3;1 2;2 0 formula 2 4 G .1;4//:  . C  D 2 A  2;2 3;1 2 This is a good example of the method of dynamic specialization, in which we consider not a special pair of cycles representing given Chow classes and intersecting generically transversely, but a family of representative pairs specializing from pairs that do intersect transversely to a pair that may not. We then must describe the limit of the intersections (not the intersection of the limits). This technique is a starting point for s ̧ kun [2009] and Vakil [2006a]. For another example of its the general algorithms of Co application, see Griffiths and Harris [1980]. Often, as in the examples cited above, to carry out the calculation of an intersection of Schubert cycles we may have to specialize in stages; see Exercise 4.30 for an example. To see this idea carried out in a much broader context, see Fulton [1984, Chapter 11]. 4.5 Young diagrams For many purposes, it is convenient to represent the Schubert class  by a a ;:::;a 1 k ; that is, as a collection of left-justified rows of boxes with the i -th row of Young diagram a would be represented by . For example,  length 4;3;3;1;1 i  ! : 4;3;3;1;1

167 Young diagrams Section 4.5 153 (Warning: there are many different conventions in use for interpreting the correspondence between Schubert classes and Young diagrams!) The condition that k  a n  a  0 1 k means that the Young diagram fits into a box with n k columns, and the rows and k rows of the diagram are weakly decreasing in length from top to bottom. As another  ,   and the dual Schubert class example, the relation between a Schubert class a a described in Proposition 4.6, could be described by saying that the Young diagrams of ı  k/ and  D  , are complementary in the , after rotating the latter 180  D k  .n  a a is as shown in the following:  box; if 2 A.G.5;10// , for example, then   D 4;3;3;1;1           I                that is,  ! : As a first application of this correspondence, we can count the Schubert classes as follows: n / . k A.G.k;n// Z as abelian groups. Corollary 4.13. Š The number of Schubert classes is the same as the number of Young diagrams that Proof: k  .n k/ box of squares B fit into a . To count these, we associate to each Young diagram Y B its “right boundary” L : this is the path consisting of horizontal and vertical in k segments of unit length which starts from the upper-right corner of the .n k/ box  and ends at the lower-left corner of the box, such that the squares in Y are those to the left of L . (For example, in the case of the Young diagram associated to  ,  G.5;10/ 4;3;3;1;1 illustrated above, we may describe by the sequence h;v;h;v;v;h;h;v;v where h L v and denote horizontal and vertical segments, respectively, and we start from the upper-right corner.) h Of course the number of n k , the width of terms in any such boundary must be the box, and the number of v terms must be k , the height of the box. Thus the length of the boundary is n k steps , and giving the boundary is equivalent to specifying which  n will be vertical; that is, the number of Young diagrams in B is , as required. k The correspondence between Schubert classes and Young diagrams behaves well with respect to many basic operations on Grassmannians. For example, under the duality G.k;n/ Š G.n k;n/ , the Schubert cycle corresponding to the Young diagram Y is

168 154 Grassmannians in general Chapter 4 that is the trans- taken to the Schubert cycle corresponding to the Young diagram Z ı , that is, the diagram obtained by flipping Y around a pose line running Y of 45 northwest-to-southeast. For example, if  2 ! ; A.G.4;7// 3;2;1;1 is then the corresponding Schubert cycle in G.3;7/ : 2 !  A.G.3;7// 4;2;1 This is reasonably straightforward to verify, and is the subject of Exercise 4.31. Pieri’s formula can also be described in terms of Young diagrams: It says that for any   and any special Schubert class  , the Schubert classes D Schubert class a;0;:::;0 a b   (all with coefficient 1) correspond to Young diagrams appearing in the product a b by adding a total of obtained from the Young diagram of a boxes, with at most one  b , as long as the result is still a Young diagram: for example, if box added to each column we want to multiply the Schubert class 2  A.G.4;8// ! 4;2;1;1 by the Schubert class , we can add a box in either the first, second, third or fifth row,  1 to obtain the expression  D :  C  C  C   1 4;2;2;1 5;2;1;1 4;3;1;1 4;2;1;1;1 4;2;1;1 The combinatorics of Young diagrams is an extremely rich subject with many applications. For an introduction, see for example Fulton [1997]. 4.5.1 Pieri’s formula for the other special Schubert classes V be an n -dimensional vector space and V D V    Let V V D V  1 n n 1 a flag in V . As we observed in Section 4.1, for any integer a with 1  a  n k the  isomorphism Š G.n k;V / carries the special Schubert cycle G.k;V / † 0 Df ƒ 2 g j ƒ \ V ¤ G.k;V / a k C 1 a n

169 Linear spaces on quadrics Section 4.5 155 V defined relative to the flag to the Schubert cycle ?   / Df ƒ 2 G.n k;V . / j dim .ƒ \ V V g a † / 1;:::;1 C k a n 1 ? V V . The Schubert classes formed by the annihilators of the defined relative to the flag i a   (often written ) 1;:::;1 1  ! 5 1 . The correspondence between Schubert are also referred to as special Schubert classes classes and Young diagrams makes it easy to translate Pieri’s formula into a formula for a a multiplication by : The Schubert classes appearing in the product   (all with  1 1 b  coefficient 1) correspond to Young diagrams obtained from the Young diagram of by b adding a total of , as long as the result a boxes, with at most one box added to each row is still a Young diagram: Theorem 4.14 (Pieri’s formula, part II) For any Schubert class  D  . 2 ;:::;b b b 1 k and any integer a with 1  a  n k , A.G.k;n// X a   D  c 1 b jD Cj b j c j a b c  b  C 1 8 i i i i 4.6 Linear spaces on quadrics We can generalize the calculation in Section 3.6 of the class of the locus of lines on a quadric surface to a description of the class of the locus of planes of any dimension on a smooth quadric hypersurface of any dimension. To begin with, since our field k has characteristic ¤ 2 , a nonsingular form Q.x/ of degree 2 on P can be written in the form Q.x/ D q.x;x/ , where q.x;y/ is a V W V ! k . A linear subspace P V  P V lies on nonsingular symmetric bilinear form  Q.x/ D 0 if and only if W is isotropic for the quadric , that is, q.W;W / D 0 . Thus, q we want to find the class of the locus  G D G.k;V / of isotropic k ˆ q . -planes for To start, we want to find the dimension of ˆ . There are a number of ways to do this; probably the most elementary is to count bases for isotropic subspaces. To find a basis for an isotropic subspace, we can start with any vector v , then choose with q.v 0 ;v D / 1 1 1 ? ? 2h v 0 i v nh v D i with q.v / ;v ;v / D 0 , v q.v 2h v with ;v i i v nh , and ;v 3 1 1 3 3 2 2 2 1 1 2 2 ? h v ; ;:::;v n=2 i  h v  ;:::;v i i so on. Since , this necessarily terminates when i 1 i 1 in other words, a nondegenerate quadratic form will have no isotropic subspaces of

170 156 Chapter 4 Grassmannians in general . (We could also dimension strictly greater than half the dimension of the ambient space  V see this by observing that carrying any q with its dual defines an isomorphism of V  ? into its annihilator  V V .)  ƒ isotropic subspace ƒ correspond to points on the quadric v In this process, the allowable choices for 1 0 v Q.x/ correspond to the points on the quadric Q j ; those for D , and so forth. In ? 2 v 1 of dimension form a locally closed subset of V general, the n i . Thus the space of v i k all bases for isotropic -planes has dimension   k .n D k.n k/ C CC k/ .n 1/ : 2 2 -dimensional family of bases for a given isotropic k -plane, the space k Since there is a of such planes has dimension     1 C k k 2 ; k.n k/ D k k/ k.n C 2 2  k C 1  ˆ in G.k;V / when k has codimension n=2 , and is or in other words the cycle 2 empty otherwise. Having determined the dimension of ˆ , we ask now for its class in A.G.k;V // . Following the method of undetermined coefficients, we write X ;  D Œˆç a a 1 k C jD a j / . 2 with // D # .ˆ \ † V . a ;:::;n k a a n k 1 k D # f ƒ j q j :  0 and dim .ƒ \ V g i for all i /  a i C ƒ i ƒ  V is a k -plane in this intersection. The subspace To evaluate , suppose that a j V  q being general, the restriction q to it will again be nondegenerate. V of C i a V i i C a i q j , or in other words 2i Since  i has an isotropic i -plane, we must have a C i V C a i i i for all i: a  i  P k C 1 a But by hypothesis , so we must have equality in each of these inequalities. D i 2 In other words, . D 0 for all a except the index a D .k;k 1;:::;2;1/ a It remains to evaluate the coefficient / (4.3) ; D # f ƒ j q.ƒ;ƒ/ D 0 and dim .ƒ \ V g  i for all i 2i 1;:::;2;1 k;k k where the equality holds by Kleiman’s theorem. We claim that this number is 2 . q j q to the two- of We prove this inductively. To start, note that the restriction V 2 has two one-dimensional isotropic spaces, and ƒ will necessarily V dimensional space 2 ˆ is disjoint from any Schubert contain exactly one of them: it cannot contain both, since  k C 1 cycle with j b j > k.n k/ † / . V . b 2

171 Giambelli’s formula Section 4.6 157 ƒ We may thus suppose that V contains the isotropic subspace , so that ƒ is W  2 ?  0 , q.W;W / induces a nondegenerate quadratic form . Now, since W contained in q ? 0 0 =W -dimensional quotient D W .n W , and the quotient space q on the 2/ ? 0 ƒ W  =W D ƒ=W 0 .k . Moreover, since the spaces V q are -dimensional isotropic subspace for is a 1/ 2i V general subspaces of , the subspaces V containing 2 0 ? ? .V =W \ W D /=W  W V 2i 2 2i 0 ? D W form a general flag in =W , and we have W 0 0 .ƒ V dim i: /  i \ 1 for all 2 2i k 1 0 0 .k 1/ -planes ƒ Inductively, there are  W 2 satisfying these conditions, isotropic k and so there are 2 planes ƒ  W satisfying the conditions of (4.3). We have proven: Proposition 4.15. Let q be a nondegenerate quadratic form on the n -dimensional vector space V  G.k;V / the variety of isotropic k -planes for q . Assuming k  n=2 , and ˆ is ˆ the class of the cycle k D  2 Œˆç : 1;:::;2;1 k;k As an immediate application of this result, we can answer Keynote Question (c). To 4 P begin with, we asked how many lines lie on the intersection of two quadrics in . To 4 0  P Q;Q be two general quadric hypersurfaces and X D Q answer this, let \ Q . 2 1 0 0 ˆ \ ˆ Q of the cycles of lines lying on The set of lines on and Q X ; is the intersection by Kleiman’s theorem these are transverse, and so we have 0 2 \ ˆ # / D deg .4 16: / .ˆ D 2;1 2n 0 are general quadrics, we ask how many and  P Q Q .n 1/ - More generally, if planes are contained in their intersection; again, this is the intersection number n 2 n 0 : ˆ D deg .2 \  / # 4 / .ˆ D 1;:::;1 n;n 4.7 Giambelli’s formula Pieri’s formula tells us how to intersect an arbitrary Schubert class with one of the special Schubert classes  . Giambelli’s formula is complementary, in D  b;0;:::;0 b that it tells us how to express an arbitrary Schubert class in terms of special ones; the two together give us (in principle) a way of calculating the product of two arbitrary Schubert classes. We will state Giambelli’s formula and indicate one method of proof; see Chapter 12 for some special cases and Fulton [1997] for a proof in general.

172 158 Chapter 4 Grassmannians in general . Proposition 4.16 (Giambelli’s formula) ˇ ˇ ˇ ˇ      2 C a 1 a a C a 1 k C ˇ ˇ 1 1 1 1 ˇ ˇ      ˇ ˇ a a C 1 a 1 2 k C a 2 2 2 2 ˇ ˇ ˇ ˇ      1 a 2 a a a 3 C k : D  3 3 3 3 a ;a ;:::;a 1 2 k ˇ ˇ : : : : : ˇ ˇ : : : : : : ˇ ˇ : : : : ˇ ˇ ˇ ˇ      a 3 1 a a k C 2 k a C k C k k k k Thus, for example, we have ˇ ˇ ˇ ˇ   2 3 ˇ ˇ    D ; D  2 1 3 2;1 ˇ ˇ   0 1 2 , for example. Giambelli’s  which we can then use together with Pieri to evaluate 2;1 formula also reproduces some formulas we have derived already by other means: For D example, when a a D 1 it gives 2 1 ˇ ˇ ˇ ˇ   2 1 2 ˇ ˇ D    ; D 1;1 2 1 ˇ ˇ   1 0 2 or in other words D   C  . 1;1 2 1 As the last two examples suggest, we could deduce Giambelli’s formula from Pieri’s formula. For example, in the 2  case, we can expand the determinant and apply Pieri’s 2 formula to obtain ˇ ˇ ˇ ˇ   a a C 1 ˇ ˇ  D    a 1 C a b 1 b ˇ ˇ   b b 1 . C  CC / / . CC  D  a C b a C 1;b a C 1;b 1 1 a C b a;b D  : a;b More generally, we could prove Giambelli’s formula inductively by expanding the determinant in Proposition 4.16 by cofactors along the right-hand column; Exercise 4.39 3  3 case. asks the reader to do this in the Giambelli’s formula implies that the Chow ring A.G/ of a Grassmannian G is generated as a ring by the special Schubert classes, and we can ask about the polynomial relations among these classes. There is a surprisingly simple and elegant description of these relations, which we will derive in Section 5.8, from the fact that the special Schubert classes are exactly the Chern classes of the universal bundles on the Grassmannian. Giambelli’s formula and Pieri’s formula together give an algorithm for calculating the product of any two Schubert classes: Use Giambelli to express either as a poly- nomial in the special Schubert classes, and then use Pieri to evaluate the product of this polynomial with the other. But this is a terrible idea for computation except in low-dimensional examples: Because Giambelli’s formula is determinantal, the number of products involved increases rapidly with k and n . Nor is it easy to use Giambelli’s

173 Generalizations Section 4.7 159 formula to prove qualitative results about products of Schubert classes; it is not even clear from this approach that such a product is necessarily a nonnegative linear combination s ̧ of Schubert classes. The algorithms of Co kun and Vakil referred to earlier (Co kun s ̧ [2009] and Vakil [2006a]) are far better in these regards. 4.8 Generalizations Much of the analysis we have given here of the Chow rings of Grassmannians applies more generally to any compact homogeneous space for a semisimple algebraic group. In this section, we will describe some of these spaces, and indicate how the analysis goes in some of the simplest non-Grassmannian cases. 4.8.1 Flag manifolds n and .k Let ;:::;k be a vector space of dimension / any sequence of integers V m 1 < with  < k 0 < k < n . We define the flag manifold F.k to be the ;:::;k V / I m 1 m 1 space of nested sequences of subspaces of V of dimensions k ;:::;k ; that is, 1 m n o Y ˇ ˇ D I .ƒ ;:::;k ;:::;ƒ ƒ / 2 : G.k  ;V / F.k V / ƒ 1 i m 1 m 1 m matters we also use V As in the case of the Grassmannian, when only the dimension of I ;:::;k F.k the symbol n/ ; also as in the case of Grassmannians, we will sometimes m 1 use the projective notation n o Y ˇ ˇ P V / D F .ƒ ;:::;ƒ .k 2 ;:::;k G .k I ; P V / / ƒ ƒ  i m 1 m 1 m 1 k <  < k 0  dim P V . We leave as an exercise the verification that the for < 1 m Q .    ƒ condition defines a closed subscheme of the product ƒ G.k ;V / i m 1 m 2 , which is the content of Exercise 3.21.) (This follows immediately from the case D F.k In particular, ;:::;k is a projective variety. I V / m 1 At one extreme we have the case m D n 1 , that is, .k ; ;:::;k 1/ / D .1;2;:::;n m 1 the variety n 1 Y F.1;2;:::;n G.k;V / 1 I V /  1 k D full flag manifold , and maps to all the other flag manifolds F.k V / ;:::;k is called the I 1 m via projections to subproducts of Grassmannians. At the other, the cases with m D 1 are just the ordinary Grassmannians, and the cases with m D 2 are called two-step flag manifolds . We have already encountered some of these: the variety F .0;k I V / Df .p;ƒ/ 2 P V  G .k; P V / j p 2 ƒ g is often called the universal k -plane in P V .

174 160 Chapter 4 Grassmannians in general Many of the aspects of the geometry of Grassmannians we have explored in the last two chapters hold more generally for flag manifolds. In particular, a flag manifold F admits an affine stratification, the classes of whose closed strata (again called Schubert classes A. F / as a group. It is possible to describe the ) freely generate the Chow ring F / in terms of these generators (see Co s ̧ kun [2009]), but there is ring structure on A. an alternative: as we will see in Chapter 9 it is possible (and easier in some settings) to A. / by realizing the flag manifold as a series of projective bundles. determine the ring F 4.8.2 Lagrangian Grassmannians and beyond There are important generalizations of flag manifolds that are homogeneous spaces for semisimple algebraic groups other than GL . For example: n Lagrangian Grassmannians : If V is a vector space of dimension (a) with a nondegen- 2n erate skew-symmetric bilinear form W V  V ! k , the Lagrangian Grassmannians Q parametrize V -dimensional subspaces ƒ  LG.k;V / that are isotropic for Q ; k , that is, such that 0 . More generally, we have Lagrangian flag manifolds D Q.ƒ;ƒ/ parametrizing flags of such subspaces. Orthogonal Grassmannians : As in the previous case, we consider a vector space (b) V with nondegenerate bilinear form Q , but now Q is symmetric. The orthogonal Grassmannian parametrizes isotropic subspaces, and likewise the orthogonal flag manifolds parametrize flags of isotropic subspaces. (In case V is even, we have dim to allow for the fact that the space of maximal isotropic planes has two connected components.) n that fixes a full flag in k GL is the group B of upper-triangular The subgroup of n Borel matrices. This is called a GL acts transitively on flags, . Since GL subgroup of n n the set of all such flags is GL =B ; it can be given the structure of an algebraic variety by n taking the regular functions on the quotient to be B -invariant functions on GL . (With n this structure, it is isomorphic to the flag manifold as we have defined it.) More generally, one could look at partial flags (for example, a single k -dimensional subspace); these are fixed by groups of block upper-triangular matrices, called parabolic subgroups . Thus for example the ordinary Grassmannian G.k;n/ has the form GL is a =P , where P n parabolic subgroup. It turns out that there is a natural way of defining Borel subgroups and parabolic subgroups in any semisimple group, and the Lagrangian and orthogonal Grassmannians may similarly be defined as quotients of the groups SO and Sp . The theory of general n n flag manifolds to which this leads is an extremely rich branch of mathematics. See for example Fulton and Harris [1991].

175 Exercises Section 4.8 161 4.9 Exercises Exercise 4.17. Use the description of the points of the Schubert cells given in Theorem 4.1 to show that Theorem 4.3 holds at least set-theoretically. Let G.2;4/ be an irreducible surface, and suppose that X Exercise 4.18.  2 2   C A .G.2;4//: D ŒXç 1;1 2 2 1;1 Show that . (In general, it are nonnegative, and that if 1 D 0 then D and 1;1 1;1 2 2 . / occur!) is not known what pairs ; 2 1;1 4 S  P G be a surface of degree d , and Ä the variety of  Let .1;4/ Exercise 4.19. S S . lines meeting 1 D ŒÄ ç 2 A . . G .1;4// (a) Find the class S S 4 (b) ;:::;S S Use this to answer the question: if P  are general translates (under 6 1 4 GL d will meet all six? ;:::;d ) of surfaces of degrees , how many lines in P 6 5 1 4 Let  P Ä be a curve of degree d , and Exercise 4.20. C  G .1;4/ the variety of C lines meeting C . 2 . D ŒÄ .1;4// ç 2 A . (a) Find the class G C C 4 Use this to answer the question: if ;C C and C are general translates of  P (b) 3 1 2 4 ;d will meet all three? and curves of degrees d , how many lines in P d 3 2 1 T .S/ The following exercise is the first of a series regarding the variety of lines 1 n S P . More will follow in Exercises 7.30, 10.39 and 12.20. tangent to a surface in n S  P d be a smooth surface of degree Exercise 4.21. Let whose general hyperplane section is a curve of genus , and T . .S/ g G .1;n/ the variety of lines tangent to S  1 To find the class of the cycle T  .S/ , we need the intersection numbers ŒT  .S/ç 1 1 3 ŒT .S/ç   . Find the latter. and 2;1 1 Let Z  G .k;n/ be a variety of dimension , and consider the variety m Exercise 4.22. n ; that is,  swept out by the linear spaces corresponding to points of X P Z [ n X D ƒ P :  Z 2 Œƒç -plane x X lies on a unique For simplicity, assume that a general point 2 ƒ 2 Z . k and degree equal to the intersection number Show that X has dimension k C m (a) deg .  ŒZç/ . m (b) Show that this is not in general the degree of Z .

176 162 Grassmannians in general Chapter 4 Exercises 4.23-4.25 deal with the geometry of the surface described in Keynote 3 X swept out by the lines corresponding to a general  Question (b): the surface P G , whose degree we worked out in Section 4.2.3. To make life  .1;3/ twisted cubic C easier, we will assume that C is general, and in particular that it lies in a general 3-plane . See also Section 9.1.1. .1;3/ G section of 5 .1;3/  P To start, use the fact that the dual of has degree 2 to show Exercise 4.23. G .M/  .1;3/ lies on the Schubert cycles † that a general twisted cubic C and † G .L/ 1 1 3 for some pair of skew lines L;M  P . 3 L;M  P .L/ , the intersection † Exercise 4.24. Show that for skew lines \ † .M/ 1 1 .L/  via the map sending a point Œƒç 2 † L M \ † is isomorphic to .M/ to the pair 1 1 \ L;ƒ \ M/ 2 L  M , and that it is the intersection of G .1;3/ with the intersection .ƒ † .L/ and † .M/ . of the hyperplanes spanned by 1 1 Using the fact that C  † Exercise 4.25. .L/ \ † in .M/ has bidegree .2;1/ 1 1 1 1 .L/ \ † † .M/ Š L  M Š P (possibly after switching factors), show  P 1 1 ' W ! M the family of lines corresponding to C may be that for some degree-2 map L realized as the locus ̊ j p 2 L D : C p;'.p/ Show correspondingly that the surface [ 3 X P D ƒ  2 C Œƒç C swept out by the lines of is a cubic surface double along a line, and that it is the 4 projection of a rational normal surface scroll S.1;2/  P . n -planes in In Section 4.3 we calculated the number of lines meeting four general 2n C 1 . In the following two exercises, we will see another way to do this (analogous to P 3 the alternative count of lines meeting four lines in P given in Section 3.4.1), and a nice geometric sidelight. 1 C 2n n Exercise 4.26. Let  P ƒ P ;:::;ƒ be four general n -planes. Calculate Š 1 4 the number of lines meeting all four by showing that the union of the lines meeting 1 C 2n 1 n P is a Segre variety S and using the calculation D and ƒ  P ;ƒ  P ƒ 1 1;n 3 2 in Section 2.1.5 of the degree of S . 1;n By the preceding exercise, we can associate to a general configuration Exercise 4.27. 2k C 1 1 of k -planes in P ƒ cross-ratios. Show that ;:::;ƒ an unordered set of k C 4 1 0 f ƒ are projectively equivalent if and only if the g and f ƒ two such configurations g i i corresponding sets of cross-ratios coincide. The next two exercises deal with the example of dynamic specialization given in Section 4.4, and specifically with the family ˆ of cycles described there.

177 Exercises Section 4.9 163 Show that the support of ˆ is all of † Exercise 4.28. .P/ [ † . .p / ;H 0 0 2;2 0 3;1 2  Verify the last assertion made in the calculation of Exercise 4.29. ; that is, show that 2 ˆ has multiplicity 1 along each component. 0 4 Hint: we can assume that the Argue that by applying a family of automorphisms of P H plane is constant, and use the calculation of the preceding chapter. t Exercise 4.30. A further wrinkle in the technique of dynamic specialization is that to carry out the calculation of an intersection of Schubert cycles we may have to specialize in stages. To see an example of this, use dynamic specialization to calculate the intersection 2 in the Grassmannian G . .1;5/  2 Hint: You have to let the two 2-planes specialize first to a pair intersecting in a point, then to a pair intersecting in a line. Exercise 4.31.  corresponds to the 2 Suppose that the Schubert class A.G.k;n// a Y in a  .n k/ box B . Show that under the duality G.k;n/ Š k Young diagram k;n/ the class  is taken to the Schubert class  corresponding to the Young G.n a b Z that is the diagram of Y , that is, the diagram obtained by flipping Y around transpose ı a line running northwest-to-southeast. For example, if 45 A.G.4;7// ! ;  2 3;2;1;1 G.3;7/ then the corresponding Schubert class in is  2 A.G.3;7// ! : 4;2;1 Exercise 4.32. Let i W G.k;n/ ! G.k C 1;n C 1/ and j W G.k;n/ ! G.k;n C 1/ be the n inclusions obtained by sending k ƒ to the span of ƒ and e  respectively. ƒ and to 1 n C  d d A Show that the map .G.k C 1;n C 1// i A ! .G.k;n// is a monomorphism if and W d  d  d , and that j only if W A .G.k;n C 1// ! A k .G.k;n// is a monomorphism n if and only if k  d . (Thus, for example, the formula 2  D  ; C  11 2 1 A. G which we established in , holds true in every Grassmannian .) .1;3// r Let C  P d be a smooth, irreducible, nondegenerate curve of degree Exercise 4.33. , as defined in and genus , and let S g .C/  G .1;r/ be the variety of chords to C 1 . ŒS .1;r// .C/ç 2 Section 3.4.3 above. Find the class G . A 2 1

178 164 Chapter 4 Grassmannians in general n Let P Q be a smooth quadric hypersurface, and let T Exercise 4.34. .Q/  G .k;n/  k n P be the locus of planes such that ƒ \ Q is singular. Show that  ƒ .Q/ç D 2 : ŒT 1 k 2  as a linear combination of Schubert classes in Exercise 4.35. Find the expression of 2;1 A.G.3;6// . This is the first example of a product of two Schubert classes where another > 1 . Schubert class appears with coefficient Using Pieri’s formula, determine all products of Schubert classes in the Exercise 4.36. .2;5/ . Chow ring of the Grassmannian G 8 0 00 Q and Exercise 4.37. Let Q be three general quadrics in P , . How many 2-planes Q lie on all three? (Try first to do this without the tools introduced in Section 4.2.4.) k.n k/ Exercise 4.38. Use Pieri to identify the degree of  with the number of standard 1 tableaux, that is, ways of filling in a k  .n k/ matrix with the integers 1;:::;k.n k/ in such a way that every row and column is strictly increasing. Then use the “hook formula” (see, for example, Fulton [1997]) to show that this number is 1 k Y iŠ .k.n : k//Š .n k C i/Š D i 0 Deduce Giambelli’s formula in the 3  3 case (that is, the relation Exercise 4.39. ˇ ˇ ˇ ˇ    1 a C a 2 a C ˇ ˇ ˇ ˇ D     a;b;c b 1 C b 1 b ˇ ˇ ˇ ˇ    2 c c c 1 for any a  b  c ) by assuming Giambelli in the 2  2 case, expanding the determinant by cofactors along the last column and applying Pieri.

179 Chapter 5 Chern classes Keynote Questions 3 3 P (a) be a smooth cubic surface. How many lines L  Let S are contained  P S ? (Answer on page 253.) in (b) F and G be general homogeneous polynomials of degree 4 Let in four variables, 3 S of quartic D V.t g F C t and consider the corresponding family f  P G/ 1 1 0 t t 2 P 3 P surfaces in S . How many members of the family contain a line? (Answer on t page 233.) (c) F and Let be general homogeneous polynomials of degree d in three variables, G 2 f C be the corresponding family of plane D V.t g F C t G/ and let  P 1 1 t 0 2 P t d . How many of the curves C will be singular? (Answer on curves of degree t page 268.) In this chapter we will introduce the machinery for answering these questions; the answers themselves will be found in Chapters 6 and 7. 5.1 Introduction: Chern classes and the lines on a cubic surface Cartier divisors — defined through the vanishing loci of sections of line bundles — are of enormous importance in algebraic geometry. More generally, it turns out that many interesting varieties of higher codimension may be described as the loci where sections of vector bundles vanish, or where collections of sections become dependent; this reduces some difficult problems about varieties to easier, linear problems.

180 166 Chapter 5 Chern classes Chern classes provide a systematic way of treating the classes of such loci, and are a central topic in intersection theory. They will play a major role in the rest of this book. We begin with an example of how they are used, and then proceed to a systematic discussion. To illustrate, we explain the Chern class approach to a famous classical result: 3 Each smooth cubic surface in contains exactly 27 distinct lines. P Theorem 5.1. 3 Sketch: P X determined by the vanishing of a cubic Given a smooth cubic surface  in four variables, we wish to determine the degree of the locus in .1;3/ of F form G . lines contained in X the problem using the observation that, if we fix a particular line L We linearize 3 , then the condition that L lie on in can be expressed as four linear conditions on P X . To see this, note that the restriction map from the 20-dimensional F the coefficients of 3 0 to the four-dimension vector space vector space of cubic forms on P D H V . O .3// L L 1 3 L P  P is a linear surjection, and the condition for the Š of cubic forms on a line L  X is that F maps to 0 in V . inclusion L As the line varies over G .1;3/ , the four-dimensional spaces V L of cubic forms on L L fit together to form a vector bundle V of rank 4 on G .1;3/ . A cubic the varying lines 3 F on form , through its restriction to each V P , defines an algebraic global section L  of this vector bundle. Thus the locus of lines contained in the cubic surface X is F the zero locus of the section  . Assuming for the moment that this zero locus is zero- F A. G .1;3// the fourth Chern class of V , denoted c dimensional, we call its class in . V / . 4 At this point all we have done is to give our ignorance a fancy name. But there are powerful tools for computing Chern classes of vector bundles, especially when those bundles can be built up from simpler bundles by linear-algebraic constructions. 0  . H In the present situation, the spaces .1// fit together to form the dual S O of the L tautological subbundle of rank 2 on G .1;3/ , and the bundle V is the symmetric cube 3   in terms of those of Sym , which allows us to express the Chern classes of V of S S  , as in Example 5.16. At the same time, it is not hard to calculate the Chern classes S  c S . / directly; we do this in Section 5.6.2. Putting these things together, we will show i in Chapter 6 that deg c . V / D 27: 4 Of course, to prove Theorem 5.1 one still has to show that the number of lines on any  all occur with multiplicity 1; this smooth cubic surface is finite, and that the zeros of F will also be carried out in Chapter 6. There are proofs of Theorem 5.1 that do not involve vector bundles and Chern classes. For example, one can show that any smooth cubic surface X can be realized 2 as the blow-up of P in six suitably general points, and using this one can analyze the geometry of X in detail (see for example Manin [1986] or Reid [1988]). But the Chern class approach applies equally to results where no such analysis is available.

181 Characterizing Chern classes Section 5.1 167 For example, we will see in Chapter 6 how to use the Chern class method to show that a 4 general quintic threefold in P contains exactly 2875 lines (a computation that played an important role in the discovery of mirror symmetry; see for example Morrison [1993]), 20 and that a general hypersurface of degree 37 in P contains exactly 4798492409653834563672780605191070760393640761817269985515 lines, a fact of no larger significance whatsoever. 5.2 Characterizing Chern classes Let E X of dimension n . We will introduce Chern be a vector bundle on a variety .X/ . E / 2 A in classes L c , extending the definition of c for a line bundle . L / 1 n i i Section 1.4. As with our treatment of the intersection product, we will give an appealingly intuitive characterization rather than a proof of existence. of a line bundle . Recall that we defined the first Chern class / c L on a variety X L 1 to be c . L / D Œ Div ./ç 2 A .X/ 1 n 1 D  . We define c for any rational section . L / L 0 for all i  2 . In this section we will of i c E . characterize Chern classes / for any vector bundle E and any integer i  0 . i We first sketch the situation in the case of a bundle generated by its global sections E (this circumstance is in fact the case in most of our applications, and in particular in the example of the 27 lines given above). Let D rank E . r In the case r D 1 already treated, the class c may be regarded as a measure of . L / 1 nontriviality: if c . L / D 0 , then L has a nowhere-vanishing section, whence L Š O . 1 X We extend this idea of measuring nontriviality using the idea of the “degeneracy locus” of a collection of sections — roughly, this is the locus where the sections become linearly dependent in the fibers of E . To make the meaning precise, we use multilinear algebra. E is trivial if and only if it has r everywhere-independent global sections The bundle  ; in this case, any set of r general sections will do. Thus a first measure ;:::; 0 1 r r general sections  are dependent. If we ;:::; of nontriviality is the locus where 1 0 r r  W O , then this is the locus write ! E for the map sending the i -th basis vector to  i X fails to be a surjection, or, equivalently, where the determinant of  is zero. We  where : It is the can interpret this as the vanishing of a special section of an exterior power of E zero scheme of the section V r  : E ^^ 2  1 r 0 V r r E rank Since E has rank 1 and the class of the zero locus is by , the bundle D V r E / c .X/ depending only on the isomorphism ; this is a class in A definition . 1 1 dim X c class of . We call it the first Chern class of E , written E / . E . 1

182 168 Chapter 5 Chern classes the scheme where i i general sections r More generally, we can consider for any of E fail to be independent, defined by the vanishing of V C i r 1 2  ^^  : E r i 0 This is called the of the sections  ;:::; . Since these degeneracy degeneracy locus i 0 r loci are central to our understanding of (and applications of) Chern classes, we should first say what we expect them to look like. To see how this should go, consider first the “degeneracy locus of one section.” A  is locally given by r functions f E ;:::;f of , so that by the principal ideal section r 1 is at most r . Moreover, if E is theorem the codimension of each component of V./  is a general section, then the function f generated by global sections and will 1 i C f not vanish identically on any component of the locus where ;:::;f vanish, and it i 1 follows that every component of has codimension exactly r . Under our standing V./ assumption of characteristic 0, a version of Bertini’s theorem tells us that V./ is reduced , for example in the case of a line bundle whose as well. (This may fail in characteristic p complete linear system defines an inseparable morphism.) It turns out that this is typical. Suppose that E is a vector bundle of rank r on a variety X , and let i Lemma 5.2. 1  i  r . Let  be an integer with ;:::; D D , and let be global sections of E i r 0 V. ^^  / be the degeneracy locus where they are dependent. i 0 r D has codimension > i . (a) No component of 0 If the  / are general elements of a vector space W  H of global sections . E (b) i , then generating i in X . is generically reduced and has codimension D E (a) This is Macaulay’s “generalized unmixedness theorem.” He proved it for the Proof: case of polynomial rings, and the general case was proved by Eagon and Northcott — see for example Eisenbud [1995, Exercise 10.9]. be an m -dimensional vector space of global sections of E that generate E , and W (b) Let ' W X ! G.m r;W / be the associated morphism sending p 2 X to the kernel of let W the evaluation map . If U  W is a subspace of dimension r E i C 1 spanned ! p 1 , then the locus V. ^^  by /  X is the preimage ' of the ;:::; .†/  r 0 r i i 0 Schubert cycle † .U/ Df ƒ 2 G.m r;W / j ƒ \ U g 0 ¤ i .m r/ of W meeting U nontrivially. By Kleiman’s theorem (Theorem 1.7), -planes in if U  W is general this locus is generically reduced of codimension i . i for vector bundles on . E / 2 A We can now characterize the Chern classes .X/ c E i smooth varieties X and integers i  0 :

183 Characterizing Chern classes Section 5.2 169 There is a unique way of assigning to each vector bundle E Theorem 5.3. on a smooth a class c. E / D 1 C c C2 . E / C c quasi-projective variety . E / X A.X/ in such a 1 2 way that: L is a line bundle on X then the Chern class of L is 1 (a) c C . L / , (Line bundles) If 1 1 is the class of the divisor of zeros minus the divisor of poles . L / where A c .X/ 2 1 L . of any rational section of  ;:::; (b) are global sections of E , and the (Bundles with enough sections) If 0 r i D where they are dependent has codimension i , then c D . E degeneracy locus / i i 2 .X/ . ŒDç A (c) (Whitney’s formula) If ! E 0 F ! G ! 0 ! is a short exact sequence of vector bundles on X then c. F / D c. E /c. G / 2 A.X/: (d) (Functoriality) If ' Y ! X is a morphism of smooth varieties, then W   // D c.' .c. E E //: ' . Although we will not prove Theorem 5.3 completely, we will explain some parts of the proof in Section 5.9 below. We will see below that these properties make many Chern class computations easy. Here are two tastes: , or more . If E is the direct sum of line bundles L (Sums of line bundles) Corollary 5.4 i generally has a filtration whose quotients are line bundles L , then i Y Y / D . c. L I / D c. .1 C c // E L i 1 i c . E / is the result of applying the i -th elementary symmetric function to the that is, i classes . . L / c i 1 Proof: This follows from a repeated application of Whitney’s formula. Corollary 5.5. If E is a vector bundle on X of rank > dim X , and E is generated by its global sections, then has a nowhere-vanishing global section. E Proof: By part (b), the degeneracy locus (vanishing locus) of one generic section has codimension > dim X . The strong Bertini theorem Corollary 5.5 has an interesting geometric consequence in the following strengthen- ing of Bertini’s theorem:

184 170 Chapter 5 Chern classes (Strong Bertini) Proposition 5.6 be a smooth, n -dimensional quasi-projective . Let X X the linear system of divisors on corresponding to the subspace variety, and D 0 . W / of sections of a line bundle L  D — that is, the H L . If the base locus of scheme-theoretic intersection \ D Z D D 2 D -dimensional subscheme of — is a smooth k < n=2 , then the general member of k , and X is smooth everywhere. the linear system D 4 X D P The inequality and D the linear system k < n=2 is sharp. For example, take 4  2 containing a fixed 2-plane Z . If of all hypersurfaces of degree D V.F/  P d Y is any hypersurface of degree containing Z , then the three partial derivatives of F d > 1 Z are identically zero and the two remaining partial corresponding to the coordinates on must have a common zero somewhere along is singular at ; thus Y derivatives of F Z . For an extension of this example, see Exercise 5.45. some point of Z Proof of Proposition 5.6: To begin with, the classical Bertini theorem tells us that the D of the linear system D is smooth away from Z . general member Z , suppose that D is the zero locus of a general To see that it is also smooth along 0  W  H  . section / . Since 2 vanishes on Z , it gives rise to a section d of the L   2 with the line ̋ L of the conormal bundle N tensor product N D I I = Z=X Z=X Z=X Z=X L as the differential of  along Z . The hypothesis that the d ; we can think of bundle 2 W generate the sheaf I sections ̋ L , together with Lemma 5.2 and the fact  Z=X  D k < n k D rank . that dim Z is nowhere zero. / , shows that d N Z=X 5.3 Constructing Chern classes A construction of Chern classes for a bundle of rank that is generated by global r c general . E / sections is implicit in Theorem 5.3 (b): r i C 1 is the degeneracy locus of i global sections. An alternative way of stating the same thing is often useful. We have already proved this in Lemma 5.2, but it is worth stating it here explicitly: on the smooth, quasi-projective Let E be a vector bundle of rank r Proposition 5.7. 0 , and let W  H variety . E / be an m -dimensional vector space of sections generating X r;W / E W X ! G.m ' denotes the associated morphism sending p 2 X to the . If kernel of the evaluation map W ! E is the pullback , then the i -th Chern class c / . E i p  E / c ' / . . D i i i of the Schubert class  2 A .G.m r;W // . i

185 Constructing Chern classes Section 5.3 171 This allows us to construct Chern classes for globally generated bundles, and we will see in Section 5.9.1 how to prove basic facts about Chern classes, such as Whitney’s formula, from this construction. To construct Chern classes for arbitrary bundles we use a different technique, the projectivization of a vector bundle. We will have much more to say about this construction in Chapter 9; for now we will simply state what is necessary to construct the Chern classes and to make use of a fundamental tool for computing with Chern classes, introduced in Section 5.4: the “splitting principle.” be a scheme, and let E be a vector bundle of rank r C 1 on Definition 5.8. . By X Let X E we will mean the natural morphism of projectivization the  P W WD Proj . Sym E  / ! X: E E over X we mean a morphism  projective bundle Y ! X that can be realized as By a W . for some vector bundle E over  X E P E .x;/ with x 2 X and  a one- Thus the closed points of correspond to pairs   dimensional subspace of the fiber E E of E . Ordinary projective space is of course x x X is a point and the special case in which is a vector space. E The bundle W P E ! X comes equipped with a tautological line bundle   WD S . O   1/ E ; E E P  Sym E -module obtained by shifting the constructed as the sheafification of the graded grading by , just as in the case of ordinary projective space. 1 Here is the result about projectivized vector bundles that serves to define the Chern classes in general: E be a vector bundle of rank r on a smooth variety X , and let Theorem 5.9. Let P E ! X be the projectivized vector bundle. Let  be the first Chern class of the  W  S dual of the tautological bundle on P E . S E E  (a) The flat pullback map W A.X/ ! A. P E / is injective.  (b) The element  2 A. P E / satisfies a unique monic polynomial f./ of degree r with  coefficients in .A.X// .  Let . The Chern be a vector bundle of rank r on a smooth variety X Definition 5.10. E classes . E / are the unique elements of A.X/ such that c i  1 r r  .  I . E / C  D CC  f./ c / c E r 1 that is, A. P E / D A.X/Œç=.f.//: In fact, this definition of Chern classes may be extended to singular varieties, as in Fulton [1984, Chapter 3], and this is a crucial element of the intersection theory of singular varieties: as we have seen (Example 2.22), it is simply not possible to define

186 172 Chapter 5 Chern classes products of arbitrary classes on singular varieties in general, but it is possible to define products with Chern classes of a vector bundle by restricting the vector bundle. For a proof of Theorem 5.9 in the smooth case, see Theorem 9.6; for the proof in general, see Fulton [1984, Chapter 3]. 5.4 The splitting principle For more complicated examples, we will use Whitney’s formula in conjunction with splitting principle a result called the , which may be stated as: (Splitting principle) . Theorem 5.11 Any identity among Chern classes of bundles that is true for bundles that are direct sums of line bundles is true in general. This remarkable result is really a corollary of the construction of projectivized vector bundles, applied via the next result: Let X be any smooth variety and E a vector . (Splitting construction) Lemma 5.12 on X . There exists a smooth variety Y and a morphism ' W Y ! X bundle of rank r with the following two properties:  ' W A.X/ ! A.Y/ is injective. (a) The pullback map  E on Y admits a filtration ' (b) The pullback bundle  E E  0 D  E ' D E  E 1 r 1 0 r  = ' E E with successive quotients E by vector subbundles  E locally free of 1 i i i rank 1. We may construct ' W Y Proof: X by iterating the projectivized vector bundle !  Writing WD P E we have a tautological subbundle S /: Y  construction: First, on E .  1 1 E Q we have exact sequences for the quotient, we next construct Y Y WD P Q . On 1 2 1 2     ! . S Q /   !  0  0  E ! 1 1 E Q Q Q 1 1 1 and  0 0: !  S ! ! Q Q ! 1 2 2 Q 1 Continuing this way for r 1 steps we get a space Y WD Y E such that the pullback of r to admits a filtration whose successive quotients are line bundles. Y 1  A / . P E Finally, by Theorem 5.9, there is a class in the Chow ring of any 2  W P E ! X that restricts to the hyperplane class on each fiber. By projective bundle the push-pull formula, if E has rank r then for any class ̨ 2 A.X/ we have  r 1  ̨/ D ̨; .     is injective. W A.X/ ! A. P E / from which we see that the pullback map

187 Using Whitney’s formula with the splitting principle 173 Section 5.4 We will study projective bundles much more extensively in Chapter 9; in particular, we give a more fleshed out version of this argument in the proof of Lemma 9.7. Proof of Theorem 5.11: With notation as in the theorem, we can use Whitney’s formula (part (c) of Theorem 5.3) and our a priori definition of the Chern class of a line bundle to describe the Chern class of the pullback: r Y  E D c.' / / c. E = I E i i 1 D 1 i E . by the first part of the lemma, this determines the Chern classes of 5.5 Using Whitney’s formula with the splitting principle We will now illustrate the use of Whitney’s formula with the splitting principle. A first consequence is that the Chern classes of a bundle vanish above the rank, something we saw already in the case of bundles with enough sections. Example 5.13 . If E is a vector bundle of rank r , then c i > r . E / D 0 for (Vanishing) . i L r then, since Reason: If , / L for line bundles L split as c. L / D 1 C c . L E i 1 i i i 1 i D Whitney’s formula would imply that r Y c. E / D //; L .1 C c . 1 i 1 D i which has no terms of degree > r . L (Duals) . If E D Example 5.14 L , then i Y Y   c. E C c / . L D //; // D L .1 c .1 . 1 1 i i  c . L since / D c is a line bundle. Given this, Whitney’s formula gives . L / when L 1 1 us the basic identity  i . E / c D . 1/ /: c E . i i By the splitting principle, this identity holds for any bundle. (Determinant of a bundle) . By the determinant det E of a bundle E we Example 5.15 V rank E WD det mean the line bundle that is the highest exterior power E . We have E already observed that if E is globally generated, then c ; this was one . det E / D c / . E 1 1 of our motivating examples. The splitting principle and Whitney’s formula allow us to

188 174 Chapter 5 Chern classes L N deduce this for arbitrary bundles: If we assume that , then det E D D L E and L i i hence X det E / D c c . . L I / D c / . E 1 1 1 i the splitting principle tells us this identity holds in general. (Symmetric squares) . E is a bundle of rank 2. If E splits as Suppose that Example 5.16 E D L ̊ M of line bundles L and M with Chern classes c a direct sum . L / D ̨ and 1 / M / D ˇ then, by Whitney’s formula, c. E . D .1 C ̨/.1 C ˇ/ , whence c 1 E . c / D ̨ C ˇ and c ̨ˇ: . E / D 2 1 Further, we would have 2 ̋ 2 ̋ 2 Sym M ̊ . L ̋ M / ̊ E D L ; from which we would deduce 2 E / D .1 C 2 ̨/.1 C c. C ˇ/.1 C 2ˇ/ Sym ̨ 2 2 2. ̨ C ˇ/ C .2 ̨ ˇ/: C 8 ̨ˇ C 2ˇ D / C 4 ̨ˇ. ̨ C 1 C E This expression may be rewritten in a way that involves only the Chern classes of : As the reader may immediately check, it is equal to  2 . E C . 2c . E / 2c C 4c C / E / 1 C 4c . E /c . E /: 2 2 1 1 1 2 By the splitting principle, this is a valid expression for c. E / whether or not E Sym actually splits. We could use the same method to give formulas for the Chern classes of any symmetric or exterior power — or of any multilinear functor — applied to vector bundles whose Chern classes we know. Together, the splitting principle and Whitney’s formula give a powerful tool for calculating Chern classes, as we will see over and over in the remainder of this text; see Exercises 5.30–5.35 for more examples. 5.5.1 Tensor products with line bundles As an application of the splitting principle, we will derive the relation between the Chern classes of a vector bundle E of rank r on a variety X and the Chern classes of the tensor product of with a line bundle L . E To do this, we start by assuming that E splits as a direct sum of line bundles r M E I M D i D 1 i

189 Using Whitney’s formula with the splitting principle Section 5.5 175 1 ̨ . M / 2 A c .X/ be the first Chern class of M , so that let D i 1 i i r Y / c. E D .1 ̨ C /: i 1 D i In other words, the elementary symmetric polynomials in the are the Chern classes ̨ i : of E C CC ̨ D c . E /; ̨ ̨ r 2 1 1 X c /; ̨ E ̨ . D 2 j i  1 i

190 176 Chapter 5 Chern classes This is just a matter of collecting the terms of degree l ̨ Proof: and degree k l in the i ˇ in the expression (5.1): we write in r Y X l r C ̨ C .1 D ˇ/ ̨  ̨ C .1 ˇ/ i i i 1 l 1 i D r 1  i < 

191 Tautological bundles Section 5.5 177 There is one other case in which we can give a closed-form expression for a Chern class of a general tensor product: We will see, in Chapter 12, a formula for the top Chern . E ̋ F / of a tensor product of bundles of ranks e and f c class . ef There is also a different approach that allows us to express the characteristic classes of a general tensor product more comprehensibly: The Chern character Ch . E / of a is a certain formal power series in the Chern classes of E , with rational vector bundle E coefficients, that satisfies the attractive formulas E ̊ F / D Ch . . / C Ch . F /; Ch E . E ̋ F / D Ch . E /  Ch . F /: Ch See Section 14.2.1 for more information. 5.6 Tautological bundles We have seen how the splitting principle, in conjunction with Whitney’s formula, allows us to express the Chern classes of bundles in terms of simpler ones. To apply this, of course, we need to have a roster of basic bundles whose Chern classes we know; in this section we will calculate the Chern classes of some of these. 5.6.1 Projective spaces r O on projective space We start with the most basic of all bundles: the bundle .1/ P r P . We have r 1 r . D  2 A .1// . P O /; c 1 P  is the hyperplane class; similarly, where 1 r r c / .n// D n   2 A . . P O 1 P for any 2 . n Z This in turn allows us to compute the Chern class of the universal quotient bundle r r P Q on D P V , from the exact sequence : If P r r D O ! 0 . 1/ ! V ̋ O 0; ! S Q ! P P we have 1 1 r 2 D  Q CC :  C D c.  / C D 1 r  1 1// . O c. P Note that we could also arrive at this directly from the description of Chern classes v 2 V gives rise to a global section  of as degeneracy loci of sections: An element the bundle Q ; given k elements v ;:::; ;:::;v  2 V , the corresponding sections 1 1 k k

192 178 Chapter 5 Chern classes r k 1 Q exactly when x lies in the P 2 P will be linearly dependent at a point of x spanned by the Dh ;:::;v W i V corresponding to the subspace v . Thus v i 1 k 1 C k r 1 C k k r 1 r D . ç P Œ D / 2 A Q  c /: P . r k C 1 5.6.2 Grassmannians G D G.k;n/ of k Let us consider next the case of the Grassmannian -planes in an n V , and its universal sub and quotient bundles S and Q . -dimensional vector space Q , since this bundle is globally generated, so that we can determine We will start with 2 give rise its Chern classes directly as degeneracy loci. Specifically, elements V v of Q simply by taking their images in each quotient of V to sections  ; that is, for a -plane ƒ  V , we set k D v 2 V=ƒ: .ƒ/ v ;:::;v Now, given a collection , the corresponding sections will fail to be 2 V m 1 ƒ 2 G exactly when the corresponding v independent at a point 2 V=ƒ are dependent, i ;:::;v intersects the span W D h v in a nonzero which is to say when V i  ƒ m 1 subspace — that is, when P ƒ \ P W ¤ ¿ : We may recognize this locus as the Schubert cycle † , from which we .W / m 1 k C n conclude that the Chern class of Q is the sum / D 1 c.  Q C  : CC  C 2 1 n k Q , the universal subbundle S does not have nonzero global sections, so we Unlike cannot use the characterization of Chern classes as degeneracy loci. But the dual bundle    V S is a linear form, we can define a section  does: If S l by restricting l to 2 of k -plane ƒ  V in turn; in other words, we set each D l j : .ƒ/ ƒ  m independent linear forms l , the corresponding sections ;:::;l Now, if we have 2 V m 1  S will fail to be independent at the point ƒ 2 G — that is, some linear combination of l fails to intersect the common will vanish identically on of the — exactly when ƒ ƒ i zero locus U of the l properly, that is, when i dim . P ƒ \ P U/  k m: Again, this locus is a Schubert cycle in , specifically the cycle † .U/ , and we G 1;1;:::;1 conclude that  S I / c. 1 C   C  CC D 1;1 1 1;1;:::;1

193 Chern classes of varieties Section 5.6 179 from this we can deduce in turn that k S  c. C  / CC . 1/ 1  D : 1;1;:::;1 1 1;1 / can also be deduced from our knowledge of c. c. / Note that this description of S Q and Corollary 4.10. 5.7 Chern classes of varieties are its tangent bundle T The most important vector bundles on a smooth variety X X  . Their Chern classes are so important in geometry and its dual, the cotangent bundle X Chern class of . that the Chern class of the tangent bundle is usually just called the X For example, if X is a smooth curve then its tangent bundle is a line bundle, so its C c is the anticanonical class, . T Chern class has the form 1 . Here c / . T / D c . / 1 1 1 X X X X 2 , where g is the genus of 2g . In general, if X is a smooth complex whose degree is n then Theorem 5.21 below says that deg c / . T is projective manifold of dimension n X X . the topological Euler characteristic of 5.7.1 Tangent bundles of projective spaces n of projective We start by calculating the Chern classes of the tangent bundle T P space. This is straightforward, given the Euler sequence of Section 3.2.4: We have n C 1 n n n O ! ! .1/ O 0 0 ! T ! P P P and hence n C 1 n .1 T / c. / D ; C P n 1 A . where P  / is the hyperplane class. 2  D Hom . S ; Q / D S T ̋ Q , We could also derive this from the identification n S D O are the universal sub and quotient bundles, by applying where . 1/ and Q P Proposition 5.17. Note that this calculation implies the algebraic/projective version of the “hairy n n , there does not exist a nowhere-zero . T 0 coconut” theorem: Since / D .n C 1/ c ¤ n P n vector field on P . 5.7.2 Tangent bundles to hypersurfaces We can combine the formula above for the Chern classes of the tangent bundle to r P and Whitney’s formula to calculate the Chern classes of the tangent projective space n of degree . d P X bundle to a smooth hypersurface 

194 180 Chern classes Chapter 5 To do this, we use the standard normal bundle sequence n n T T ! j 0 ! N 0 ! ! X X P P X= and the identification n n D O D .d/ .X/ j O N X X P P X= established in Section 1.4.2. Letting X of the hyperplane denote the restriction to  X n P class on , we can write n c. T j / X P T / c. D X n N X= P 1 C n C .1 /  X D d C 1 X     C 1 n 2 2 2 C D 1/  C 1  .n C C d .1 C d /: C X X X X 2 We can generalize this calculation to complete intersections: (Chern classes of complete intersections) . Suppose that Example 5.19 n D \\ Z  P Z X 1 k k hypersurfaces of degrees d is the complete intersection of ;:::;d defined by forms 1 k F of degrees d . The relations among the F are generated by the Koszul relations i i i . This means that if we restrict to F F F D 0 Y , where the F vanish, we get F j i i j i M 2 j = D I I I Y D /; d O . i Y Y=X Y=X Y=X i L n n .d . D of X in P so the normal bundle is a direct sum N D N O / N i X P X= Applying Whitney’s formula, we get 1 C n  .1 / C X Q : c. / D T X .1  / C d i X 5.7.3 The topological Euler characteristic of a manifold M is by definition Recall that the topological Euler characteristic P i i i  1/ .M/ dim WD H is the singular cohomology .M I Q / , where H . .M I Q / top Q group. When M is a smooth projective variety over C , it may be regarded as a manifold with respect to the classical, or analytic, topology, so  .M/ makes sense in this case. top

195 Chern classes of varieties Section 5.7 181 ́ (Poincar . If M is a smooth compact orientable manifold Theorem 5.20 e–Hopf theorem) is a vector field with isolated zeros, then and  X  .M/ D ./: index top x g .x/ 0 j f D x A beautiful account of this classic result can be found in Milnor [1997]. Now is a smooth complex projective variety. If the tangent bundle T is M suppose that X that vanishes at only finitely many generated by global sections, then it has a section  points, and vanishes simply there. Since this section is represented locally by complex analytic functions, its index at each of its zeros will be 1, and we may replace the sum ́ in the Poincar e–Hopf theorem by the number of its zeros — in other words, the degree of the top Chern class of . An elementary topological argument (see, for example, T X Chapter 3 of Griffiths and Harris [1994]) shows that this is true more generally: X n -dimensional projective variety, then If Theorem 5.21. is a smooth .X/ D  c deg . T /: n top X n 1 C n n Since / T c. . ) D .1 C / P (Euler characteristic of Example 5.22 , where  D P n O is the class of a hyperplane, we deduce that c . .1// 1 P n n P  / D deg .c 1: . T C D // . n top n P n 2i . Of course this is immediate from the fact that H ; Q / D Q for i D 0;:::;n while P n 2i 1 C P H ; Q / . 0 for all i . D (Blow-up of a surface) . Sometimes one can use Theorem 5.21 to compute Example 5.23 a Chern class. For example, the blow-up Y of a complex surface X at a point p can be described topologically as the union of X D with a tubular neighborhood of the n 1 P . Thus exceptional curve, which is a copy of 1 .X/ D  C .X/  .X/ .p/ C  D . P  / D   .X/ 1 C 2 1; top top top top top top c deg T . (One can generalize this formula / D deg . and we deduce that . T c / C 1 2 2 Y X algebraically, and identify the class c. T / , by using the Chern classes of coherent sheaves Y that are not vector bundles; see for example Section 14.2.1, and, for the computation, Fulton [1984, Section 15.4].) be a smooth hyper- (Euler characteristic of a hypersurface) . Now let X Example 5.24 n P . From the normal bundle sequence surface of degree d in n n ! 0 T j T ! ! 0 N ! X X P X= P n and the fact that , we have N Š O .d/ X X= P 1 C n  / .1 C X 2 2 1 C n T / c. D /: ..1 C  C / / . d /.1 D d C X X X X C .1 d / X

196 182 Chapter 5 Chern classes X Taking the component of degree dim 1 , we get D n n 1   X C 1 n i 1 i n D T / c . 1/ .  : d n 1 X X 1 n i i 0 D n 1  general is the number of points of intersection of n 1 Since the degree of X n 1 D d . Thus, finally,  hyperplanes on the 1/ .n -dimensional variety X , we have X n 1   X n C 1 i 1 i C  .X/ D . T deg // .c D 1/ . : d top n 1 X i n 1 i D 0 We can get still more from this formula: The Lefschetz hyperplane theorem (see Section C.4) tells us that the integral cohomology groups of X are all equal to the corre- n 1 .X/ ; sponding cohomology groups of projective space, except for the middle one H i / dim that is, the Betti numbers H b .M I Q D other than b are 1 in even dimen- 1 n Q i sions and 0 in odd. (In fact, the analogous statement is true for any smooth complete intersection: All the cohomology groups except the middle are equal to those of projec- tive space.) Thus the Euler characteristic determines the middle Betti number b . In 1 n Table 5.1, we give the results of this calculation in a few of the cases where it is most frequently used. hypersurface  b n 1 4 quadric surface 2 cubic surface 9 7 quartic surface 24 22 quintic surface 55 53 quadric threefold 4 0 cubic threefold 10 6 60 quartic threefold 56 200 204 quintic threefold quadric fourfold 2 6 cubic fourfold 27 23 Table 5.1 Euler characteristics of favorite hypersurfaces. It is interesting to compare this computation with what we already knew: A smooth 1 1 3 is isomorphic to P  P quadric surface in , from which we can see directly both P 3 P is the Euler characteristic and the second Betti number; a smooth cubic surface in 2 P C at six points, so the Euler characteristic is 3 the blow-up of 6 ; the quadric fourfold ̈ may also be viewed as the Pl ucker embedding of the Grassmannian G .1;3/ , whose cohomology has as basis its six Schubert cycles, and whose middle cohomology in .   and particular has basis given by the two Schubert cycles 1;1 2

197 Generators and relations for A.G.k;n// 183 Section 5.7 5.7.4 First Chern class of the Grassmannian In theory, we should be able to use the identification of the Chern classes / c. S Q to derive the Chern class of the tangent bundle T , which by Theorem 3.5 is c. / and G  S ; Q isomorphic to D S Hom ̋ Q . In general, unfortunately, this knowledge remains . / theoretical: As we indicated in Section 5.5.2, the formula for the Chern class of the tensor product of two bundles of higher rank is complicated. But we can at least use  , . T  / ; since c D . S Q / D c Proposition 5.18 to give the first Chern class . c / 1 1 1 1 G we have: The first Chern class of the tangent bundle of the Grassmannian Proposition 5.25. G D G.k;n/ is : . T c / D n   1 1 G K We see from this also that the canonical class of G is n . Note that this 1 G n 1 P k D 1 agrees with our prior calculations in the case , and in the of projective space case k D 2 and n D 4 , where the Grassmannian G.2;4/ may be realized as a quadric 5 hypersurface in and we can apply the results of Section 5.7.2. P A.G.k;n// 5.8 Generators and relations for We have seen in Corollary 4.7 that the Chow ring of the Grassmannian is a free abelian group generated by the Schubert cycles. It follows moreover from Giambelli’s special Schu- formula (Proposition 4.16) that it is generated multiplicatively by just the , which are the Chern classes of the universal subbundle. We will now see that bert cycles Whitney’s formula and the fact that the Chern classes of a bundle vanish above the rank of the bundle provide a complete description of the relations among the special Schubert cycles, and that these form a complete intersection. The Chow ring of the Grassmannian G.k;n/ has the form Theorem 5.26. D Z A.G.k;n// ç=I; ;:::;c Œc 1 k i c 2 A where .G.k;n// and the i -th Chern class of the universal subbundle S is the i ideal I is generated by the terms of total degree n k C 1;:::;n in the power series expansion 1 2 ;:::;c ŒŒc Z 2 C / D 1 .c c CC c CC / çç: .c 1 1 1 k k k CC c 1 c C 1 k Moreover, I is a complete intersection.

198 184 Chapter 5 Chern classes G.3;7/ Z For example, the Chow ring of ;c is ;c Œc ç=I , where I is generated by 1 3 2 the elements 5 3 2 2 c 4c C 3c C c c C 3c 2c ; c c C 1 2 3 2 3 1 1 2 1 6 4 2 2 3 3 2 c 5c C 6c C c c C c C 4c 6c c C c c c C ; 2 1 3 3 2 1 1 1 2 2 1 3 7 5 3 2 3 4 2 2 2 c C 6c c 10c C c C 4c c C 5c 12c C c c c 3c C 3c c c C ; 2 1 3 3 2 1 3 1 1 1 2 2 1 1 2 3 and these elements form a regular sequence. The proof of Theorem 5.26 uses two results from commutative algebra, Proposi- tion 5.27 and Lemma 5.28, which are variations on some frequently used results; readers may wish to familiarize themselves with them before reading the proof of Theorem 5.26. of a finite-dimensional graded algebra T Recall that the socle is the submodule of el- ements annihilated by all elements of positive degree. In particular, if d is the largest T . For a somewhat different ¤ 0 , then the socle of T degree such that T contains d d proof, and the generalization to flag bundles of arbitrary vector bundles, see Grayson et al. [2012]. Set A D A.G.k;n// Proof: t part of the power series for the degree- i and write i 2 1=.1 C CC c c / , so that t . D 1; t ;::: D c c ; t D c expansion of 1 2 1 0 1 2 k 1 D .t Let . ç=J ;:::;c Œc ;:::;t J / , and let R D Z n 1 C k k 1 n Corollary 4.10, which is the special case of Whitney’s formula (Theorem 5.3, part (c)) applied to the tautological sequence of vector bundles n S ! O 0 ! ! Q ! 0 G.k;n/ Q , shows that c. Q / D 1=c. S / . Since Q has rank n k , the classes c on . G.k;n/ / i vanish for all k , and it follows that there is a ring homomorphism i > n W R ! A; t ' 7! c /: . Q i i i c  goes to the Schubert cycle c Under this homomorphism, the class . S / D . 1/ i i i 1 1 i times). Recall from Corollary 4.2 (where the subscript denotes a sequence of repeated n k is the class of a point. that  k 1 F the sequence t We will show that for any field is a regular sequence ;:::;t n C k 1 n R ̋ in F , and the induced map Z 0 ̋ ' WD F ' Z 0 0 A WD R ! R F WD A ̋ ̋ F Z Z is an isomorphism. Since is a finitely generated abelian group, the surjectivity of ' A follows from this result using Nakayama’s lemma and the two cases D Z =.p/ and F F D Q . On the other hand, by Corollary 4.7, A is a free abelian group so, as an abelian group, is free. Thus the kernel of ' is a summand of R , so the injectivity of ' '.R/ 0 F ' follows from the injectivity, for every choice of . Using Lemma 5.28 inductively, , of this also follows that t is a regular sequence, proving the theorem. ;:::;t n k C 1 n

199 Generators and relations for A.G.k;n// 185 Section 5.8 0 t have is a regular sequence in R ;:::;t it suffices, since the t To show that i n C 1 n k positive degree, to show that FŒc ç=J ;:::;c 1 k was arbitrary it suffices, by the Nullstellensatz, to F has Krull dimension zero. Since F are substituted for the c in such a way that show that, if f 2 , 0 D DD t t i i n 1 C k n then all the f are zero. i 2 k Indeed, after such a substitution we see that x C f D x 1=.1 CC f C x f / 2 1 k n p.x/ is a polynomial of degree  is a rational , where k and q.x/ p.x/ C q.x/ n C 1 function vanishing to order at least at 0. We may rewrite this as 2 k C f x C f / x 1 CC f p.x/.1 x 2 1 k q.x/: D k CC f C x f x 1 1 k However, the denominator of the left-hand side is nonzero at the origin, and the numerator . Since q.x/ vanishes to order at least has degree at most C 1 at the origin, both sides n n p.x/ D 1;q.x/ D 0 , and thus each f , as required. must be identically zero; that is 0 D i Combining this information with Proposition 5.27, we get:  n 0  . ) is R F (as a vector space over The dimension of k 0  The highest degree R d . ¤ 0 is k.n k/ such that d  Since a complete intersection is Gorenstein (Eisenbud [1995, Corollary 21.19]), 0 0 R contains every nonzero ideal of R . k/ k.n  n 0 ' ; . By Corollary 4.13, the rank of A.G.k;n// We now return to the map is also k 0 ' is an isomorphism, it suffices to show that its kernel is zero. We thus to show that n k 0 0 . Since know that is in the image of ' / , so Ker ' . does not contain R k k.n k/ 1 0 0 0 is the socle of R , the kernel of ' must be zero. R k/ k.n We have used the following two results from commutative algebra: Proposition 5.27. F is a field and that Suppose that D FŒx T ;:::;x ;:::;g ç=.g / 1 1 k k deg x D D ı is a zero-dimensional graded complete intersection with > 0 and deg g i i i  is > 0 . The Hilbert series of T i Q 1 k  X i / d .1 1 D i u : WD H d dim D .d/ T u T F Q k ı i .1 d / 0 u D 1 i D P P k k is , and the dimension of ı T  T is The degree of the socle of i i i D 0 D i 0 Q k . 1/ i i D 1 D T dim : F Q k .ı 1/ i i 1 D

200 186 Chapter 5 Chern classes FŒx We begin with the Hilbert series. The polynomial ring ç is the ;:::;x Proof: 1 k , so FŒx tensor product of the one-variable polynomial rings ç i 1 : WD .d/ H Q FŒx ;:::;x ç 1 k k ı / d .1 1 D i i We can put in the relations one-by-one using the exact sequences g i C 1 ç=.g / ;:::;g ;:::;g /. ! ç=.g / FŒx ;:::;x ;:::;x 0 ! FŒx  1 1 i 1 i i 1 k k FŒx ! ;:::;x 0; ç=.g ! ;:::;g / 1 C 1 1 i k and using induction we see that Q k  i .1 / d 1 D i D .d/ D H .d/ H : T Q ;:::;x / ;:::;g ç=.g FŒx 1 1 k k k ı i / .1 d 1 i D P P k k ı WD s  A priori this is a rational function of degree . Since we know i i i D i D 1 1 from the computation above that T F , the is a finite-dimensional vector space over s , so the largest Hilbert series must be a polynomial. Thus it is a polynomial of degree T is nonzero is s . degree in which k H 1 .d/ at d D The dimension of . The product .1 d/ T obviously is the value of T divides both the numerator and the denominator of the expression for the Hilbert series above. After dividing, we get Q P 1  k i j d D i 1 j D 0 H D .d/ : T Q P 1 ı k i j d 1 D i j D 0 Setting d D 1 in this expression gives us the desired result. The other result from commutative algebra that we used is a version of the fact that regular sequences in a local ring can be permuted (Eisenbud [1995, Corollary 17.2]). The same result holds in the local case when every element of the regular sequence has positive degree, but the case we need is slightly different, since one element of the regular sequence is an integer. The result may also be viewed as a variation on the local criterion of flatness (Eisenbud [1995, Section 6.4]). Lemma 5.28. R is a finitely generated graded algebra over Z Suppose that , with algebra generators in positive degrees, and that f 2 R is a homogeneous element. If R is free as a Z -module and f ̋ is a Z =.p/ is a monomorphism for every prime p , then f Z monomorphism and is free as a Z -module as well. R=.f / Proof: R is free, so is every submodule; in particular fR is free, and the kernel Since K of multiplication by f is a free summand of R . It follows that K ̋  Z =.p/ Z R ̋ Z =.p/ . Since this ideal is obviously contained in the kernel of multiplication by f Z

201 Steps in the proofs of Theorem 5.3 Section 5.8 187 0 Z , we see that K ̋ ̋ Z =.p/ R =.p/ . Since K is free, this implies that K D 0 on D Z is a nonzerodivisor on R , and the diagram as well; that is, f f - - - - 0 R. R=fR 1/ R 0 p p p ? ? ? f - - - - R 0 R. 0 R=fR 1/ ) shows that is a nonzerodivisor on snake lemma has exact rows. A diagram chase (the p p was an arbitrary prime, R=fR is a torsion-free abelian group. Since R R=fR . Since is homogeneous, R=fR is a direct sum of finitely generated is finitely generated and f abelian groups, and torsion-freeness implies freeness. 5.9 Steps in the proofs of Theorem 5.3 Though the locus D in item (b) of Theorem 5.3 depends very much on the sections  chosen, Theorem 5.3 asserts that the class ŒDç does not, so long as it has the “expected” i k r of D codimension. This point is worth understanding directly: We start with the case 0 and  r are two sections of E whose zero loci are of codimension the top Chern class. If ,  0 V./ and V. / with the family then we can interpolate between 1 0 0 D .p/ t  X j s.p/ C : g P 2 Df ˆ .Œs;tç;p/ 0 V./ / : Since ˆ has codimension V. and This gives a rational equivalence between 1 ˆ intersecting the fibers over 0 or 12 P at most must r everywhere, components of 1 , and taking the union of these components we get a rational equivalence P dominate 0 and that of between the class of the zero locus of  .  0 0 ;:::;  The same argument works in the general case: If both  and ;:::; i r 0 0 r i are collections of sections with degeneracy loci of codimension i , we set ̊ 1 0 0 t / p 2 V.s C j X ^^ s  C t P .Œs;tç;p/ D : ˆ 2 i 0 r 0 i r 1 ˆ dominating P Using Lemma 5.2, one can show that the components of give a rational 0 0 ^^ ^^  / equivalence between  / and V. . V. r 0 i 0 i r 5.9.1 Whitney’s formula for globally generated bundles Though we will not prove the existence of Chern classes satisfying the properties of Theorem 5.3, it is instructive to see how Whitney’s formula (property (c) in Theorem 5.3) follows in the case of a globally generated bundle from facts about the Grassmannian.

202 188 Chern classes Chapter 5 F Suppose that . Denote the and are globally generated bundles on a variety E X by F and f respectively. We will show that and E ranks of e F / D c. E /c. F / 2 A.X/; c. E ̊ or equivalently X ̊ F / D . c / F . /c E E . c j i k k C j D i  0 . for i D i i D e C f we can see this at once: In the first of In the extreme cases 1 and these cases, Whitney’s formula says that / F . D c . E ̊ C c c . F /: E / 1 1 1 0 0  H  . E / and ;:::; ;:::; If 2 H 2 . F / are general sections, then the degener- 1 e 1 f acy locus of the f sections e C 0 / F . E ̊ H 2 ;0/;.0; . /;:::;.0; ;0/;:::;. / e 1 1 f V. . ^^  is the sum, as divisors, of the degeneracy loci / and V. / ^^  e 1 1 f Here we are using the identification V V V f e C e f E D E ̋ ̊ F F : . / In the second case, Whitney’s formula says that /: . c ̊ F / D c F . E /c . E e f f e C To see this, let  be general sections of E and F respectively. The zero locus and  0 F .;/ . E ̊ H / is then the intersection of the zero loci of the section V..;// 2 V./ ; by Lemma 5.2 applied to F j V./ and , it will have the expected codimension V./ f and the equality above follows. e C For the general case we adopt the alternative characterization of Chern classes 0 is an V of Proposition 5.7: If  n -dimensional subspace generating E , we H . E / of sections ' W X ! G.n e;V / sending p to the subspace V have a map  V p V  ; the k -th Chern class of E is then the pullback vanishing at ' p  of the Schubert class k V k . 2 A  .G.n e;V // k 0 0 H Let . E / and W  V  . F / be generating subspaces, of dimensions n and m ; H 0 ' is and ' / be the corresponding maps. The subspace V ̊ W  H ̊ . E let F W V again generating, and gives a map ' W /: ̊ f;V W X ! G.n C m e W V ̊ Let '  ' G.m f;W / X ! G.n e;V /  W W V

203 Steps in the proofs of Theorem 5.3 Section 5.9 189 be the product map. We have ' D  ı .'  ' /; W V ̊ V W G.n e;V /  G.m W ! G.n C m e f;V ̊ W / is the map sending where  f;W / and W to their direct sum. V a pair of subspaces of Lemma 5.29. and W be vector spaces of dimensions n and m . For any s Let t , let V and W  G.t;W / ! G.s C t;V ̊ W /  G.s;V / and to Ä . If ̨ ̊ ˇ are the projection maps on .ƒ;Ä/ be the map sending a pair ƒ G.t;W / , then, for any k , G.s;V /  X    .  ̨ / :   ˇ D  i j k C j D k i ' , Given Lemma 5.29, Whitney’s formula (in our special case) follows: with ' W V and ' as above, we have W ̊ V X X    D ' ̊ . E c F / D / . ' . /' D . E c . F /: . / /c i i j j k k ̊ V W V V j k D C i k j C D i Note that Lemma 5.29 is a direct (and substantial) generalization of the calculation r r P  in Section 2.1.4 of the class of the diagonal P Å . Specifically, if V D W ,  r r n D r C 1 and s D t D 1 , then the diagonal Å  P m  P D is the preimage under the map  V  P V ! G.2;V ̊ V / of the Schubert cycle † W .V / of 2-planes P n intersecting the diagonal V ̊ V . Thus Lemma 5.29 in this case yields the formula V  of Section 2.1.4. r r P  P , we will use As in the earlier calculation of the class of the diagonal in Proof: G.s;V /  G.t;W / can be the method of undetermined coefficients. Note that the product   ̨   ˇ  stratified by products of Schubert cells; thus, by Proposition 1.17 the products a b A.G.s;V /  G.t;W // . (In particular, we have A span .G.s;V /  G.t;W // D Z .) 0 Moreover, intersection products in complementary dimensions between classes of this type again have a simple form: We have 8 i for all s n D c C a if 1 and < s i 1 i C      t for all j; D b m C d  ˇ / . ̨  deg /. ̨ D  ˇ  C 1 m j j a c d b : 0 otherwise : From this, we see that A.G.s;V /  G.t;W // is freely generated by the classes   ̨ ˇ  , and that the intersection pairing in complementary dimensions is nondegen-  a b erate. Thus, to prove the equality of Lemma 5.29 it will be enough to show that both sides      ˇ ̨ have the same product with any class . Specifically, we need to show that for a b   ) products k  ˇ ̨  of dimension k (that is, with j a jCj b jD s.n s/ C t.m t/  a b

204 190 Chapter 5 Chern classes we have 8 i/ s 1 if a D .n s;:::;n s;n and ˆ ˆ < t j/ t;m t;:::;m b D .m    deg / D   .  ˇ   ̨ a b k C i D j for some k; ˆ ˆ : : 0 otherwise j a jCj b jD We start with the “otherwise” half. Note that, by the dimension condition b and i/ s D s;n , the condition a D .n s;:::;n k t/ t.m C s/ s.n i t;:::;m t;m t j/ for some C j D k is equivalent to saying that the sum .m a ; in all other cases it and b k is a C b t of the last two indices n s C m D t t s s will be strictly greater. Start by choosing general flags    V V D V , W and    W W D m n 1 1  U U . Then W D V ̊ m 1 n C . † / f ƒ  V V ƒ  V j g n a a s and / . W † f Ä  W j ƒ  W g ; m a b t so 1 1 \ ˇ  † † g / f  : V ̊ . ̨ j   V W ̊ W a n m a a b t s But † ; . U / Df   V ̊ W j  \ U g 0 ¤ C m t k C 1 s n k a 0 b D > n s C m t k , then .V C and, if U \ ̊ W / ; a m s a t n s C m t k C 1 n t s thus 1 1 1 ̨ † ¿ \ ˇ \ † † D  a b k and the product of the corresponding classes is zero. a D .n s;:::;n s;n s Similarly, in case and b D .m t;:::;m t;m t j/ i/ for some i C j D k , the intersection U D .V will U \ / ̊ W m a a n s C m t k C 1 n s t be one-dimensional. Since   ˇ and ƒ  V 1 s ˇ / D † ƒ  V . V ˇ a ƒ V  n a s and   ˇ and ƒ  W 1 t ˇ ;  Ä D / W . W † ˇ b  W ƒ m a t 1 1 1  \ † will consist of the single point \ ˇ † we see that the intersection † ̨ a b k W ƒ  V is the span of V .ƒ;Ä/ and the projection  .U/ and likewise Ä  , where s 1 1  is the span of . That the intersection is transverse follows .U/ W and the image 1 2 t from Kleiman’s theorem in characteristic 0, and from direct examination of the tangent spaces in general.

205 Exercises Section 5.9 191 5.10 Exercises Many of the following exercises give applications of Whitney’s formula and the splitting principle. We will be assuming the basic facts that if f e M M E D F D and L M i i 1 i D 1 i D are direct sums of line bundles, then M k ; Sym D E L ̋ ̋ L i i 1 k   r 1 i i  1 k M V k D E L ̋ ̋ L ; i i 1 k  < 1 

206 192 Chapter 5 Chern classes Find all the Chern classes of the tangent bundle Exercise 5.38. T of a quadric hyper- Q 5  . Check that your answer agrees with your answer to the last exercise! Q surface P m n  Exercise 5.39. Calculate the Chern classes of the tangent bundle of a product P P of projective spaces Find the Euler characteristic of a smooth hypersurface of bidegree .a;b/ Exercise 5.40. m n P P . in  n n  2 the tangent bundle Exercise 5.41. Using Whitney’s formula, show that for T P of projective space is not a direct sum of line bundles. Exercise 5.42. Find the Betti numbers of the smooth intersection of a quadric and a 4 5 , and of the intersection of three quadrics in cubic hypersurface in . (Both of these P P 3 , which are diffeomorphic to a smooth quartic surface in P .) are examples of K3 surfaces 5 Find the Betti numbers of the smooth intersection of two quadrics in Exercise 5.43. . P quadric line complex , about which you can read more in Griffiths and This is the famous Harris [1994, Chapter 6]. 4 Exercise 5.44. Show that the cohomology groups of a smooth quadric threefold Q  P 3 P are isomorphic to those of Z in even dimensions, 0 in odd), but its cohomology ( 2 H .Q; Z / is twice the generator of ring is different (the square of the generator of 4 .Q; ). (This is a useful example of the fact that two compact, oriented manifolds / H Z can have the same cohomology groups but different cohomology rings, if you are ever teaching a course in algebraic topology.) 4 Exercise 5.45. S  P be a smooth complete intersection of hypersurfaces of Let 4 . Show d , and let Y  P degrees be any hypersurface of degree f containing S e and that if f is not equal to either d or e , then Y is necessarily singular. Hint: Assume Y is smooth, and apply Whitney’s formula to the sequence ! 0 N 0 ! N ! j N ! 4 4 S S=Y Y= P P S= to arrive at a contradiction.

207 Chapter 6 Lines on hypersurfaces Keynote Questions 4 4 (a) L be a general quintic hypersurface. How many lines Let  P X does X  P contain? (Answer on page 228.) 3 f X be a general pencil of quartic surfaces. How many of the  P Let g (b) 1 t t P 2 X surfaces contain a line? (Answer on page 233.) t 3 Let f X g  P be a general pencil of cubic surfaces, and consider the locus (c) 1 t t P 2 3  G C of all lines L  P that are contained in some member of this family. .1;3/ 3 C ? What is the degree of the surface  P What is the genus of swept out by S these lines? (Answers on pages 233 and 233.) 4 (d) Can a smooth quartic hypersurface in P contain a two-parameter family of lines? (Answer on page 238.) In this chapter we will study the schemes parametrizing lines (and planes of higher Fano schemes . There are two phases to dimension) on a hypersurface. These are called the treatment. It turns out that the enumerative content of the keynote questions above, and many others, can be answered through a single type of Chern class computation. But there is another side of the story, involving beautiful and important techniques for working with the tangent spaces of Hilbert schemes, of which Fano schemes are examples. These ideas will allow us to verify that the “numbers” we compute really correspond to the geometry that they are meant to reflect. We will go even beyond these techniques and explore a little of the local structure of the Fano scheme. There are many open questions in this area, and the chapter ends with an exploration of one of them.

208 194 Chapter 6 Lines on hypersurfaces 6.1 What to expect n d should we expect a general hypersurface X  P For what of degree d n and to contain lines? What is the dimension of the family of lines we would expect it to contain? When the dimension is zero, how many lines will there be? To answer these questions, we introduce in this chapter a fundamental object, the .X/  G .k;n/ Fano scheme k -planes on X , and then study F parametrizing k its geometry. , and F We will defer for a moment a discussion of the scheme structure on .X/ k start by answering the first two of the questions above, since these have to do only with . Even so, the answers may not be apparent at first,  G .k;n/ F the underlying set .X/ k since (as we shall see) the equations on the Grassmannian of lines that describe the n locus of lines contained in a given hypersurface X  P may be complicated. But L N and — that is, ask for the locus, in the space P L of all if we reverse the roles of X n in P hypersurfaces of degree , of the hypersurfaces X that contain a given line L — d L the equations are much simpler; in fact, given that contain L is simply , the locus of X N . a linear subspace of P  C n d 1 D N To capitalize on this, we use an incidence correspondence: We set d N be the projective space parametrizing all hypersurfaces of degree and let d , and P ˆ ˆ.n;d;k/ given by the formula consider the variety D N g X   : G .k;n/ j L P 2 .X;L/ Df ˆ (As the title of this chapter suggests, our primary focus will be on the case k D 1 of lines, but many of the constructions we make can be carried out just as readily for arbitrary-dimensional linear spaces, as here.) N ˆ   G .k;n/ is a closed subset may be seen by a number of elementary That P arguments (see for example Harris [1995, Chapter 6]); in any event, we will give explicit ˆ in the next section. The variety ˆ will be quite useful in many ways; we equations of n of k -planes on hypersurfaces of degree d in P , since call it the universal Fano scheme N of P X is the Fano scheme F -planes .X/  G .k;n/ 2 k the fiber over any point k on X . To start, we have:  n C d Proposition 6.1. Let N  D 1 . The universal Fano scheme ˆ D ˆ.n;d;k/ d n N G .k; P P / is a smooth irreducible variety of dimension    k C d 1/.n k/ D ˆ.n;d;k/ N C dim : C .k k Proof: As we said, the fibers of ˆ over G .k;n/ are readily described: For any plane n k ƒ  P P , the restriction map Š 0 0 n H O .d// .d// ! H . . O ƒ P

209 What to expect Section 6.1 195 ˆ over the point 2 G .k;n/ is simply the projectiviza- is a surjection, and the fiber of ƒ k C d N / . k P tion of the kernel, that is, a projective space . Proposition 6.1 gives us an “expected” answer to the questions raised at the begin- ning of this section, and also allows us to deduce the answer in some cases: We would ex- n X  P of degree pect the family of d — that is, the k -planes on a general hypersurface N N 2 P over a general point — to have dimension '.n;d;k/ WD dim ˆ ŒXç , fiber of ˆ ' < 0 a general X will contain no k -planes. In fact, Proposition 6.1 and that in the case immediately implies the second statement, and while it does not imply the first, it does imply a lower bound on the dimension of the family of such planes, should there be any. We collect these consequences in the following corollary: k (a) The dimension of any component of the family of -planes on any Corollary 6.2. n in is at least d hypersurface of degree P   C d k '.n;d;k/ WD .k C 1/.n : k/ k n '.n;d;k/ < 0 , then the general hypersurface of degree d in P (b) contains no If k -planes. k If  0 and the general hypersurface of degree d (c) '.n;d;k/ -planes, contains any then every hypersurface of degree d contains k -planes, and every component of the family of k -planes on a general hypersurface of degree d has dimension exactly '.n;d;k/ . N Part (b) is immediate: If , then ˆ cannot dominate P dim . For part (a), Proof: ˆ < N N over P is cut out by N equations, the we observe that since a fiber of ˆ.n;d;k/ principal ideal theorem (Theorem 0.1) gives the desired lower bound. As for part (c), d contains a k if the general hypersurface of degree ˆ.n;d;k/ dominates -plane, then N N ˆ.n;d;k/ is projective, the map to P P is surjective with general fiber , and, since . ˆ N D ' of dimension dim We shall eventually show (Corollary 6.32 and Theorem 6.28) that if '.n;d;k/  0 then, except in some cases where k > 1 and d D 2 , the general hypersurface actually does contain k d  2n 3 , every -planes, so the results above apply. For example, if n d P hypersurface of degree contains lines, and the family of lines on the general in '.n;d;1/ such hypersurface has dimension 2n 3 d . D 3 P d  4 contains no of degree Corollary 6.2 shows that the general surface in lines. But we can say more, using the same sort of incidence correspondence argument 3  P made above. For example, we will see in Exercise 6.64 that a general surface of S N d  4 containing a line contains only one. This implies that the locus †  P degree of surfaces that do contain a line has codimension d 3 .

210 196 Chapter 6 Lines on hypersurfaces 6.1.1 Definition of the Fano scheme F of k -planes on We begin by giving a direct definition of the Fano scheme .X/ k n  G .k;n/ . We will return to the definition twice later a scheme P X that is local on in this chapter to give a global description and a universal property that justifies the idea that we are taking the “right” scheme structure. Note that it will suffice to give the n when F is a hypersurface in P definition of the Fano scheme ; for an arbitrary .X/ X k n Y  P we define scheme \ D .Y/ F F .X/: k k n Y   P X is a hypersurface X of degree .X/ for a hypersurface X To define F given by an equation g D 0 , we d k lies on X if and only if the restriction of g L L is zero. If we use the idea that a plane to k W P , then we can pull back ! L of L have a parametrization g via ̨ ; the condition ̨   ̨ L .g/ . X is given by the vanishing of the coefficients of 2 In fact, we can give such parametrizations simultaneously for all planes G L .k;n/ n G .k; of the open cover of / described in Section 3.2.2. Recall U lying in an open set P is defined as the set of all k that such an open set .n k 1/ - U -planes not meeting a fixed k C 1 coordinates, then U may be plane. If the latter is given by the vanishing of the first .k C 1/  .n k/ matrices: Any k -plane L belonging identified with the affine space of U is the row space of a unique matrix of the form to 0 1 0 a  a 1 0  C 1 0;n C 1 0;k B C 0 1  0 a  a 1;n C 1 1 1;k C B C A D : B C : : : : : : : : : : : : : : @ A : : : : : : : a 0 0  1 a  1 k;n 1 C C k;k L of the form We can thus give a parametrization of   X X ̨ k P / s ! .s 3  s a /A D .s s ; ;:::;s ;:::; ; ;:::;s s a C i 0 i;n 0 1 0 i i;k C k k k 1 i i k/ 1/.n C .k are homogeneous A are coordinates on Š a and the s U where the i i;j k coordinates on our fixed source P . n  P Now suppose that is the hypersurface V.g/ given by the polynomial X X ı ;:::;z g.z / D : z c 0 n ı d ı j jD for the variables n 1 We substitute the ̨ C z ;:::;z coordinates of the parametrization 0 n of g , and arrive at a homogeneous polynomial of degree d in s : ;:::;s 0 k X  ı D ̨ : .g/ s e ı d jD j ı

211 What to expect Section 6.1 197 g The coefficients f a e of this polynomial are polynomials in the coordinates on f g i;j ı .k;n/ , which we take as defining equations for U  .X/ . G F k We should check, of course, that the scheme structure defined in this way agrees on the overlap U \ V of two such open sets. This is straightforward, but we will skip it: In what follows we will see a remarkable intrinsic characterization of the Fano scheme that will imply it. N ˆ  G .k;n/ discussed in the P Finally, we return to the universal Fano scheme  e , and we have: The coefficients ˆ preceding section. We promised to give equations for ı  .g/ above may be viewed as polynomials in both sets of variables ̨ of the polynomial N f a g f c to be the subscheme defined locally by g , and we take ˆ  P and  G .k;n/ i;j ı n P these polynomials. Note that for any hypersurface the Fano scheme F  .X/ is X k N ˆ the scheme-theoretic fiber of P 2 . X over the point : it is reduced. In fact, there is a simpler way to characterize the scheme structure of ˆ This follows from the fact that the scheme-theoretic fibers of the projection ˆ ! G .k;n/ e ). For the same are homogeneous linear in the variables c are projective spaces (the ı ı is smooth. reason, ˆ to say that the Fano scheme not .X/ is either smooth or reduced This is very much F k X . It does imply that F .X/ for a given is smooth and reduced for a general hypersurface k n X P , but we will see many examples of nonreduced and/or singular Fano schemes;  part of the challenge of the subject is to figure out under what circumstances this may F happen. As a first example, you may wish to consider the Fano scheme .Q/ of lines 1 3 Q  P on a quadric surface ; as you can see from the equations, F Q .Q/ is smooth if 1 is smooth, but everywhere nonreduced if is singular. Apart from this being the first Q nontrivial example of such phenomena, what makes this interesting is that we will also be able to see this from two other viewpoints, without coordinates: once when we describe the class .Q/ç 2 A. G .1;3// in Section 6.2, and again at the end of Section 6.4.2 ŒF 1 when we introduce the notion of first-order deformations. Of course special hypersurfaces may well contain families of planes of dimension . We can easily give an upper bound on the possible dimension: '.n;d;k/ greater than n is an m -dimensional variety, then X If P Proposition 6.3.  F .k;m/; .X/  dim k/.k C 1/ D dim G .m k X m -plane. with equality if and only if is an Proof: X is nondegenerate. Let U  We may assume without loss of generality that k C 1 be the open set consisting of .k C 1/ -tuples of linearly independent points, and let X Ä Df ..p : ;:::;p g /;L/ 2 U  F i .X/ j p for all 2 L i 0 k k  Ä U , we see that Via the projection Ä ! m.k C 1/ . Since the fibers of the dim projection Ä ! F  .X/ have dimension k.k C 1/ , we conclude that dim F .X/ k k m.k C 1/ k.k C , as required. D .m k/.k C 1/ 1/

212 198 Chapter 6 Lines on hypersurfaces Ä ! is dominant, that is, Equality of dimensions can hold only if the projection U contains the plane spanned by any k 1 general points of X , and this can happen if X C X only if is a linear space. In Section 6.8 we will discuss some open questions about these dimensions. 6.2 Fano schemes and Chern classes To get global information about the Fano scheme of a hypersurface, we will express it as the zero locus of a section of a vector bundle on the Grassmannian. To understand the n P g is the hypersurface g D 0 , where idea, suppose that is a homogeneous form X  . As we have seen, the condition that d contain a particular k -dimensional of degree X is that is sent to 0 by the restriction map L linear space g 0 0 n . H ! H O . O .d//: .d// L P F , we will realize the family .X/  G .k;n/ of k To describe the scheme X -planes on k 0 H O of vector spaces .d// (with varying k -planes L ) as the fibers of a vector bundle . L g  in such a way that the images of in these vector spaces are the values of a section g of the bundle: V be an .n Proposition 6.4. 1/ -dimensional vector space, and let S  V ̋ O Let C G be the tautological rank- C 1/ subbundle on the Grassmannian G D G .k; P V / of .k n P V Š P k . A form g of degree -planes in on P V gives rise to a global section  of d g d  S Sym whose zero locus is F . .X/ , where X is the hypersurface g D 0 k  k C d d  / S D rank . Sym Thus, when F has expected codimension .X/ in G , k k we have  d A.G/: 2 / S Sym . D ŒF .X/ç c k d C k / . k Proof: S over the point ŒLç 2 G .k; P V / representing the subspace L Š The fiber of k 1/ P V is the corresponding .k C  -dimensional subspace of V . The fiber of the P  0 L at ŒLç is thus the space of linear forms on dual bundle , that is to say H S . O , .1// L P    O V ! S V evaluated at a point ŒLç takes a linear form ' 2 and the dual map ̋ , G  , to the restriction of ̋ O thought of as a constant section of the trivial bundle V ' G d  0 d on P V is Sym to V L D H . The vector space of forms of degree . O , and .d// V P the induced map on symmetric powers d  d  ! Sym S V Sym evaluated at L takes a form g of degree d to its restriction to L , as required.

213 Fano schemes and Chern classes 199 Section 6.2 d d  0  2 S . / be the global section of Sym Sym S  that is the image of H . Let g g .X/  G .k; F V / is the zero locus of this section. It is enough to check We claim that P k .k; P V / , and we use the open covering by basic this locally on an open covering of G described in Section 3.2.2. On such an open set, the bundle is trivial, affine sets U S with the inclusion k C 1 S j V D ! O ̋ O U U U . It follows that the dual map given by the transpose of the matrix A  0   n O W A .1// ̋ O j D V H ̋ O . ! S U U U P V d -th symmetric power is the restriction P , and its is the restriction of linear forms from . Thus the value of  U at the point of of forms of degree corresponding to a plane d g is the restriction of g to L , or in other words the result of the substitution given in L Section 6.1.1, as required. n ŒF depends only on d and .X/ç (assuming it has the The fact that the class k expected dimension) has consequences by itself, even without calculating the actual 3  P class. For example, consider the lines on a quadric surface . As we saw in Q Q is smooth the Fano scheme F Section 3.6.1, when .Q/ consists of two disjoint 1 5 smooth conic curves in the Grassmannian P G . But what happens when Q .1;3/  is a single conic F is a cone over a smooth conic curve? Here the support of .Q/ 1 .1;3/ , and we may deduce from this and Proposition 6.4 that F curve in .Q/ is G 1 . everywhere nonreduced 6.2.1 Counting lines on cubics 3 We want to see how this works for the case of lines on a cubic surface X  P . F In the language above, we want to compute the class of the Fano scheme .X/ in the 1  S .1;3/ . We saw in Section 5.6.2 that the Chern class of D Grassmannian is G G  D / : 1 C   C S c. 1 1;1 3 3   S Since S is 4, so we want to compute c . To . Sym has rank 2, the rank of S Sym / 4 do this, we will apply the splitting principle (Section 5.4), which implies that to compute  S splits into a direct sum of two line bundles L the Chern class we may pretend that and M . Suppose that L / D c. C ̨ and c. M / D 1 C ˇ: 1 By the Whitney formula,  C / D .1 c. ̨/.1 C ˇ/; S so that ̨ : ˇ D   and ̨  ˇ D C 1 1;1

214 200 Chapter 6 Lines on hypersurfaces 3   S would split as well: S were to split as above, then the bundle Sym If 3  2 2 3 3 D . ̊ ̋ L / ̊ . L ̋ M L / ̊ M S ; Sym M so that we would have 3  S c. Sym D .1 C 3 ̨/.1 C 2 ̨ C ˇ/.1 C ̨ C 2ˇ/.1 C 3ˇ/: / In particular, the top Chern class could be written 3  C c S . / D 3 ̨.2 ̨ C ˇ/. ̨ Sym 2ˇ/3ˇ 4 2 2 C 5 ̨ˇ 9 ̨ˇ.2 ̨ 2ˇ D / C 2 9 ̨ˇ.2. ̨ C ˇ/ D C ̨ˇ/:  Re-expressing this in terms of the Chern classes of itself, we get S 2 3  C S Sym / D 9 / .2 c  . 4 1;1 1;1 1 27 D ; 2;2 so  3  .. Sym deg S 27 // c D I 4 by the splitting principle, these formulas hold even though does not in fact split. S 3  Sym S can also be computed by hand in this way, or The whole Chern class of Macaulay2 with the following commands in : loadPackage "Schubert2" G = flagBundle({2,2}, VariableNames=>{s,q}) -- sets G to be the Grassmannian of 2-planes in 4-space, -- and gives the names $s_i$ and $q_i$ to the Chern classes -- of the sub and quotient bundles, respectively. (S,Q)=G.Bundles -- names the sub and quotient bundles on G chern symmetricPower(3,dual S) which returns the output 2 2 o4 = 1 + 6q + (21q - 10q ) + 42q q + 27q 1 1 2 1 2 2 QQ[][s , s , q , q ] 1 2 1 2 o4 : -------------------------------------------- (s + q , s + s q + q , s q + s q , s q ) 1 1 2 1 1 2 2 1 1 2 2 2 The answer on the first output line “o4” is written in terms of the Chern classes q WD i c .Q/ , which generate the (rational) Chow ring of the Grassmannian, described on the i second output line “o4.”

215 Definition and existence of Hilbert schemes Section 6.2 201 3  . S / is nonzero, we deduce that every cubic surface must Since the class c Sym 4 , and thus that a general cubic surface contains only finitely many. Moreover, contain lines 3 X  P contains only finitely many lines, then the number if a particular cubic surface of these lines, counted with the appropriate multiplicity (that is, the degree of the ), is 27. As we will soon see, the corresponding component of the zero scheme of  g F .X/ of a smooth cubic surface X is necessarily of dimension zero and Fano scheme 1 reduced, so the actual number of lines is always 27. In the next section we will develop a general technique that will allow us to prove this statement, and much more. We will X is singular. also see, in Section 6.7, how to count lines in cases where 6.3 Definition and existence of Hilbert schemes It was Grothendieck’s brilliant observation that the Grassmannian and the Fano Hilbert scheme . Hilbert scheme are special cases of a very general construction, the schemes are defined by a universal property that we will explain in this section, after making the property explicit for the Grassmannian and Fano schemes. One of the useful properties of Hilbert schemes is a general formula for tangent spaces, which we will explain in the next section. For more remarks about Hilbert schemes in general, see Section 8.4.1. 6.3.1 A universal property of the Grassmannian P D 1;V / D G .k; C V / of G G.k Recall from Theorem 3.4 that the Grassmannian 1/ -planes in an .n C 1/ -dimensional vector space V , with its tautological subbundle .k C  V ̋ O S , has the following universal property: Given any scheme B and any G V C 1/ subbundle F of the trivial bundle .k ̋ O rank- , there is a unique morphism B  W B ! G such that F D ' ' S . We could, of course, just as well express this in terms of a universal property of the quotient bundle Q V ̋ O D = S , or of the .n k/ -subbundle B   Q O ̋  V , which is the most convenient for what we will do in this section. B Similarly, the universal -plane k Df .ƒ;p/ 2 G ˆ P V j p 2 ƒ g  is a universal family of k -planes in P V in the following sense: For any scheme B , we flat family of will say that a subscheme B  P V is a  k -planes in P V if the restriction L  W L ! B of the projection  is flat, and the fibers over closed points W B  P V ! B 1 of B are linearly embedded k -planes in P V . We have then:

216 202 Chapter 6 Lines on hypersurfaces If Proposition 6.5.  B  P V ! B is a flat family of k -planes in P V , then there  W L B L G such that W is equal, as a subscheme of B  P V , to the is a unique map ̨ ! L ̨ pullback of the family : via - B  D ˆ ˆ L G   - ? ? - G B ̨ We will prove the proposition by showing that the desired property can be Proof: reduced to the universal property of Theorem 3.4. Though the reduction may appear technical, it is really just an application of Theorem B.5, together with the remark that -plane in P V is generated by n k independent linear forms. the ideal of any k and   P V G L  ˆ To simplify the notation we denote the ideal sheaves of  P V by I and J B J .1/ for J ̋   O  , where .1/ respectively, and we write V P P denotes the projection onto I .1/ similarly. For any scheme B , we denote V . We define   B by V the trivial bundle with fiber ̋ V O and base . B The proof consists of the following steps: We will begin by showing that  J .1/   L ̋ V is a subbundle of . Since ˆ satisfies the same hypotheses as O , the same B   .1/ is a subbundle of O reasoning will show that the sheaf ̋ V I . We will see  G  that this subbundle is equal to the subbundle Q . It follows that there is a unique map W ̨ ! G such that B  . .1/: I .1// ̨  J D   Finally, we will show that this last equation is equivalent to the equality L D B  ˆ ̨ as families of P V , where B  k L denotes the pullback B  -planes in L defined ̨ G using the map . ̨  J .1// is a bundle follows from Theorem B.5 and the remark The fact that .  .n . J .1// to a fiber b is the  k/ -dimensional linear space of that the restriction of  0 -plane L forms vanishing on the k  f b g P V Š P V . The natural map  ! J .1/  b  O ̋ V  O .1/ D is an inclusion on fibers, so  J .1/ is a subbundle,   B  B V P as claimed.   , and I .1/ with Q To identify , we remark that both are subbundles of O  ̋ V  B b B their fibers are the same subspace — namely, the space of linear 2 at each point forms vanishing on L . It now follows from the universal property of Theorem 3.4 that b   W ! G such that ̨ ̨ there is a unique morphism as subbundles I .1/ D B .1/ J     of O ̋ V . B We claim that this property of ̨ implies the equality L D B  . To prove this, ˆ ̨   D . ̨  1/ 1/ I , or equivalently J .1/ D . ̨  it suffices to show that J I .1/ . If we restrict L to the fiber over b 2 B , we get a subspace of P V whose ideal is generated by

217 Definition and existence of Hilbert schemes Section 6.3 203 b B the linear forms it contains. For 2 , Theorem B.5 identifies this space of linear forms J at b . Thus there is a surjection with the fiber of .1/    J  ! J .1/:  .1/  I . Thus the commutative diagram Similar remarks hold for      .1/  ̨ .1/   I .1/ D . ̨  1/ J    D I    -  O B  P V and B  shows that the ideal sheaves of ˆ are equal. L ̨ 0 Finally, we prove the uniqueness of L D B  for some ̨ ˆ . Suppose that ̨ 0 0 0 .1/ D ̨ by showing that  ̨ J . We will show that D ̨ ̨  . But the I .1/ morphism    0 D 1/ .1/  J hypothesis implies that I .1/ . From the definition of the pushforward, . ̨ we get a natural map 0 0  1/ .1/ !  J . ̨   .1/ I I .1/ D  ̨    that is an isomorphism fiber-by-fiber, so we are done. 6.3.2 A universal property of the Fano scheme We realized the Fano scheme of a projective variety X as the subscheme of the Grassmannian consisting of planes lying in X , and as such it inherits a universal property: n X If is a subscheme, then the scheme F P .X/ represents the Proposition 6.6.  k -planes on X functor of k , in the sense that the correspondence above induces a one- B F to-one correspondence between morphisms of schemes .X/  ! .k;n/ and G k n -planes L  B  X  B  P families of that are flat over B . k n This is a corollary of the statement for the Grassmannian: Suppose that D P V Proof: P d  i 2 and Sym X is defined by some homogeneous forms V g . Let S be the universal i  G .k;n/ , so that the fiber of S subbundle on at a point ŒLç 2 G .k;n/ is the space of ı linear forms on the corresponding  P V . Writing L k for the section of -plane g i d  i vanishes on that is the image of the form g Sym , we see that g if and only if S L i i the sections  . vanish at the point ŒLç g i 6.3.3 The Hilbert scheme and its universal property X and Grothendieck’s idea was to ask, more generally: given any projective scheme a subscheme Y Y can move within X ?” More precisely and ambitiously: Can we , “How describe all flat families B  X  Y ! B including Y as a fiber? Is there a universal such family?

218 204 Chapter 6 Lines on hypersurfaces B When is reduced, a family Y as above is flat if and only if the fibers all have the same Hilbert polynomial; in particular, any family over a reduced base whose fibers -planes is automatically flat. (See for example Eisenbud and Harris [2000, k are all Proposition III-56].) Grothendieck’s idea was to define “the family of all subschemes” P . .d/ , the Hilbert polynomial of Y of X with Hilbert polynomial equal to Y We might worry that this goes too far to be a generalization of the Fano scheme — X -plane but whose Hilbert polynomial is could there be a subscheme of k that is not a k equal to that of a -plane? The following result shows that all is well: n k P Proposition 6.7. is a linearly embedded k -plane P A subscheme if and only Y  if the Hilbert polynomial of is Y    1/ C .d 1/ .d k/.d C k C d k C D D P.d/ : k 1/  k.k 1 d C k Since the dimension of the 1 Proof: -th graded component of a polynomial ring on  C k d variables is , the Hilbert polynomial of a linearly embedded k -plane is P.d/ . k Conversely, suppose that has Hilbert polynomial P . From the degree and lead- Y ing coefficient of we see that Y is a scheme of dimension k and degree 1. Thus P L WD Y -plane. This inclusion induces a surjection of  Y is a linearly embedded k red homogeneous coordinate rings S ! S , and the equality of Hilbert polynomials shows L Y n L  Y  P can be that it is an isomorphism in high degrees. Since the inclusions Proj .S is the homogeneous coordinate /  Proj .S S /  Proj .S/ recovered as , where Y L n P , and since Proj .S , this shows / ring of S depends only on the high degree part of Y Y L  Y is actually an equality. Here is the general definition and existence theorem for Hilbert schemes, showing that there is a unique “most natural” scheme structure: n Let X P Proposition–Definition 6.8. be a closed subscheme, and let P.d/ be a  X H polynomial. There exists a unique scheme , called the Hilbert scheme of for .X/ P the Hilbert polynomial P , with a flat family  .X/ .X/  X  Y H ! H P P with Hilbert X universal family of subschemes of of subschemes of , called the X polynomial P , having the following properties:  The fibers of  all have Hilbert polynomial equal to P.d/ .  For any flat family  0 X B Y B  ! 

219 Definition and existence of Hilbert schemes Section 6.3 205 P.d/ whose fibers have Hilbert polynomial B ! , there is a unique morphism ̨ W 0 such that is equal to the pullback of Y : H Y .X/ P 0 - Y  Y B Y D .X/ H P   - ? ? - B .X/ H P ̨ ! H X  Y  H .X/ .X/ A compact way of stating the existence of the family P P and its universal property is to use the language of representable functors. Consider the contravariant functor from schemes to sets that is defined on objects by ̊ n 0 F 7! W X  B  Y B ! B of subschemes of X  P flat families X;P P whose fibers over closed points all have Hilbert polynomial 0 B to the map of sets taking a flat family over B to its and that takes a map ! B 0 . In this language, the existence theorem says that F pullback to a family over is B X;P by the scheme H .X/ representable , in the sense that P F Š Mor . ; H .X// P X;P F as functors. The universal family in H .X// then corresponds to the identity map . X;P P in Mor . H See for example Eisenbud and Harris [2000, Chapter VI] for .X/; H .X//: P P more about this idea. Proof of uniqueness in Proposition 6.8: As with any object with a universal property, W ! H .X/ with the given properties is easy: Given  the uniqueness of a map Y P 0 0 B W ! B , the universal properties of the two produce maps  ! another such map Y .X/ H is the unique and .X/ .X/ ! B whose composition H H .X/ ! B ! H P P P P  W map guaranteed by the definition that corresponds to the family ! H .X/ itself — Y P that is, the identity map — and similarly for the composite B ! H . .X/ ! B P 6.3.4 Sketch of the construction of the Hilbert scheme .X/ and the universal family is also relatively easy to de- H The construction of P scribe, though the proofs of the necessary facts are deeper. There are several approaches, all along the lines of Grothendieck’s original idea (see Grothendieck [1966b]), but the following (from Bayer [1982]) is perhaps the most explicit. n D P We first treat the case when , since (as in the case of the Fano schemes) we X S D k Œx ;:::;x shall see that the general case reduces to this. Let ç be the homogeneous n 0 n coordinate ring of . The Hilbert scheme H .X/ is constructed as a subscheme of P P the Grassmannian of P.d/ -dimensional subspaces of S , the space of homogeneous d forms of degree d , for suitably large d . The possibility of doing so is provided by the following basic result from commutative algebra, which combines ideas of Macaulay and Gotzmann (see Green [1989] for a coherent account).

220 206 Chapter 6 Lines on hypersurfaces Theorem 6.9. .P/ (explicitly computable d With notation as above, there is an integer 0 ) such that if  d .P/ P then the saturated homogeneous ideal from the coefficients of d 0 of any subscheme of with Hilbert polynomial P is generated in degrees  d and I X S =I dim / D P.d/ . Further, a subspace U  .S generates an of dimension P.d/ d d d if and only if ideal with Hilbert polynomial P.d/ U/ 1/; =S dim .S  P.d C 1 1 C d in which case D =S U/ P.d C 1/: .S dim 1 C 1 d n X is any hypersurface of degree s in P If , then the Hilbert function Example 6.10. of is S X     n C d s d n C ; dim D .S / X d n n P.d/ of degree n 1 for all d such that d  s , as one can which is equal to a polynomial n X check immediately. Conversely, given any scheme with this Hilbert polynomial,  P X D dim 1 , so X is a hypersurface, and the leading coefficient of we see that n deg X D s . It follows that the saturated ideal of the Hilbert polynomial tells us that S is generated by a single form of degree . In this case, every subspace U  s X d generates an ideal with this Hilbert polynomial; the growth condition of the theorem is automatically satisfied. n d  d to be the closed .P/ , and take H / . P Given Theorem 6.9, we choose 0 P WD G. dim S subscheme of the Grassmannian P.d/;S G / defined by determinantal d d n U P equations saying that / consists of those H 2 G such that the vector space S . U 1 P dim S . Writing for S P.d C 1/ has the smallest possible dimension, which is 1 d C n ̋ on G the universal subbundle of the trivial vector bundle H O . P , / is the S G P d subscheme defined by the condition that the composite map ̋ ̋ ! S ̋ S S O O ! S S ̋ 1 1 G G 1 C d d has corank C 1/ .  P.d n n P as follows. Let  H Further, we can construct the universal family . P Y /  P   1 2 n n G P P ! G   n n ̋  be the projection maps. There is a natural map S O O , and . d/ ! d P  G P 1 composing this with the inclusion we get a map of sheaves  n n ! d/ : O . ̋ U  O P  G P 2

221 Definition and existence of Hilbert schemes Section 6.3 207 n z  G defined by the image of this map, and let Y ! be the subscheme of Let P Y n of the (non-)flat family given by the / . H H .X/  G P be the restriction to P P composite  2 n z G Y  P G: !  The universal property (which we will not prove) shows that these construction are d d independent of the choice of (up to canonical isomorphism).  0 n P H / , but we can use this to construct . So far we have only defined .X/ H P P n for any X . Let I D I.X/  S be the ideal corresponding to X , and suppose  P I is generated in degrees  e . Given the Hilbert polynomial P , we choose d  that n implying .P/;e g . Then to define H d .X/ we simply add equations to H max . P f / 0 P P   n n . d/ d/ O O . is contained in I ̋  U ̋  that . This can be translated into a d P P 1 1 rank condition on a map of vector bundles, as before. Example 6.11 (Example 6.10, continued) The argument above shows that the Hilbert . n N P scheme of hypersurfaces of degree is the projective space P s of all homogeneous in s , and the universal family is the universal hypersurface forms of degree n N .x;X/ 2 P  X P Df j x 2 X g ; as one would hope. that plays a central role in the Here is one way to understand the integer d .P/ 0 can be ordered d construction. Recall that the set of monomials of given degree lexico- graphically , where f e f f e e 0 0 n n x WD x  < x  x x x DW n n 0 0 e x > f involves more of for the smallest i such that e if ¤ f e — informally put, if i i i i f S . A monomial ideal I  the lowest-index variables than is called lexicographic if, x f e f e whenever are monomials of degree d x x and 2 I , then x < x 2 I too. It follows easily that the saturation of a lexicographic ideal is lexicographic. Proposition 6.12. S D k Œx Let ;:::;x . ç n 0 If I is any homogeneous ideal of S , then there is a lexicographic ideal (a) such that J the Hilbert function of S=J is the same as that of S=I . , then there is a unique saturated lexico- If P D P S=I is the Hilbert polynomial (b) I graphic ideal J with Hilbert polynomial P . P The integer d . .P/ may be taken to be the maximal degree of a generator of J 0 P For example, if I is the principal ideal generated by a form of degree s as in the example above, this proposition gives d D s . See Green [1989] for further information. 0

222 208 Chapter 6 Lines on hypersurfaces 6.4 Tangent spaces to Fano and Hilbert schemes In order to use the Chern class calculation of Section 6.2.1 to count the number of distinct lines on a cubic surface, we need to know when the Fano scheme is reduced. In the zero-dimensional case, this is the same as being smooth, and the question can thus be approached through a computation of Zariski tangent spaces. Happily, we can give a simple description of the Zariski tangent spaces of any Hilbert scheme. We first state the main assertions for Fano schemes. They will allow us to deduce n of degree d  2n P 3 , the exact number of lines on a general hypersurface X D along with other geometric facts. We will then compute the tangent spaces in the general setting of Hilbert schemes (Theorem 6.21). In Section 6.7 below, we will show how to F at L by writing down explicit local equations for calculate the multiplicity of .X/ 1 .X/ G .1;n/ . F  1 6.4.1 Normal bundles and the smoothness of the Fano scheme We will make use of the universal property of Fano schemes to give a geometric F condition for the smoothness of .X/ at a given point. Recall that if Y  X is a k smooth subvariety of the smooth variety , then the normal bundle N X of Y in X Y=X is the cokernel of the map of tangent bundles D coker . T induced by ! T j N / X Y Y Y=X the inclusion of Y  X . Recall also that the Zariski tangent space of a scheme F at a 2 p is by definition Hom is the maximal ideal of m . m , where = m m / ; O point = p p p O p p p at of F the local ring p . O p The following theorem is a special case of a general result on Hilbert schemes, Theorem 6.21, which we will prove in the next section: n L  X is a k -plane in a smooth variety X Theorem 6.13. P Suppose that , and let  ŒLç F is .X/ be the corresponding point. The Zariski tangent space of F 2 .X/ at ŒLç k k 0 H . N / . L=X The result is intuitively plausible if we think of a section of N as providing a L=X normal vector at each point in X . X , with a corresponding infinitesimal motion of D 0 . The Hilbert scheme of points on For a case that is easy to understand, take k X is X itself, as one checks from the definition. The tangent space at a point a variety 2 X is thus the Zariski tangent space to X at x , and this is — identifying sheaves x 2 f x g with vector spaces — equal to Hom . N . m D = m on the space / ; O x O X;x x=X X X;x Before introducing the general machinery of the proof, we explain how the result can be used.

223 Tangent spaces to Fano and Hilbert schemes Section 6.4 209 X L Figure 6.1 A tangent vector to the Fano scheme .X/ at ŒLç corresponds to a normal F 1 L X . vector field along in n Corollary 6.14. X is a k -plane in a smooth variety X  P Suppose that L  , and ŒLç F let .X/ be the corresponding point. The dimension of F 2 .X/ at ŒLç is at most k k 0 .X/ . N if and only if equality holds. / . Moreover, F dim H is smooth at ŒLç L=X k By the principal ideal theorem, the dimension of the Zariski Proof of Corollary 6.14: tangent space of a local ring is always at least the dimension of the ring, and equality holds if and only if the ring is regular. See Eisenbud [1995]. To apply Corollary 6.14, we need to be able to compute normal bundles, and this is often easy. For example, we have: Proposition–Definition 6.15. Suppose that  X are schemes. Y 2 X and Y are smooth varieties then N om D H For arbitrary If /: I (a) = I . O ; O Y Y Y=X Y Y Y schemes X , we define N by this formula.  Y=X (b) If Y  X  W are schemes, and X is locally a complete intersection in W , then there is a left exact sequence of normal bundles ̨ N ! ! N : 0 ! N j Y X=W Y=W Y=X If all three schemes are smooth, then ̨ is an epimorphism. If Y is a Cartier divisor on X then N is the zero D O Y (c) . More generally, if .Y/ X Y=X Y of rank e on X locus of a section of a bundle E has codimension e in X , then , and N : D E j Y Y=X (a) For any inclusion of subschemes  X , there is a right exact sequence Proof: Y involving the cotangent sheaves of and Y : X d 2 = I j ! I !   0; ! Y Y X Y=X Y=X where d is the map taking the class of a (locally defined) function f 2 I to its Y=X differential df 2  j ; see for example Eisenbud [1995, Proposition 16.12]. Since X Y

224 210 Chapter 6 Lines on hypersurfaces 2 and Y is locally a complete intersection in X , so I X = I Y are smooth, is a Y=X Y=X X dim Y D rank  locally free sheaf on j dim of rank equal to rank  . If Y Y Y X the left-hand map d were not a monomorphism of sheaves, then the image of d would have strictly smaller rank, so the sequence could not be exact at j  . Thus d is a Y X monomorphism, and we have an exact sequence d 2  I ! 0 0 !  ! j = ! I Y Y X Y=X Y=X of bundles. Since Y is smooth, is locally free, so dualizing preserves exactness, and  Y we get an exact sequence 2 H om 0; . I T = I O 0 ; / T j O X Y Y Y Y=X Y Y=X Y  , proving that where the right-hand map is the differential of the inclusion X 2 D om N H / . . = I I ; O O Y Y Y=X Y Y Y X  W , we derive an exact sequence of ideal sheaves (b) From the inclusions  ! I 0: ! I 0 ! I ! Y=X Y=W X=W H om Applying the functor . ; / gives a left exact sequence O O Y W ! N /: 0 N O ! H om . I ; ! Y Y=X Y=W Y=X 2 H om . I , ; O Š / Since H om . I I ̋ O = ; O I / and I Š ̋ O Y Y Y Y Y=X Y=X Y=X Y=X Y=X we get the desired sequence. In the smooth case, we start with the exact sequence T 0 ! T ! j 0 ! N ! X W X X=W that defines . We restrict to Y and factor out the subbundle N and from both T T X Y X=W T j to get the required exact sequence X W ! N ! N 0 ! N 0: j ! Y Y=X Y=W X=W I (c) The first formula follows at once from part (a), since in that case D O , . Y/ X Y=X and taking the dual of a bundle commutes with restriction. For the second statement of part (c) we first give a geometric argument that works in the smooth case, and then a proof in general. Let Z be the total space of the bundle E . The tangent bundle to Z restricted to the zero section X  Z is T . ̊ E X Y  , the derivative D of  of T Along the zero locus j ! is thus a map Y X T j , ̊ E Y . Since the component of D that maps T is zero along j E to Y X Y Y Y X the composite D T E T ! ! T j ̊ E ! j Y Y Y Y X X Y / is zero. Locally at each point 2 Y , the image of . T is the tangent / E in . T ̊ y y y y X X space to .X/ , the Z . Since Y is smooth of codimension equal to the rank of E 

225 Tangent spaces to Fano and Hilbert schemes Section 6.4 211 E vector in Z D Z zero section 0 Š tangent vector to Z X 0 X restricted to Z ̊ Š X is T . Figure 6.2 The tangent bundle to Z E X 0 / meets the zero locus  Z transversely. This means that . T manifold X .X/ y X projects onto E , and tells us that the composite map of bundles y D j E T ! T ! j E ̊ Y Y X Y X Y is surjective. Considering the ranks, it follows that the sequence ! 0 ! T 0 j T ! E ! Y X Y Y N E . is exact; that is, D Y Y=X X or Y is smooth. With a more algebraic approach, we can avoid the hypothesis that 2 We may think of O . Dualizing, ! E that sends 1 2 O E to  as defining the map  X X Y is the zero locus of  the statement that I is the image means that the ideal sheaf Y=X   e W E of the map ! O is locally a . Since the codimension of Y is  , we see that Y X  is generated by the Koszul relations; that is,  complete intersection. Thus the kernel of the sequence V  2   ! E ! I E ! 0 ! Y=X   f / D  lie .e/f  is exact, where .f /e . Because the coefficients in the map  .e ^ , so we get the right exact I O in D O , they become zero on tensoring with = I Y Y X Y=X sequence V 0 2   2 ! E ! j j ! I 0: = I ! E Y Y Y=X Y=X 2   E Š I . = I E N , whence This shows that j D D E j j Y Y Y Y=X Y=X Y=X n Y is a complete intersection of X with divisors on P In the special case where L .d , so the last statement of , the normal bundle is N / D of degrees O d i i X Y=X Proposition 6.15 takes a particularly simple form. We can make it even more explicit when both X and Y are complete intersections:

226 212 Lines on hypersurfaces Chapter 6 n  P are (not necessarily smooth) complete  Y Corollary 6.16. Suppose that X intersections of hypersurfaces with homogeneous ideals X I ;:::;g : /  I f D .f a ;:::;f .g /; g D D i 1 1 i;j t j s X Y j f If D ' deg and deg g , then D i i i i t s M M n n D N .' D O N /; . / O i i Y X Y= P X= P i D 1 1 i D n n ̨ W N N given by the matrix and ! is the kernel of the induced map j N Y Y=X P X= P Y= a . / , where a . denotes the restriction of a Y to j;i j;i j;i The complete intersection X of the .g Proof: ;:::;g / is the zero locus of the section 1 s n n / O / ̊ ̊ O . Using the formula of part (c), bundle . Y . , and similarly for s 1 P P we see that s M n N D /; . O i X P X= 1 D i Y ̨ follows at once from part (a). and similarly for . The identification of As an immediate application, we can finally show that there are exactly 27 distinct lines on every smooth cubic surface (pending, of course, the proof of Theorem 6.13): 3 Corollary 6.17. X Let be a smooth surface of degree d  3 P F  .X/ ¤ ¿ , then . If 1 F .X/ is smooth and zero-dimensional. In particular, X contains at most finitely many 1 lines, and if d 3 then X contains exactly 27 distinct lines. D See Corollary 6.27 for a strengthening. Proof: L  X is a line. As we saw in Section 2.4.2, the self-intersection Suppose L number of X is negative, so the normal bundle N is a line bundle of negative on L=X 0 dim H . degree. It follows that N is / D 0 , and Corollary 6.14 now implies that L L=X F .X/ is smooth at ŒLç . isolated and 1 In particular, in the case of the cubic surface the fact that the class of the Fano scheme is 27 points implies, with this result, that the Fano scheme actually consists of 27 reduced points. We will also be able to see Corollary 6.17 geometrically once we have introduced the notion of first-order deformation in Section 6.4.2 6.4.2 First-order deformations as tangents to the Hilbert scheme The proof of Theorem 6.13 and its generalization involves the idea of a first-order deformation of a subscheme, which is the main content of this section. Suppose that

227 Tangent spaces to Fano and Hilbert schemes Section 6.4 213 is a closed subscheme of a scheme X deformation of Y , defined over the field k . By a T T over a scheme k 2 with distinguished point we mean a subscheme X  Y Spec  X , flat over T , whose fiber over the distinguished point Spec k is equal to Y , Y  T that is, a diagram T  X Y Y ̨ ˇ projection T k Spec ! T Spec T , and we will denote We think of the image of , k as a distinguished point of it by ŒYç . T is the spectrum of a local ring of if its base first-order A deformation is called 2 k Œ ;:::; ç=. . Note ;:::; R / D for some m . We set T WD Spec R the form m m m 1 m 1 m that this is a scheme with a unique closed point, which we shall denote by 0. We think of as a first-order neighborhood of a point on a smooth -dimensional variety. T m m It follows from the universal property of the Hilbert scheme that a first-order defor- . T Y is the same thing as a morphism T over ! mation of sending 0 to ŒYç H m m into a In general, we will denote the set of morphisms of -scheme Z sending 0 T k m z to a point Z by Mor so we have .T ;Z/; 2 m z deformations of Mor  X over T ;H/: gD f Y .T m m ŒYç For simplicity we restrict ourselves for a while to the case m D 1 , and consider deforma- tions over T . 1 T to H is the The identification of first-order deformations with morphisms from 1 H key to identifying the tangent space of (and thus, in our case, of the Fano scheme). -rational point z on any scheme Z we can identify the set Indeed, for any closed k .T ;Z/ with the Zariski tangent space to Z at z . To describe the identification, Mor 1 z t W T we have a pullback map on ! Z sending 0 to z recall that for any morphism 1  W O t ! R functions, denoted . Restricting this map to m , we get 1 Z;z Z;z  ! j W m : t m k D k  Š m T;0 Z;z Z;z 2   sends Since t with the induced map to zero, we may identify t m j m Z;z Z;z 2  : k W m Š = m t j ! m  D k m T;0 Z;z Z;z Z;z Lemma 6.18. Let z 2 Z be a k -rational point on a k -scheme. The map 2 m .T / ;Z/ ! T k D Hom ; . m Mor = z k 1 z=Z z=Z z=Z  t to the restriction of the pullback map on functions t sending a morphism j is m Z;z bijective.

228 214 Chapter 6 Lines on hypersurfaces Giving a morphism t T Proof: ! Z is equivalent to giving the local map of k -alg- W 1  = ! R that induces the identity map k Š O t O m ebras ! R W =. / D k . 1 1 1 Z;z Z;z Z;z 2  R . . /  ! Thus t = m m is determined by the induced map of vector spaces 1 1 Z;z Z;z 2 ! ./ extends to a local algebra homomorphism Conversely, any map = m m Z;z Z;z  2 W ! k Œç=. t / D R O . 1 Z;z Y  As we have explained, the universal property of the Hilbert scheme of also X allows us to identify Mor .T X ;H/ with the set of first-order deformations of Y  1 ŒYç T . Such deformations admit another very concrete description: over 1 Suppose that Y  X are schemes. There is a one-to-one correspondence Theorem 6.19. over the base T between flat families of subschemes of with central fiber Y and homo- X m m 2 ! O = I -modules . In particular, flat families of deformations I morphisms of O Y Y Y Y Y in X over T of correspond to global sections of the normal sheaf of Y in X . 1 We will use the following characterization of flatness over T : m Lemma 6.20. M is a (not necessarily finitely generated) module over the ring R , If m M is flat if and only if the map then . ;:::; / m 1 m M ! M m .M=. . ;:::; /M /M/ . Š induces an isomorphism ;:::; 1 m m 1 M is flat The general criterion of Eisenbud [1995, Proposition 6.1] says that Proof: ! I ̋ if and only if the multiplication map M W IM is an isomorphism for  I R . But every nontrivial ideal of R . is a summand of all ideals , I ./ / D ;:::; m m 1 m ./ , the map  .R=.// may be identified with the given map Š and, since ./ m ! ./M . .M=./M/ The problem is local, so we may assume that X and Y are Proof of Theorem 6.19: m 2 I I affine. Since any homomorphism of sheaves must annihilate ! O , we may Y Y Y identify a homomorphism / ;:::;' .' m 1 2 n ! I I W ' O = Y Y Y 2 m with the composition I = I I be the ideal ! O R ̋ . Let I O  ! ' m X Y Y Y Y o n X ˇ ˇ and mod .g/ '  g ; I g 2 I g WD g  I C j j j ' j Y Y j P . I  ./ I D I and note that  j ' Y Y j From I , we construct the family ' Y  X T Y m T m ̨ ˇ projection Spec k T m

229 Tangent spaces to Fano and Hilbert schemes Section 6.4 215 Y where is defined by I ̨ . If we set all the  , so D 0 , then I I becomes equal to ' ' j T Y m is indeed the pullback of ˇ . m in I We may identify / with the graph of ' W I I ! O =../ ' Y Y Y   M m  O O I Š I ̊ ̊ j Y Y Y Y   M O  O ̊  j Y X 2 /: Œç=../ D C ./ I O X Y I ./ O Thus D ./ I \ , and it follows that ' X Y ./. = I / / D ./ O O =. I O \ ./ ' ' X X X D ./ O =./ I Y X m m . = I Š / O Š O : Y X Y . By Lemma 6.20, I O is flat over R = m ' X Conversely, given an R -algebra of the form m 2 C I /; S WD O Œç=../ X the statement that Y is the pullback of Y T  WD Spec S over the morphism Spec k m T m 2 and using that is congruent to modulo ./ . Multiplying by ./ I ./ I D 0 , means that Y I  ./ I . . If S is flat over R we see that , then we must have I \ ./ D ./ I m Y Y I I is the graph of a homomorphism =./ Putting these facts together, we see that Y m ! Š O ./ =./ I , and this is the inverse of the construction above. I O Y Y X Y T These results identify both the Zariski tangent space of the Hilbert scheme H ŒYç;H Y  X at the point corresponding to Y , and the vector space of global sections of the of normal sheaf, with the set of first-order deformations of Y in X , which we have already identified with the set Mor .T ;H/ . Since our goal is to compute the dimension of 1 ŒYç one of these two vector spaces in terms of the dimension of the dimension of the other, we must also ensure that the identification of sets preserves the vector space structure. This is the new content of the following result: n  X is a subscheme of a Theorem 6.21. -scheme X  P Suppose that , and let H Y k Y . If ŒYç 2 H denotes the point corresponding to Y , then be the Hilbert scheme of 0 2 // T . H om ; Š . I = I H O O Y ŒYç=H Y=X Y Y=X as vector spaces. Theorem 6.13 is the special case where Y is a k -plane in X . Proof of Theorem 6.21: We will show how to give the set of morphisms T ! H the 1 structure of a vector space, and prove that this third structure is compatible with the bijections we have already given.

230 216 Chapter 6 Lines on hypersurfaces Mor ;H/ are similar, The rules for addition and scalar multiplication in the set .T 1 ŒYç and the one for addition is more complicated, so will define addition, and check that it is compatible with the identifications of Lemma 6.18 and Theorem 6.19. We leave the analogous treatment of scalar multiplication to the reader. 2 D Œ As before, we set ;:::; (we will ç=. R ;:::; Spec R k and T D / m m 1 m m m 1 D 1 and m D 2 ). A morphism of schemes ‰ only use the cases T m ! H sending W 2 corresponds to a homomorphism W m the closed point to ! k  or, ̊ k  ŒYç 1 2 H;ŒYç ; m W equivalently, a pair of homomorphisms , or a pair of morphisms ! k 1 2 H;ŒYç with itself in the ;‰ T W T is the coproduct of ! H (in fancier language: T ‰ 1 2 1 1 2 category of pointed schemes). Moreover, there is an addition map plus / . T ! T 1 2  .  / T R . This map has that embeds as the closed subscheme with ideal 2 1 1 2 ı ‰ plus / W T . ! H is the morphism corresponding to the sum the property that 1 . C k W ! m 1 2 H;ŒYç Y be the family obtained by pulling back the universal family along ‰ , and Let i ' i 2 ! I W = I ' be the homomorphism corresponding to this flat family. We have let O i Y Y Y a pullback diagram  - Y Y Y 2 ' ' 1 2 ? ? ? -  T T T 2 1 1 . To show ! T of flat families, where is the family obtained by pulling back along ‰ Y 2 2 ;H/ .T that the addition law on the set Mor agrees with addition in the vector space 1 ŒYç 2 0 H om along the H , it suffices to show that the pullback of Y . = // I . ; O I 2 O Y Y=X Y Y=X T plus . T ! / is the family Y map W . 2 ' C 1 ' 2 1 2 2 ! , so that the ideal = I ' Let W O I Y be the homomorphism corresponding to 2 Y Y Y of Y R is the ideal I ! . If we compose ' with the map induced by the projection R 2 ' 1 2 and  ' annihilating , and similarly for  is in we get the map ' ' . It follows that 2 1 1 2 fact the map  ' 1 ' 2 2 2 = I I ! O : Y Y Y Thus if we pull back Y from the structure along the map . plus / , that is, factor out   1 2 2 sheaf of Y , the resulting algebra corresponds to the map ' C ' , as required. 2 s 1 Associated to any family   X  B Y ! B of subschemes of is the union of the schemes in the family , defined to be the image X of Y B X . In this spirit, if Y  X are projective schemes, and under the projection to

231 Tangent spaces to Fano and Hilbert schemes Section 6.4 217 Y in , then we define the subscheme swept is a subscheme of the Hilbert scheme of X 0 0 to be the union Y out by Y of the schemes in the restriction to B of the universal B D B . H family over 0 Y in the case where Y We can now give a bound on the Zariski tangent spaces to are smooth. Suppose that 2 Y is a point of one of the schemes Y represented X and p 0 Y contains the tangent space to at p by points of Y at B , so it is . The tangent space to p 0 Y T in T enough to bound the image of X=T of the normal Y , which is the fiber at p p p p N / . bundle of Y in X . p Y=X T Y “moves” as Intuitively, the amount the tangent space moves in B is measured Y p B at ŒYç , although some tangent vectors to B by the tangent space to may produce trivial motions of T Y . Of course T , and by Theorem 6.21 the latter is B  T H p ŒYç ŒYç 0 N / . Let ' H be the evaluation map . p;Y Y=X 0 T X=T . N Y: / ! . N D / W ' H p p p p;Y Y=X Y=X Proposition 6.22. Y  X be smooth projective schemes, and let B  H be a Let Y Y X containing the point ŒYç . If p 2 closed subscheme of the Hilbert scheme of in 0 Y is the subscheme swept out by B , then and 0 Y T =T B/: Y  ' .T p p p;Y ŒYç This will follow directly from the following lemma: Lemma 6.23. Let Z  Y be closed subschemes of a scheme X , and let Z  ;Y   2 Y /  X be first-order deformations of Z and k in X corresponding to the Œç=. Spec 0 0 H if and . N sections / and  2 H  . N 2 / . The scheme Z Y is contained in   Y=X Z=X  and  are equal under the maps only if the images of 0 2 . N / D Hom H  / O . I ; O Z Z=X Z=X X ? O . I Hom ; / O Z Y=X X 6 0 H I . N  / D Hom / O . 2 ; O Y Y=X Y=X X . If I  I and and the projection O Y induced by the inclusion O ! Z Y Z=X Y=X X are smooth, or more generally if  X is locally a complete intersection, then Y 0 om . / . I j ; O N / Š N . j Hom , and thus H H Š . I / ; O O O Z Z Z Z Y=X Y=X Y=X Y=X X X See Figure 6.3. Proof: The statement is local, so we can assume Z;Y and X are affine. We regard the O global sections and  as module homomorphisms I O !  ! and I . Z Y Y=X Z=X

232 218 Chapter 6 Lines on hypersurfaces Y T D T Y Z .Z/ .Z/  Z .Z/ modulo the tangent line T to  at Z , so the deformation Figure 6.3 .Z/ Y corresponding to  keeps Z inside the deformation of of the point corresponding Z Y  to .   The schemes Z X ;Y T are given by the ideals 1   0 0 f f Df C j f 2 I g and f I  .f / mod I  Z=X Z=X and 0 0 g C g g j I 2 I I and g g  Df mod .g/  Y=X Y=X O D ̋ R . in O  ̊ O 1 X X X Accordingly, we have Z — if and only if  Y I — that is, I      .f / mod I ; for all f 2 I .f /  Z=X Y=X which is the first statement of the lemma. 2 = I The second statement holds because, with the given hypothesis, is a I Y=X Y=X vector bundle, and thus 2 . I O ; H / / D H om . I O = I om ; Z Z Y=X Y=X Y=X 2 om . I ̋ I D O ; O H / = Z Y X Y=X Y=X  N j D : Z Y=X Finally, we use the notion of first-order deformation to see Corollary 6.17 geometri- 3 3  W X ! P cally, via the Gauss map G sending p 2 X to the tangent plane T P X  p X 3 G to the dual line to a line L  X  P (see Section 2.1.3). The restriction of sends L X 3 ?  j H P L Df 2 L  H g ;

233 Tangent spaces to Fano and Hilbert schemes Section 6.4 219 Figure 6.4 The tangent planes to a smooth quadric surface along a line wind once around the line, but in the case of a smooth cubic surface they wind around twice. , has and this map, being given by the partial derivatives of the defining equation of X degree 1 d . Thus, for example, as we travel along a line on a smooth quadric surface rotate once around the line; on a smooth cubic surface , Q X , the tangent planes to Q z by contrast, they wind twice around the line (see Figure 6.4). But if is a first-order L 3 deformation of L , the direction of motion of a point p 2 L — that is, the 2-plane in P L and the normal vector .p/ , where  is the section of the normal bundle spanned by , X on — is linear in p N L . It is thus impossible to find a first-order deformation of 3 L= P or on any smooth surface of higher degree. Note that if X is singular at a point of L , the partial derivatives of the defining ? have a common zero along will , and so the degree of G L W L ! L X equation of X be less than 1 . Thus, for example, the tangent planes to a quadric cone are constant d L is a line on a cubic surface with an ordinary X along a line of its ruling, and if  L double point on L . In this case, there will exist the Gauss map will have degree 1 on first-order deformations of L on X — as we will see shortly in Section 6.7 6.4.3 Normal bundles of k -planes on hypersurfaces T F .X/ to a Fano scheme In order to apply the description of the tangent space L k , we need to know something about .X/ of k -planes on a hypersurface X at a point L F k the normal bundle of in X . L n L Suppose that  P  is a k -plane on a (not necessarily smooth) hypersurface X n of degree d in P L . Choose coordinates so that the ideal of X is I , D .x / ;:::;x n L 1 k C I D and let .g/  I . There is a unique expression L X n X D g C / ;:::;x .x h x g i i 0 k 1 C k D i

234 220 Chapter 6 Lines on hypersurfaces 2 h 2 .x , is the ;:::;x L / with . Differentiating, we see that g , as a form on i n C k 1 restriction to of the derivative @g=@x . L i n linear forms, the normal bundle of  n k is generated by L Since the ideal of P n n k n .1/ O L P , and, similarly, the normal bundle of X in P in is O is .d/ . Thus the X L n , and the left exact sequence of part (b) of Proposition 6.15 restriction j .d/ N O is L L P X= takes the form ;:::;g .g D ̨ / n k C 1 n k N .1/ O ! ! (6.1) 0 ! O .d/: L L=X L With notation as above, let Proposition 6.24. n k O ;:::;g ! / D ̨ .g W .d/: .1/ O n L 1 C k L The map ̨ is a surjection of sheaves if and only if the hypersurface X is smooth (a) L along . The map ̨ is surjective on global sections if and only if the point ŒLç is a smooth (b) F ŒLç .X/ and the dimension of F is equal to the “expected .X/ at point on k k  C k d . dimension” C k/ .k 1/.n k (c) The map ̨ is injective on global sections if and only if the point ŒLç is an isolated reduced (that is, smooth) point of .X/ F . k Proof: are all zero, so , the derivatives of L along L X X is smooth at a (a) Since  g 2 L if and only if at least one of the normal derivatives g @g=@x D point p , for i > k , i i is nonzero at . This is the condition that ̨ is surjective as a map of sheaves. p at any point is at least .X/ (b) By Corollary 6.2, the dimension of F k   k C d ; D WD .k C 1/.n k/ k T .X/ is smooth of dimension D at ŒLç if and only if the tangent space so D F .X/ F k ŒLç k 0 N has dimension D . Since H L=X   k C d 0 k n 0 D .k C 1/.n k/ and dim H O . O H .d// D dim . ; .1// L L k 0 D H (before the proposition) that N we see from the exact sequence in dim D (6.1) L=X ̨ is surjective on global sections. if and only if ŒLç is an isolated reduced point of F (c) The condition that .X/ is the condition that k 0 H .X/ D T F N D 0 , and by the argument of part (b) this happens if and only k ŒLç L=X if is injective on global sections. ̨ We can unpack the conditions of Proposition 6.24 as follows: The condition of part (a) is equivalent to saying that the components g of the map ̨ do not all vanish i simultaneously at a point of L .

235 Tangent spaces to Fano and Hilbert schemes Section 6.4 221 (6.1) , and assuming that is smooth along L so that ̨ is Using the exact sequence X is surjective on global ̨ a surjection of sheaves, we see that the condition of part (b) that 1 . N H D 0 . On the other hand, the global sections is equivalent to the condition / L=X 1 n .1/ of the -th summand O sections .1/  O i ;:::;x to the sections x ̨ map by 0 L k L g x , so the condition of surjectivity on sections is also equivalent to g ;:::;x 0 k k C i C k i the condition that the ideal . .g ;:::;g d / contains every form of degree n C k 1 Similarly, it follows from the exact sequence that the condition of part (c) is equiv- 0 D . N alent to the condition / H 0 . This means that there are no maps O to the L L=X ̨ or, more concretely, that the kernel of have no linear syzygies. g i L , we can Although part (a) of Proposition 6.24 tells only about smoothness along do a little better: Bertini’s theorem tells us that the general member of a linear series can X with only be singular along the base locus of the series, and it follows that the general n k ̨ W a given map ̨ is smooth except possibly along L . Thus if .1/ ! O is .d/ O L L X containing any surjective map of sheaves, there is a smooth hypersurface such that L N D Ker ̨ . L=X Example 6.25 . The following gives another treatment of Corol- (Cubic surfaces again) 3  P n , we have lary 6.17. In the case of a cubic surfaces D d X 3 and the expected D dimension of F .X/ is D D 0 . If we choose 1 2 2 ; g x D D g x 3 2 0 1 g and g obviously have then the conditions in all three parts of Proposition 6.24 apply: 3 2 1 g ; because and g are relatively prime quadratic forms, they no common zeros in P 3 2 have no linear syzygies; and since 2 2 3 2 2 3 ;x .x ;x / ç; .x ;x ;x ;x /.x Œx x k ;x  x D / ;x ;x 3 0 1 0 1 2 1 0 0 1 1 1 0 0 the map ̨ is surjective on global sections. Since the numbers of global sections of the source and target of ̨ are equal, the map ̨ is injective on global sections as well. We can see this directly, too: Because g are relatively prime quadratic forms, the kernel ;g 3 2 ̨ is of  g 3 g 2 2 ; 1/ O ! . .1/ O L L 0 N . D O 0 . 1/ , and we see again that H D . N so / L L=X L=X L will be an isolated smooth point of F .X/ From all this, we see that , where X is 1 2 2 x . Although this hypersurface x 0 C the hypersurface defined by the equation x D x 3 2 0 1 is not smooth, Bertini’s theorem, as above, shows that there are smooth cubics having the same map ̨ . Since the rank of a linear transformation is upper-semicontinuous as the transformation varies, this will also be true for the general cubic surface containing a 3 line. By Corollary 6.2, every cubic surface in P contains lines. One special case of Proposition 6.24 shows that a smooth hypersurface of degree d > 1 cannot contain a plane of more than half its dimension:

236 222 Chapter 6 Lines on hypersurfaces n Let Corollary 6.26. be a hypersurface of degree d > 1 . If L  X is a k -plane P X  X is smooth along L , then , and X on n 1  : k 2 4 P — For example, there are no 2-planes on a smooth quadric hypersurface in even though the “expected dimension” '.4;2;2/ is 0. This implies that all singular quadrics contain families of 2-planes of positive dimension — of course, it is easy to see this directly. k Proof: 1/=2 , then k C 1 > n k , so n If k forms on P k > .n of strictly positive degree must have a common zero, and we can apply part (a) of Proposition 6.24. Corollary 6.26 is a special case of a corollary of the Lefschetz hyperplane Remark. n P theorem (see Appendix C), which tells us in this case that if  is a smooth X C Y hypersurface and X is any subvariety of dimension k > .n 1/=2 , then  .X/ j deg deg .Y/: In the case of planes of the maximal dimension allowed by Corollary 6.26, Proposi- tion 6.24 gives us particularly sharp information; note that this applies, in particular, to 3 lines on surfaces in , and thus generalizes Corollary 6.17: P n  X Let Corollary 6.27. be a hypersurface of degree d  3 containing a k -plane P L with k D .n 1/=2 . If X is smooth along L then ŒLç is an isolated smooth point of 3 the Fano scheme .X/ . If n D d D 3 — that is, if X  P F is a cubic surface — then k the converse is also true. g are general If, in the setting of Proposition 6.24 we take an example where the i k variables vanishing at some point of 1 in forms of degree C 1 P d with d D 2;k > 1 k or d > 3;k  1 , then the g will have no linear syzygies, so the corresponding L  X i be a smooth point on the Fano scheme, though is singular at a point of L . Thus the X “converse” part of the corollary cannot be extended to these cases. Proof of Corollary 6.27: If is smooth along L , then by Proposition 6.24 the k C 1 X forms g have no common zeros. It follows that they are a regular of degree d 1 i 1  2 , so, again by d sequence, so all the relations among them are also of degree ŒLç is a smooth point of F . .X/ Proposition 6.24, k In the case of a cubic surface, g and g are quadratic forms in two variables. If 2 3 they have a zero in common then they have a linear common factor, so they have a linear syzygy.

237 Tangent spaces to Fano and Hilbert schemes Section 6.4 223 4 X P Despite the nonexistence of 2-planes on smooth quadric hypersurfaces  and other examples coming from Corollary 6.26, the situation becomes uniform for 3 . The proof for the general case is quite complicated, and hypersurfaces of degree d  we only sketch it. In the next section we give a complete and independent treatment for the case of lines.  C k d Set 1/.n C .k D ' k/ Theorem 6.28. . k k D 1 or d  3 and '  0 , then every hypersurface of degree d contains If (a) n X of degree d k P -planes, and the general hypersurface has dim F . .X/ D ' in k , then If 0 and X is a general hypersurface containing a given k -plane L  L is (b) ' F .X/ . an isolated smooth point of k See Exercise 6.59 for an example that can be worked out directly. (a) The first part follows from the second using Corollary 6.2. For the second Proof: part we use Proposition 6.24. We must show that, under the given hypotheses, a general .n k/ -dimensional vector space of forms of degree d 1 generates an ideal containing d . all the forms of degree On the other hand, for part (b) we must show that a general .n k/ -dimensional vector space of forms of degree d 1 generates an ideal without linear syzygies. These two statements together say that if g is a general collection of ;:::;g 1 n C C k k k forms of degree d 1 in k C 1 variables, then the degree- d component of the n  ̊ d C k ideal .g / has dimension equal to min ;:::;g .k C 1/.n k/; . This is a n C k 1 k C k special case of the formula for the maximal Hilbert function of a homogeneous ideal ̈ with generators in given degrees conjectured in Fr oberg [1985]. This particular case of ̈ oberg’s conjecture was proved in Hochster and Laksov [1987, Theorem 1]. Fr 6.4.4 The case of lines The case k 1 of lines is special because, very much in contrast with the general D 1 completely. The following result is situation, we can classify vector bundles on P sometimes attributed to Grothendieck, although equivalent forms go back at least to the theory of matrix pencils of Kronecker and Weierstrass: 1 E on Theorem 6.29. P is a direct sum of line bundles; that is, Any vector bundle r M E D / .e O 1 i P 1 D i for some integers e ;:::;e . 1 r n The analogous statement is false for bundles on projective space P of dimension n  2 (see for example Exercise 5.41).

238 224 Chapter 6 Lines on hypersurfaces We use the Riemann–Roch theorem for vector bundles on curves. Riemann– Proof: Roch theorems in general will be discussed in Chapter 14, where we will also discuss 1 P . The reader may wish to glance more aspects of the behavior of vector bundles on ahead or, since we will not make logical use of Theorem 6.29, defer reading this proof until then. That said, we start with a basic observation: An exact sequence of vector bundles ̨ ! E ! G ! 0 ! 0 F splits if and only if there exists a map ˇ W G ! F such that ̨ ı on any variety D Id X . ˇ G om . G ; F / ! H om H G ; G / given by composition This will be the case whenever the map . ̨ is surjective on global sections; from the exactness of the sequence with ! H om . G ; E / ! H om . G ; F / ! H 0 . G ; G / ! 0; om 1 1  H om . G ; E // D H this will in turn be the case whenever . G . ̋ E / D 0 . H 1 E P , with first Chern class of degree d . is a vector bundle of rank 2 on Now suppose By Riemann–Roch, we have 0 . /  d C 2 I h E  of E vanishing at from this we may deduce the existence of a nonzero global section 1 .m/ , m P  , or equivalently of an inclusion of vector bundles O E ! d=2 points of 1 P with m  d=2 . We thus have an exact sequence 0 ! O 0; ! .m/ ! E ! O m/ .d 1 1 P P d  0 , we have and, since 2m  1 1 d// .2m O . 0: D O H m/; D H .d H .m// om . O 1 1 1 P P P E Š O In this case, we conclude that O .m/ ̊ .d m/ . 1 1 P P E of general rank r follows by induction: If we let L  E be a The case of a bundle m , we get a sequence sub-line bundle of maximal degree ̨ ! O 0; ! .m/ ! E 0 ! F 1 P F .e L for Š O m  with by induction a direct sum of line bundles / . Moreover, e 1 i i i P 1 all : If > m for some i , then ̨ i ; . L > 2m / would be a bundle of rank 2 and degree e i i by the rank-2 case, this would contradict the maximality of m . Thus this sequence splits, and we are done. We remark in passing that vector bundles on higher-dimensional projective spaces n P remain mysterious, even for n D 2 , and open problems regarding them abound. To n P mention just one, it is unknown whether there exist vector bundles of rank 2 on , other than direct sums of line bundles, when n  6 . Interestingly, though, Theorem 6.29

239 Tangent spaces to Fano and Hilbert schemes Section 6.4 225 provides a tool for the study of bundles on higher-dimensional projective spaces, via the notion of , which we will discuss in Section 14.4 jumping lines n  P To return to our discussion of linear spaces on hypersurfaces, suppose that X  a line. We choose coordinates so that L is L is a hypersurface of degree d X and x x defined by 0 . As before, we write the equation of X in the form DD D n 2 n X x h; g C .x / ;x 1 i 0 i 2 i D 1 n 2 ! O .d/ h .g ;:::;g / W O 2 .x ;:::;x . / with , and we let ̨ be the map 2 2 n n L L .X/ is ' WD 2n In this situation, the expected dimension of the Fano scheme 3 d . F 1 We will make use of this notation throughout this subsection. We can say exactly what normal bundles of lines in hypersurfaces are possible. 1 Since any vector bundle on L is a direct sum of line bundles, we may write Š P L n 2 Š N . O / .e 1 i L=X 1 P . There exists a smooth hypersurface  d  1 and n 3 Suppose that Proposition 6.30. L n 2 n 1 n Š Š  X such that N , and a line L d of degree P in , X P .e O / i L=X P D i 1 if and only if n 2 X and 1 e for all i  d: 1 e D n i i D i 1 L , then, from the fact that there is Š Proof: O N If the normal bundle is .e / 1 i L=X P n 1 n .1/ i for all e Š O an inclusion N . Computing 1  ! , it follows that N i L=X P L= 1 P 1 Chern classes from the exact sequence of sheaves on P n 2 M 1 n 0 ! O .e 0; / ! O ! .d/ O ! .1/ 1 1 i 1 P P P 1 i D P e . D n we get 1 d i e Conversely, suppose the satisfy the given conditions. To simplify the notation, i L n 2 n 1 / .e O and be ̨ be any map, and let G G D O D F let ! F .1/ . Let ˇ W 1 i 1 1 i D P P the map G ! O minors of the matrix 2/ .d/ given by the matrix of .n 2/  .n 1 P 1 ̨ˇ is zero because the i -th entry of the , with appropriate signs. The composition of ˇ composite matrix is the Cauchy expansion of the determinant of a matrix obtained from -th column. i by repeating the ˇ 1 is the composite map More formally and invariantly, ̨ V 2 n  V V O ̋ ˇ 1/ .n 1 2 n 2 n P   O Š G ̋ .n G 1/ O ! 1/ ̋ F Š .n O .d/; 1 1 1 P P P V n 2  where we have used an identification of .n 1/ ̋ G with O corresponding to a global section of G 1 P V 1 n O O G . : n C 1/ ̋ D 1 1 P P

240 226 Lines on hypersurfaces Chapter 6 ˇ of the form If we take 1 0 1 e 1  0 0 0 x 0 0 C B 1 e 1 e 2 2 B C 0  x 0 0 x C B 0 1 B C : : e 1 e 1 C B 3 3 : : : x 0 x  : B C 1 0 C B : 1 e C B 4 : D ˇ ; : 0 x 0 0 0 C B 1 C B : : : : C B e 1 n 4 : : : : : 0 : x : : C B 0 B C e 1 1 e 3 n n 3 C B x 0 0 0  x 0 1 A @ e 1 2 n 0 x  0 0 0 1 1 d  .n 2/ minor will be x and the bottom then the top .n 2/ .n 2/  .n 2/ 0 d 1 minor will be . This shows that the map ̨ will be an epimorphism of sheaves, so x 1 X containing L will be smooth. By Eisenbud [1995, that the general such hypersurface Theorem 20.9], the sequence ˇ ̨ F 0 ! G ! ! O 0 ! .d/ 1 P is exact, so Š F . N L=X n 2n 3 , then there exists a pair .X;L/ with   P If a smooth Corollary 6.31. d X and hypersurface of degree  X a line such that F d .X/ is smooth of dimension L 1 2n 3 d in a neighborhood of ŒLç . Using Proposition 6.30, we observe that, if d  2n 3 , we can choose all Proof: 0 and hence to be  1 . With this choice, dim H the . O e i .e for all // D e 1 C 1 i i i P 0 d . N everywhere, / D 2n dim 3 d . Since F H .X/ has dimension at least 2n 3 1 L=X the result follows. n in 2n 3 , then every hypersurface of degree d  P If contains Corollary 6.32. d a line. The universal Fano scheme ˆ.n;d;1/ . N d C 2n 3 is irreducible of dimension Proof: Moreover, Corollary 6.31 asserts that at some point .X;L/ 2 ˆ the fiber dimension of the N ˆ.n;d;1/ ! P d is 2n projection 3 . It follows that this projection is surjective. 3 We have seen above that the Fano scheme of any smooth cubic surface in is P reduced and of the correct dimension. We can now say something about the higher- dimensional case as well: The Fano scheme of lines on any smooth hypersurface of degree d  3 Corollary 6.33. is smooth and of dimension 2n 3 d . But if n  4 and d  4 , then there exist smooth n hypersurfaces of degree in P whose Fano schemes are singular or of dimension d . 3 d > 2n

241 Lines on quintic threefolds and beyond 227 Section 6.4  3 , then for any e Proof: ;:::;e d We follow the notation of Proposition 6.30. If 2 1 n  allowed by the conditions of the proposition we have that all the , and thus e 1 i 0 N h / D . N / . D 2n 3 , proving that the Fano scheme is smooth and of L=X L=X expected dimension at . L  4 and d  4 then we can take e On the other hand, if D  D e n 1 D 3 n 1 0 and , so the Fano d D 2 d  2 . In this case h e . N / D 2n 6 > 2n 3 n 2 L=X . L scheme is singular or of “too large” dimension at The first statement of Corollary 6.33 is an easy case of the conjecture of Debarre and de Jong, which we will discuss further in Section 6.8. 6.5 Lines on quintic threefolds and beyond We can now answer the first of the keynote questions of this chapter: How many 4  P ? More generally, we can now lines are contained in a general quintic threefold X X of degree d D 2n 3 compute the number of distinct lines on a general hypersurface n P , the case in which the expected dimension of the family of lines is zero. in The set-up is the same as that for the lines on a cubic surface: The defining equation g d  gives a section  on the Grassmannian of the bundle Sym of the hypersurface S X g G , the zero locus of  .1;n/ is then the Fano scheme F , and .X/ of lines on X 1 g F .X/ has the expected dimension 0) the degree m of this scheme is the (assuming 1 d  d C 1 A degree of the top Chern class S c / 2 .1;n// . If we can show in . Sym . G C 1 d 0 H X . N , then it follows as in the previous / D 0 for each line L  addition that L=X section that the Fano scheme is zero-dimensional and reduced, so the actual number of X m . distinct lines on is exactly To calculate the Chern class we could use the splitting principle. The computation is n reasonable for 4 , d D 5 , the case of the quintic threefold, but becomes successively D more complicated for larger n and d . Schubert2 (in Macaulay2 ) instead deduces it from ̈ a Gr Schubert2 script that computes the numbers obner basis for the Chow ring. Here is a n for 3;:::;20 , along with its output: D loadPackage "Schubert2" grassmannian = (m,n) -> flagBundle({m+1, n-m}) time for n from 3 to 20 do( G=grassmannian(1,n); (S,Q) = G.Bundles; d = 2 n-3; * print integral chern symmetricPower(d, dual S)) 27 2875 698005 305093061 210480374951 210776836330775

242 228 Chapter 6 Lines on hypersurfaces 289139638632755625 520764738758073845321 1192221463356102320754899 3381929766320534635615064019 11643962664020516264785825991165 47837786502063195088311032392578125 231191601420598135249236900564098773215 1298451577201796592589999161795264143531439 8386626029512440725571736265773047172289922129 61730844370508487817798328189038923397181280384657 513687287764790207960329434065844597978401438841796875 4798492409653834563672780605191070760393640761817269985515 -- used 119.123 seconds The following result gives a geometric meaning to these numbers beyond the fact that they are degrees of certain Chern classes: n  P If is a general hypersurface of degree d  1 , then the Fano X Theorem 6.34. F 3 .X/ of lines on X is reduced and has the expected dimension 2n d scheme . 1 We now have the definitive answer to Keynote Question (a): 4 A general quintic threefold P X contains exactly 2875 lines. More Corollary 6.35.  generally, the numbers in the Schubert2 output above are equal to the number of distinct 3;5;:::;37 lines on general hypersurfaces of degrees . 2;3;:::;19 and dimensions smooth cubic surface has exactly 27 distinct lines. By We have seen that every contrast, the hypothesis of generality in the preceding corollary is really necessary for quintic threefolds: By Corollary 6.33, the Fano scheme of a smooth quintic threefold may be singular or positive-dimensional (we will see in Exercises 6.62 and 6.67 that both possibilities actually occur). The 2875 lines on a quintic threefold have played a significant role in algebraic geom- etry, and even show up in physics. For example, the Lefschetz hyperplane theorem (see for example Milnor [1963]) implies that all are homologous to each other, but one 2875 can show that they are linearly independent in the group of cycles modulo algebraic equiv- alence (Ceresa and Collino [1983]). On the other hand, the number of rational curves of d degree 2875 lines are the first example, is on a general quintic threefold, of which the one of the first predictions of mirror symmetry (see for example Cox and Katz [1999]). Proof of Theorem 6.34: We already know that for general X of degree d > 2n 3 the . We have Fano scheme .X/ is empty, so we henceforward assume that d  2n 3 F 1 n seen in Corollary 6.31 that there exists a pair .X;L/ with X  P a smooth hypersurface of degree d and L  X a line such that dim T is F .X/ .X/ D 2n 3 d ; that is, F 1 1 L smooth of the expected dimension in a neighborhood of . We now use an incidence L X , the lines correspondence to deduce that, for general 2 F with this property .X/ L 1 form an open dense subset of F contains just a .X/ . In particular, if d D 2n 3 then X 1 finite number of lines, every one of which is a reduced point of F .X/ . 1

243 The universal Fano scheme and families of lines Section 6.5 229 N P d in n C 1 variables, whose points Let be the projective space of forms of degree n P we think of as hypersurfaces in . Consider the projection maps from the universal WD ˆ.n;d;1/ : Fano scheme ˆ N j Df .X;L/  G .1;n/ 2 L  X g ˆ P ' N G P .1;n/ N over the point X of P so that the fiber of is the Fano scheme F ' .X/ of X . As we 1 have seen in Proposition 6.1, is smooth and irreducible of dimension N C 2n 3 d . ˆ It follows that the fiber of ˆ has dimension  2n through any point of 3 d . ' N P where the fiber dimension of The set of points of ' is equal to 2n 3 d is open; within that, the set U of points where the fiber is smooth is also open. Corollary 6.31 X shows that this open set is nonempty; given this, it follows that if is a general hypersurface of degree d , then any component of F .X/ is generically reduced of 1 N C 2n 3 d dimension F is defined by the vanishing of a section .X/ . Since 1 d F 1 , it is locally a complete intersection. Thus of a bundle of rank C .X/ cannot 1 have embedded components, and the fact that it is generically reduced implies that it is reduced. 6.6 The universal Fano scheme and the geometry of families of lines 3 in P In Keynote Question (c) we asked: What is the degree of the surface swept S out by the lines on a cubic surface as the cubic surface moves in a general pencil? consisting of the points corresponding What is the genus of the curve G .1;3/ C  to lines on the various elements of the pencil of cubic surfaces? We can answer such questions by giving a “global” view of the universal Fano scheme as the zero locus of a section of a vector bundle, just as we have done for Fano schemes of individual hypersurfaces. We will compute the degree of S as the number of times S intersects a general line. The task of computing this number is made easier by the fact that a general point of the surface lies on only one of the lines in question (reason: a general point that lies on two lines would have to lie on lines from different surfaces in the pencil, and thus would lie in the base locus of the pencil, contradicting the assumption that it was a general point). Thus the degree of the surface is the same as the degree of the curve C  G .1;3/ in the ̈ Pl ucker embedding (see Section 4.2.3 for a more general statement).

244 230 Lines on hypersurfaces Chapter 6 n N be the space of hypersurfaces of degree Let . The incidence correspon- in P P d dence N .X;L/ 2 P  ˆ D .1;n/ j L  X g ; ˆ.n;d;1/ Df G or relative Fano scheme of lines on such hypersurfaces, universal which we call the was introduced in Section 6.1. We can learn about its global geometry by realizing it as the zero locus of a section of a bundle, just as in the case of the Fano scheme of a given hypersurface. We have seen that the maps of vector spaces n P f g!f polynomials of degree d on L g polynomials of degree on d 2 G .1;n/ fit together to form a bundle map for different L d  S ! Sym O ̋ V G .1;n/ 0 n , where V D H on the Grassmannian . O G .1;n/ .d// is the vector space of all P polynomials of degree d . Likewise, the inclusions i , V h f ! 19 V P P fit together to form a map of vector bundles on Š O T D . 1/ ! V ̋ O ; 19 19 P P 19 is the universal subbundle on P where . T ˆ.n;d;1/ , but We will put these two constructions together to understand not only N also its restriction to a general linear space of forms M  P . We denote the restriction M by ˆ.n;d;1/ j of the universal Fano scheme to . M Theorem 6.36. The universal Fano scheme ˆ.n;d;1/ j of lines on a general m - M m n dimensional linear family M of hypersurfaces of degree d in P P is reduced and D m -dimensional space in the .2n 2 C m/ 1 P d  G .1;n/ . It is the zero C of codimension d    m .d  C on Sym locus of a section of the rank- S vector bundle ̋  E .1/ O 1/ D P 2 1 c . that space, so its class is E . / 1 C d Proof: The fact that ˆ.n;d;1/ j is reduced and of the expected dimension follows M from Bertini’s theorem and the corresponding statement for ˆ (Proposition 6.1). To characterize as the zero locus of a section of a vector bundle, it likewise j ˆ.n;d;1/ M d  P . Sym d V suffices to treat the case / , the space of all forms of degree M , so that D  WD dim V D N 1 . m d  Sym V Consider the product of P and the Grassmannian G .1;n/ , and its projec- tions   1 2 d  d  .1;n/: P . P V V . /  G .1;n/ / Sym ! G Sym

245 The universal Fano scheme and families of lines Section 6.6 231 On the product, we have maps     d  d  d   O  1/ . ! Š  V Sym Sym V !  Sym S : d  1 1 2 2 V Sym P d  V corresponding to f , the composite Restricted to the fiber over the point of P Sym  O  . . Thus the zero locus of the  to j map takes a generator of 1/ d  f i h f 1 V Sym P composite map is the incidence correspondence ˆ.n;d;1/ .  Let be the corresponding global section of the bundle    d E WD H om . O 1/;  . Sym / S d  1 2 P V Sym  d   Š  ̋ S Sym  O .1/: N 2 1 P  . Moreover, if we The zero locus of the composite map is the same as the zero locus of S restrict to an open subset of the Grassmannian over which the universal subbundle  is given by the local equations we originally used to is trivial, then the vanishing of define the scheme structure on ˆ . N in the Chow ring of P G  Theorem 6.36 allows us to calculate the class of ˆ , .1;n/ which immediately gives the answers to Keynote Question (c). To express this, we will 19 G P  use the symbol for the pullback to .1;3/ of the hyperplane class on the  19 34  of cubic surfaces (and for the pullback to P space P G .1;3/ of the hyperplane 34 class on the space of quartic surfaces), and the symbols  P for the pullbacks to i;j 34 19 .1;3/ and P P  G .1;3/ of the corresponding classes in A.  .1;3// . G G The class of the universal Fano scheme ˆ.3;3;1/ of lines on cubic Corollary 6.37. 3 surfaces in P is  3   . Sym .1// c S D ̋  Œˆ.3;3;1/ç O 19 4 2 1 P 4 2 3 ; C 6 C   C 21 C / 42 D  C .11 27 1 1;1 2 2;2 2;1 while the class of the universal Fano scheme ˆ.3;4;1/ of lines on quartic surfaces 3 in is P   4   . Œˆ.3;4;1/ç S c ̋ Sym D O .1// 34 5 1 2 P 5 2 3 4 C 220 C  D C .30  320 55 C / : C 10   1 1;1 2;1 2;2 2 C If C is is the curve of lines on a general pencil of cubic surfaces, then the degree of 42 and the genus of C is 70. The number of quartic surfaces in a general pencil that contain a line is 320. 19 Restricting to a point in , we see again that a general cubic surface X will contain P 19   Œˆ.3;3;1/ç D 27 lines.

246 232 Lines on hypersurfaces Chapter 6 Œˆ.3;3;1/ç The identifications of with the and Proof of Corollary 6.37: Œˆ.3;4;1/ç given Chern classes is part of Theorem 6.36. For the explicit computations of the Chern classes one can use the splitting principle or appeal to . Here is the computation, via the splitting principle, for the case Schubert2 19 E on P of ˆ.3;3;1/ G .1;3/ : , the fourth Chern class of the bundle    S as Formally factoring the Chern class of  2   c. ̨/.1 / D 1 C  ˇ/; C  C D .1 C S 1;1 1 2 we can write    3 c.  .1/ ̋ O Sym / S 19 1 2 P .1 C 3 ̨ C /.1 C 2 ̨ C ˇ C /.1 C ̨ C 2ˇ C /.1 C 3ˇ C /; D and in particular the top Chern class is given by   3  C O c C .1/ ̋  / 2ˇ Sym . S /.3ˇ / D .3 ̨ C /.2 ̨ C ˇ C /. ̨ C 19 4 2 1 P 4 19  . G .1;3//: P A 2 Evaluating, we first have 2 C / D 9 .3 ̨ C 3 C  C  /.3ˇ ; 1 1;1 and then 2 2 C /. ̨ C 2ˇ C / D 2 .2 ̨ C C  : C 3 ˇ  C  1 1;1 1 Multiplying out, we have 4 2 3 C : C 6   .11 27 C 21 42 / C Œˆç D  C 1 1;1 2;1 2;2 2 Schubert2 Here is the corresponding code: n=3 d=3 m=19 P = flagBundle({1,m}, VariableNames=>{z,q1}) (Z,Q1)=P.Bundles V = abstractSheaf(P,Rank =>n+1) G = flagBundle({2,n-1},V,VariableNames=>{s,q}) (S,Q) = G.Bundles p = G.StructureMap ZG = pˆ (dual Z) * chern_4 (ZG symmetricPower_d dual S) ** Replacing the line “ d D 3 ” with “ d D 4 ,” we get the corresponding result for ˆ.3;4;1/ . From the computation of , we see that the number of lines on members Œˆ.3;3;1/ç of a general pencil of cubics meeting a given line is 18  42; D   Œˆç  1

247 The universal Fano scheme and families of lines Section 6.6 233 C from which we deduce that the degree of , which is equal to the degree of the surface of g.C/ C swept out by the lines on our pencil of cubics, is 42. For the genus , we use j , part (c) of Proposition 6.15 to conclude that the normal bundle of C is the bundle E C 1 3    O ̋  S Sym .1/   G .1;3/ of the bundle where E is the restriction to , whose P 1 1 2 P section defines ˆ.3;3;1/ , as in Corollary 6.37. From the exact sequence 1 P ! T 0; ! T ! 0 N ! j 1 1 C C .1;3/ P .1;3/  G G P  C= T , which is 2 2g.C/ , is we deduce that the degree of C T / D deg c N . T . / D deg .ŒCçc c . T deg deg // 1 1 1 1 1 C C .1;3/ C= P   G .1;3/ P G c c . E /.4 /; C 2/ D E . E /c . 1 4 1 4 where we have used the computation c from Proposition 5.18. We can . T 4 D / 1 1 .1;3/ G c E / by the splitting principle or by calling compute . 1 symmetricPower_d dual S) chern_1 (ZG ** 2 . E / c 6 restricts to zero on the preimage of C and we get . Using the fact that  D 4 1 1 19 a line in P , this gives 2g.C/ D deg 2 4/ C 42 /.4 6 C 2 .27 1 2;1 2;2 1 deg D 138 . / D 138; 2;2 whence D 70 . Another view of this computation is suggested in Exercise 6.54. g Finally, consider a general pencil of quartic surfaces. By Exercise 6.64, no element of the pencil will contain more than one line. It is likewise true that no line will lie on more than one element of the pencil. (If a line lay on more than one element of the pencil, it would be a component of the base locus — but, since the pencil is general, the base locus is smooth and connected.) Thus, the number of quartic surfaces that contain a line in a general pencil of quartic surfaces is the number of lines that lie on some 1 P again for . Writing  \ ˆ.3;4;1/ quartic surface in the pencil, that is, the degree of 4     S ̋ the section of  Sym defined above, this is O .1/ 1 1 2 P    4  33 : O Sym c S .1// ̋  .  deg 1 5 1 2 P By the computation of Œˆ.3;4;1/ç , this is 320. The coefficients of higher powers of  in the class of ˆ.3;3;1/ computed above have to do with the geometry of larger linear systems of cubics: For example, we will see how to answer questions about lines on a net of cubics in Exercise 6.50. 6.6.1 Lines on the quartic surfaces in a pencil Here is a slightly different approach to Keynote Question (b). Given that the set of 34 Ä in the projective space P quartic surfaces that contain some line is a hypersurface of quartic surfaces, we are asking for the degree of Ä .

248 234 Chapter 6 Lines on hypersurfaces E G , To find that number, we look again at the bundle on the Grassmannian .1;3/ 2 .1;3/ is the vector space L whose fiber over a point G 0 H E . O .4//; D L L 4  that is, the fourth symmetric power S of the dual of the universal subbundle on Sym . As before, the polynomials f and g generating the pencil define sections  G .1;3/ f 3 that lie on some element of the pencil of the bundle E . The locus of lines L and P   g  and  are dependent, so the degree of is the locus where the values of the sections g f 4 . As before, . E / 2 A .1;3// . G this locus is the degree of the fourth Chern class c Z Š 4 Schubert2 this can be computed either with the splitting principle or with , and one finds again the number 320. We will see another way of calculating the genus of the curve ˆ in the following chapter (after we have determined the number of singular cubic surfaces in a general 1 ˆ as a 27-sheeted cover of P and using Hurwitz’s theorem. pencil), by expressing 6.7 Lines on a cubic with a double point Identifying the Fano scheme F has allowed .X/ as the Hilbert scheme of lines on X 1 us to give a necessary and sufficient condition for its smoothness, and to show that it is indeed smooth in certain cases. But there are aspects of its geometry that we cannot get at in this way, such as the multiplicity of .X/ at a point L where it is not smooth. F 1 n X P of We might want to know, for example, if we can find a smooth hypersurface  3 whose Fano scheme of lines includes a point of multiplicity exactly 2, as in degree 2n Harris [1979]; or, we might ask, if has an ordinary double point, how does this affect X the number of lines it will contain? To answer such questions we must go back to the F .X/ introduced (in more generality) at the beginning of this chapter. local equations of 1 We will describe the lines on a cubic surface with one ordinary double point. Other examples can be found in Exercise 6.55, where we will consider the case of cubic surfaces with more than one double point, and in Exercise 6.62, where we will show 4  P that it is possible to find a smooth quintic hypersurface whose Fano scheme X contains an isolated double point. To this end, we will adapt the notation of Section 6.1.1 to the case of cubic surfaces. We work in an open neighborhood U  G .1;3/ of the line L x D x D 0; W 2 3 where U consists of the lines not meeting the line x can be D x U D 0 . Any line in 1 0 written uniquely as the row space of a matrix of the form   1 0 a a 3 2 A D 0 1 b b 2 3

249 Lines on a cubic with a double point Section 6.7 235 4 U a ;a ;b , with coordinates ;b ). Such a line has the parametrization A Š (so 2 2 3 3 1 3 P ;s : / ! .s 3 ;s .s /A D .s P ;s 2 ;a / s s C b b s C ;a s 3 1 2 3 0 1 2 1 0 1 0 0 0 1 3 P be a cubic surface containing L , and suppose that the point Now let X  .1;0;0;0/ L is an ordinary double point of X ; that is, the tangent cone to X D p 2 is the cone over a smooth conic curve. We assume that X has no other singularities at p . along L X at p is given by the equation We may also suppose that the tangent cone to 2 D 0 can be written in x x C x . With these choices, the defining equation g.x/ of X 3 1 2 the form 2 2 2 2 2 x D C x x x g.x/ x ̨x C x C ˇx x x C x x x C ıx C x x k; C 0 1 0 3 2 3 1 2 3 1 1 2 1 1 2 3 3 2 .x L ;x says that / where . The condition that X be smooth along k except at p 3 2 ; otherwise the coefficients ̨;:::; are arbitrary. 0 ¤ ̨ N can be computed from the As we saw in Section 6.4.3, the normal bundle L=X short exact sequence / g .g 3 2 2 ! 0 O ! N ! O .3/; .1/ L L=X L g that is not contained in ;g where the are the coefficients of x g ;x in the part of 3 2 3 3 2 2 D ;x D ̨x / and g g x x C ˇx .x ; that is, . 1 3 3 2 0 2 1 1 Since the polynomial ring in has unique factorization, the syzygies between s;t ˇx g these two forms are generated by the linear syzygy .x , so C ̨x 0 /g D 2 0 1 3 1 Š N O , and the tangent space to the Fano scheme is given by T .X/ F D 1 L L=X ŒLç 0 “smooth of not , which is one-dimensional. In particular, the Fano scheme is H N . / L=X the expected dimension” at ŒLç . F We can now write down the local equations of .X/ near L : If we substitute the 1 four coordinates from the parametrization of a line in into g , we get U 3 2 2 3 C ; st c C t t s c C c s g.s;t;a D s t/ C b C s b c t;a 2 3 1 2 3 2 0 3 c where the are the polynomials in the a that define the intersection of the Fano i;j i U . Writing this out, we find that, modulo terms of higher degree, the c are scheme with i 2 c D a ; 0 2 2 2 C c 2a ; b a C a D a C ıa C a 3 2 1 2 2 3 2 3 2 ; D b b C ̨a 2a C ˇa C C b c / C 2 a b b a C ı.a C b 3 2 3 2 2 2 3 2 3 3 3 2 2 2 2 c : C ˇb D C b ̨b b C ıb C b 3 2 3 3 2 3 2 Examining these polynomials, we see that c have independent differen- ;c c and 2 1 3 tials at the origin a ; thus, in a neighborhood of the origin the D a 0 D b D D b 3 3 2 2 zero locus of these three is a smooth curve. Moreover, the tangent line to this curve is 2 a vanishes to order exactly 2 on this curve. D not contained in the plane , so c a D 0 0 2 2

250 236 Chapter 6 Lines on hypersurfaces F supported at L is zero-dimensional, and is isomorphic to Thus the component of .X/ 1 2 Œç=. . In particular, it has multiplicity 2. Spec k / 3  is a cubic surface with X Having come this far, we can answer the question: If P , and X is otherwise smooth, how many lines will X contain? one ordinary double point p L  passing through p count with multiplicity 2, and We have seen that the lines X with multiplicity 1. Since we know that the total count, with p those not passing through pass through p ? multiplicity, is 27, the only question is: How many distinct lines on X D .1;0;0;0/ as above and expand the defining equation p To answer this, take of X g.x/ p . Since p is a double point of X , we can write around g.x ;x /; ;x ;x ;x ;x / D x C A.x / ;x ;x B.x 2 1 0 0 1 3 2 2 3 1 3 where A B homogeneous of degree 3. The lines on X is homogeneous of degree 2 and through then correspond to the common zeros of A and B . Moreover, if we write a p line L through p as the span L D p;q with q D .0;x , then, by Exercise 6.61, ;x / ;x 3 2 1 be smooth along nf p g is exactly the condition that the zero loci the condition that L X and B of .x A ;x . Thus there will be exactly six lines on ;x / intersect transversely at 3 2 1 through p . Summarizing: X 3 Proposition 6.38. Let  P be a cubic surface with an ordinary double point p . If X is smooth away from p , it contains exactly 21 lines: 6 through p and 15 not passing X p . through (Compare this with the discussion starting on page 640 of Griffiths and Harris [1994].) In Exercises 6.55–6.58, we will take up the case of cubics with more than one 3 a cubic surface X  P singularity, arriving ultimately at the statement that can have at most four isolated singular points . We have used the local equations of the Fano scheme only to describe the locus of lines on a single hypersurface. A similar approach gives some information about the lines on a linear system of hypersurfaces. As a sample application, we will see in Exercises 6.65 and 6.66 how to describe the singular locus of and tangent spaces to the 34 3  † of quartic surfaces in P locus containing a line. P 6.8 The Debarre–de Jong Conjecture n X  P all have Fano schemes of degree d By Theorem 6.34, general hypersurfaces . On the other hand, it is easy to F .X/ of the “expected” dimension ' D 2n d 3 1 > 3 whose Fano schemes have dimension > ' ; any find smooth hypersurfaces of degree 3 that contains a line is such an example. in P > 3 smooth surface of degree

251 The Debarre–de Jong Conjecture Section 6.8 237 every smooth hypersurface of degree 3 has However, Corollary 6.33 shows that  (that is, the open set of hypersurfaces for which F .X/ has ' Fano scheme of dimension 1 the expected dimension contains the open set of smooth hypersurfaces when the degree  d  n every smooth ). Further, it was shown by Harris et al. [1998] that when 3 is n P has a Fano scheme of lines of the correct dimension — in hypersurface of degree d k .X/ have the expected dimension when both d and in fact, all the are much smaller F k than n given there is very large, and examples are few. In n . But the lower bound on general, we have no idea what to conjecture for the true bound required! There is a conjecture, however, for the Fano schemes of lines. To motivate it, 2m C 1 note that a general hypersurface P containing an m -plane will be smooth  X m -plane, a (Exercise 6.68) and will contain a copy of the Grassmannian of lines in the 2 . When variety of dimension C 1 , this is larger than the expected dimension 2m d > 2m ' 2d 3 of F . Another family of such examples is given in Exercise 6.67, but, .X/ D 1 d > n just as in the examples above, that construction requires . n . If X P is a smooth hypersurface of degree d (Debarre–de Jong)  Conjecture 6.39 d  n , then the Fano scheme F d .X/ of lines on X has dimension 2n 3 with . 1 d  One striking aspect of the Debarre–de Jong conjecture is that the inequality n n  X is exactly equivalent to the condition that the anti- for a smooth hypersurface P  is ample, though it is not clear what role this might play in a proof. canonical bundle ! X Conjecture 6.39 has been proven for  5 by de Jong and Debarre, and for d d 8 by Beheshti (see for example Beheshti [2006]). One might worry that proving the  conjecture, even for small d , would involve high-dimensional geometry, but as we will now show, it would be enough to prove the conjecture for n D d . If F dim .X/ D d 3 for every smooth hypersurface of degree d Proposition 6.40. 1 d in , and d  n , then dim F P .X/ D 2n d 3 for every smooth hypersurface of 1 n degree P in . d We have already treated the case of quadrics (Proposition 4.15), so we may Proof: n  d  n . Suppose that X assume that P 3 is a smooth hypersurface and L  X  n ƒ be a general d -plane in P ƒ containing L , and let Y D is a line. Let \ X . By d Lemma 6.41 below, d in ƒ D P Y . is a smooth hypersurface of degree F .Y/ is the intersection of F The Fano scheme .X/ with the Schubert cycle 1 1 .ƒ/ †  G .1;m/ ; by the generalized principal ideal theorem, n;m n m d 3 D dim d/; F .Y/  dim 2.n F .X/ 1 1 L L whence dim , as required. F 3 .X/  2n d 1 L We have used a special case of the following extension of Bertini’s theorem:

252 238 Chapter 6 Lines on hypersurfaces m k Let  P k < n < m be a smooth hypersurface and L Š P X  X Lemma 6.41. ; let m n ƒ Š P -plane contained in  P . If is a general n -plane containing L , then the k a X D X \ ƒ is smooth if and only if n 1  2k . intersection Y n 1 < 2k then Y must be singular, by Corollary 6.26. Proof: If D 1 . Bertini’s n For the converse, we may assume by an obvious induction that m is smooth away from L . On the other hand, the locus of tangent theorem implies that Y 2 T to X at points p X L is a subvariety of dimension at most k in the hyperplanes p m  dual projective space P L will be the , while the locus of hyperplanes containing m  ? D L  P 1/ -plane . Thus, if n 1 m 2  2k , so that k < m k 1 , .m k L is tangent to X at a point of L . It follows that, then not every hyperplane containing ƒ , the intersection Y D ƒ \ X is smooth. for general d 4 , and thereby give a negative answer D We can now prove Conjecture 6.39 for for Keynote Question (d): n X P If is a smooth hypersurface of degree 4 , then the Fano scheme Theorem 6.42.  . .X/ has dimension F 7 2n 1 Proof: Proposition 6.40 shows that it is enough to consider the case n D 4 . Suppose F  F be a general .X/ is an irreducible component with dim F  2 , and let L 2 F 1 point. By Proposition 6.30, the normal bundle D N N must be either O 1/ ̊ O . L L L=X .1/ ̊ O O or take values in a line bundle con- 2/ . Either way, all global sections of N . L L 0 . It follows that, for any point p 2 L , the map H / . N tained in / ! . N D N p L=X L=X 0 2 , the normal bundle L has rank at most 1. (Since dim H F .N/  dim T X=T T  p p L O O .1/ ̊ must in fact be , but we do not need this.) . 2/ L L Let Y  X be the subvariety swept out by the lines of F  F . By Propo- .X/ 1 sition 6.22, can have dimension at most 2. But by hypothesis, Y contains a two- Y Y is a 2-plane. dimensional family of lines. From Proposition 6.3 we conclude that Corollary 6.26 tells us this is impossible, and we are done. 6.8.1 Further open problems The Debarre–de Jong conjecture deals with the dimension of the family of lines on n P X a hypersurface  , but we can also ask further questions about the geometry of F .X/ : for example, whether it is irreducible and/or reduced. Exercises 6.70–6.73, in 1 F which we show that the Fano scheme .X/ of lines on the Fermat quartic hypersurface 1 4 P X is neither, shows that the Debarre–de Jong statement cannot be strengthened  d  n . But — based on our knowledge of examples — it does seem to be the case for all that the smaller d is relative to n , the better behaved F is for an arbitrary smooth .X/ 1 n X  P of degree d . For example, the following questions are open: hypersurface n Is F is a smooth .X/ is reduced and irreducible if d  n 1 and X  P (a) 1 hypersurface of degree d ?

253 The Debarre–de Jong Conjecture 239 Section 6.8 F Can we bound the dimension of the singular locus of in terms of d ? (The (b) .X/ 1 D d the Fano scheme F .X/ is smooth, while for arguments above show that for 3 1 n it may not be reduced. What about the range 4  d  n 1 ?)  d F .X/ with k > 1 are completely open. We can The analogous questions for k and k , what is the largest n such that there exists a smooth ask, for example: Given d  d k C n dim F P .X/ > .k C 1/.n k/ X  of degree d ? Again, with hypersurface k d are bounded, but the bound given there is probably Harris et al. [1998] says that such n far too large. Finally, we can ask: Why the Fano schemes instead of other Hilbert schemes? Why on a hypersurface? Here the not look, for example, at rational curves of any degree e field is wide open. Specifically, we have an “expected” dimension: Since a rational curve n 1 n is given parametrically as the image of a map f W C  ! P P , which is specified P 1 , and two such homogeneous polynomials of degree e on P n 1 .n C 1/ -tuples have C by 1 , the the same image if and only if they differ by a scalar or by an automorphism of P H .n C 1/.e C 1/ of such curves has dimension 4 . On the other hand, the condition space 1  0 V.F/ to contain such a curve C Š P H is that f for F D 0 2 X D . O , .de// 1 P ed C conditions on X . If we expect these conditions to be which may be regarded as 1 independent then we would expect the fibers of the incidence correspondence N j ‰ Df  H .X;C/ C  X g 2 P H to have dimension N .de C 1/ , and over correspondingly to have dimension ‰ .n 1/.e C 1/ 4 C N .de C 1/ D C C .n d/e C n C e 4: N This leads us to: n If X  P is a general hypersurface of degree Conjecture 6.43. d , then X contains a rational curve of degree e if and only if I WD d/e C n C e 4 .n;d;e/ 0 .n  when this inequality is satisfied the family of such curves on has dimension .n;d;e/ . X e D 1 , but the general case is difficult We proved the conjecture in this chapter for (the case n D 4 , d D 5 alone is the Clemens conjecture , which has been the object of much study in its own right). Recently, however, there has been substantial progress: see Beheshti and Mohan Kumar [2013] and Riedl and Yang [2014]. Note that the analog of the Debarre–de Jong conjecture in this setting — that the n dimension estimate of Conjecture 6.43 holds for an arbitrary smooth X  P of degree d  n — is false; one counterexample is given in Exercise 6.74. But it might hold when d satisfies a stronger inequality with respect to n , perhaps for d  n=e .

254 240 Chapter 6 Lines on hypersurfaces 6.9 Exercises Exercise 6.44. Show that the expected number of lines on a hypersurface of degree n 3 2n  c 3 2n . Sym (that is, the degree of P S ) is always / 2 A. G .1;n// in 2n 2 n 3 in P positive, and deduce that must contain a line . every hypersurface of degree 2n (This is just a special case of Corollary 6.32; the idea here is to do it without a tangent space calculation.) 4 will P Exercise 6.45. be a general quartic threefold. By Theorem 6.42, X Let X  G .1;4// of the Fano scheme A. contain a one-parameter family of lines. Find the class in 4 .X/ F Y  P swept out by these lines. , and the degree of the surface 1 Find the class of the scheme .Q/  G .2;5/ of 2-planes on a quadric F Exercise 6.46. 2 5 P . (Do the problem first, then compare your answer to the result in Proposi-  Q tion 4.15.) Exercise 6.47. Find the expected number of 2-planes on a general quartic hypersurface 7 4  , that is, the degree of c . X  P S . / 2 A. G .2;7// Sym 15 We can also use the calculation carried out in this chapter to count lines Exercise 6.48. n D Z , simply by finding the classes of the \\ on complete intersections X  P Z 1 k schemes F . .Z .1;n// / of lines on the hypersurfaces Z G and multiplying them in A. i 1 i 5 D X \ Y Y  P Do this to find the number of lines on the intersection of two general 2 1 5 cubic hypersurfaces in P . 3  3 . Sym Exercise 6.49. S Find the Chern class / 2 A c . G .1;3// as a multiple of the 3 class . Why is this coefficient equal to the degree of the curve of lines on the  2;1 cubic surfaces in a pencil? Note that this computation does not use the universal Fano . scheme ˆ 3 3 X  Exercise 6.50. P Let g . f P be a general net of cubic surfaces in 2 t P 2 t 3 Let p 2 P (a) be a general point. How many lines containing p lie on some member X of the net? t 3 (b) Let P H be a general plane. How many lines contained in  lie on some H X of the net? member t Compare your answer to the second half of this question to the calculation in Chapter 2 of the degree of the locus of reducible plane cubics! 3 Let X  P F be a surface of degree d  3 . Show that if is positive- .X/ Exercise 6.51. 1 dimensional, then either X is a cone or X has a positive-dimensional singular locus. 4 Let X  P be a smooth cubic threefold and Exercise 6.52. H f g D X \ S 1 t t t 2 P

255 Exercises Section 6.9 241 X a general pencil of hyperplane sections of . What is the degree of the surface swept out S , and what is the genus of the curve parametrizing them? by the lines on the surfaces t Prove Theorem 6.13 using the methods of Section 6.7, that is, by writing Exercise 6.53.  G .k;n/ the local equations of F .X/ k S f g Let Exercise 6.54. be a general pencil of cubic surfaces, and let ˆ be the 1 t P 2 t incidence correspondence 1 2 ˆ Df  G .1;3/ j L  S .t;L/ g : P t 1 P Using Propositions 6.38 and 7.4, show that the projection has degree 27 and ! ˆ t S is singular, and deduce has six branch points over each of the 32 values of for which t again the conclusion of Corollary 6.37 that the genus of ˆ is 70. Extending the results of Section 6.7, suppose that X is a general cubic Exercise 6.55. surface having two ordinary double points 2 X . Describe the scheme structure of p;q .X/ L D p;q , and in particular determine the F at the point corresponding to the line 1 at L . F .X/ multiplicity of 1 3  P Let be a cubic surface and p;q 2 X isolated singular points X Exercise 6.56. X ; let L D of . Show that L is an isolated point of F and that the multiplicity .X/ p;q 1 .X/ is  4 . mult F 1 L 3 Exercise 6.57.  P Let be a cubic surface and p isolated singular points ;:::;p X 1 ı of X . Show that no three of the points p are collinear. i Use the result of the preceding two exercises to deduce the statement Exercise 6.58. 3 X a cubic surface can have at most four isolated singular points .  P that .X;ƒ/ Using the methods of Section 6.7, show that there exists a pair Exercise 6.59. 7 P a quartic hypersurface and with X  X a 2-plane such that ƒ is an isolated,  ƒ F .X/ . reduced point of 2 Exercise 6.60. Using the result of Exercise 6.59, show that the number of 2-planes on a 7  P is the number calculated in Exercise 6.47 (that is, general quartic hypersurface X F the Fano scheme is reduced for general X ). .X/ 2 3 To complete the proof of Proposition 6.38, let Exercise 6.61.  P be a cubic surface X p D .1;0;0;0/ , given as the zero locus of the cubic with one ordinary double point ;Z ;Z ;Z F.Z / D Z A.Z ;Z ;Z / C B.Z ;Z ;Z /; 2 3 3 0 1 1 1 2 0 3 2 where B homogeneous of degree 3. If we write a is homogeneous of degree 2 and A L  X through p as the span L is p;q , with q D .0;Z ;Z ;Z line / , show that X D 1 3 2 intersect transversely at smooth along nf p g if and only if the zero loci of A and B L .Z ;Z ;Z / . 1 2 3

256 242 Chapter 6 Lines on hypersurfaces 4 Show that there exists a smooth quintic threefold P X whose scheme Exercise 6.62.  .X/ F of lines contains an isolated point of multiplicity 2. 1 Let ˆ be the incidence correspondence of triples consisting of a hyper- Exercise 6.63. n P X of degree d D 2n 3 , a line L  X and a singular point p of surface lying X  L ; that is, on N n 2 .X;L;p/  G .1;n/  P P j ˆ 2 L  X and p 2 X Df g : p sing ˆ Show that is irreducible. 34 3 Exercise 6.64. be the space of quartic surfaces in P P . Let Show that the closure of the locus of quartics containing a pair of skew lines (a) has dimension 32. Show that the closure of the locus of quartics containing a pair of incident lines also (b) has dimension 32. f X is a general pencil of quartics, then no member D V.t Deduce that if F C t g G/ (c) 1 t 0 of the pencil will contain more than one line. X t 3 Exercise 6.65. and G are two quartic polynomials on P F , and that Suppose that g X D V.t be the f C t  G/ and is the pencil of quartics they generate; let  F t 1 0 F G 4  corresponding to S sections of the bundle on G .1;3/ Sym F and G . Let X be a t 3  P member of the pencil containing a line . L  Find the condition on G for L to be a reduced point of V. (a) ^ and . /  G .1;3/ F G F 1 .t;L/ 2 P .1;3/  G (b) is Show that this is equivalent to the condition that the point 4     a simple zero of the map . 1/ !  introduced in the proof of O Sym S 1 2 1 P Theorem 6.36. 3 34 † P Exercise 6.66. be the space of quartic surfaces in P Let containing a line.  Interpret the condition of the preceding problem in terms of the geometry of the pencil D around the line , and use this to answer two questions: L † (a) What is the singular locus of ? (b) What is the tangent hyperplane T † at a smooth point corresponding to a smooth X X containing a single line? quartic surface The following two exercises give constructions of smooth hypersurfaces containing families of lines of more than the expected dimension. n 2 P Exercise 6.67. Z  be any smooth hypersurface. Show that the cone p;Z  Let n n 1 1 n  P over is the hyperplane section of a smooth hypersurface X Z P P , and in n d > n there exist smooth hypersurfaces X  P hence that for whose Fano scheme F .X/ of lines has dimension strictly greater than 2n 3 d . 1

257 Exercises Section 6.9 243 n Take 2m C 1 odd, and let ƒ  P n be an m -plane. Show that there Exercise 6.68. D n ƒ of any given degree d containing  , and deduce X P exist smooth hypersurfaces n X  P once more that for d > n there exist smooth hypersurfaces whose Fano scheme d of lines has dimension strictly greater than 2n 3 .X/ . F 1 Note that the construction of Exercise 6.68 cannot be modified to provide counter- examples to the Debarre–de Jong conjecture, since by Corollary 6.26 there do not exist n  P smooth hypersurfaces X containing linear spaces of dimension strictly greater than 1/=2 . The following exercise shows that the construction of Exercise 6.67 is .n similarly extremal. It requires the use of the second fundamental form (see Section 7.4.3). n . Show that P Exercise 6.69. be a smooth hypersurface of degree d > 2 X X  Let can have at most finitely many hyperplane sections that are cones. To see some of the kinds of odd behavior the variety of lines on a smooth hypersur- face can exhibit, short of having the wrong dimension, the following series of exercises 4  P , that is, the zero locus will look at the Fermat quartic X 4 4 4 4 4 D V.Z /: C Z C X C Z Z Z C 3 1 4 0 2 The conclusion is that F has 40 irreducible components, each of which is every- .X/ 1 where nonreduced! We start with a useful more general fact: 3 p D  P Let be the cone with vertex p;C over a plane curve C of S Exercise 6.70. d  2 , and L  S any line. Show that the tangent space T .S/ F has dimension degree 1 L is everywhere nonreduced. F .S/ at least 2, and hence that 1 X Exercise 6.71. Y Show that , each a cone over a has 40 conical hyperplane sections i 2 quartic Fermat curve in . P Show that the reduced locus F .Y Exercise 6.72. / has class 4 . 1 i 3;2 red Exercise 6.73. Using your answer to Exercise 6.45, conclude that 40 [ .Y D I / F F .X/ i 1 1 1 D i in other words, F .X/ is the union of 40 double curves. 1 Exercise 6.74. Show that: 5 5 2 P (a) There exist smooth quintic hypersurfaces containing a 2-plane P   P X . (b) For such a hypersurface X , the family of conic curves on X has dimension strictly greater than the number .5;5;2/ of Conjecture 6.43.

258 Chapter 7 Singular elements of linear series Keynote Questions 2 (a) If  F C t G/ C P V.t g D f , is a general pencil of plane curves of degree d 1 1 0 t P 2 t how many of the curves C are singular? (Answer on page 253.) t 2 be a general net of plane curves. What is the degree and f  P Let g C (b) 2 t 2 t P 2  Ä traced out by the singular points of members geometric genus of the curve P of the net? What is the degree and geometric genus of the discriminant curve 2 t 2 P D j C Df is singular g ? (Answer in Section 7.6.2.) t r be a smooth nondegenerate curve of degree  P (c) Let d and genus g . How many C r H  P hyperplanes have contact of order at least r C 1 with C at some point? (Answer on page 268.) 2 C If  P (d) g , how many of d is a general pencil of plane curves of degree f 1 t P 2 t the curves C have hyperflexes (that is, lines having contact of order 4 with C )? t t (Answer on page 405.) 2 P C (e)  If f g is a general four-dimensional linear system of plane curves of 4 t P t 2 degree d C , how many of the curves have a triple point? (Answer on page 257.) t In this chapter we introduce the associated with a line bundle of principal parts L bundle X . This is a vector bundle on X whose fiber at a point on a smooth variety p 2 X is the space of Taylor series expansions around p of sections of the line bundle, up to a given order. We will use the techniques we have developed to compute the Chern classes of this bundle, and this computation will enable us to answer many questions about singular points and other special points of varieties in families. We will start out by discussing hypersurfaces in projective space, but the techniques we develop are much more broadly applicable to families of hypersurfaces in any smooth projective variety X , and in Section 7.4.2 we will see how to generalize our formulas to that case. In the last section (Section 7.7) we introduce a different approach to such questions, the “topological Hurwitz formula.”

259 Singular hypersurfaces and the universal singularity Section 7.0 245 is It is important to emphasize the standing hypothesis that the ground field k of characteristic 0. In contrast to the preceding chapters, many of the theorems in this . When it makes the geometric argument chapter are false as stated in characteristic p > 0 simpler, we will allow ourselves to work over the complex numbers, appealing to the “Lefschetz principle” to say that the results we obtain apply over any algebraically closed field of characteristic 0. 7.1 Singular hypersurfaces and the universal singularity Before starting on this path, we will take a moment to talk about loci of singular d n C n 1 N . / n P P . Let D plane curves, and more generally singular hypersurfaces in P n d in . Our primary be the projective space parametrizing all hypersurfaces of degree P N D discriminant locus P , defined as the set of singular object of interest is the  hypersurfaces. universal singular point † D A central role in this chapter will be played by the n d in P , defined as follows: † of hypersurfaces of degree n;d  2 n N n - P j p 2 Y 2 g Df P  P .Y;p/ † sing  1 ? n N gD P hypersurfaces Y of degree d in P f P n I P x as F D If we write the general form of degree a and think of it as a d on I N a .1;d/ of P bihomogeneous form of bidegree and the coordinates in the coordinates I n , then of P ;:::;x x † is defined by the bihomogeneous equations n 0 @F 0 and F D 0;:::;n; D 0 for i D @x i so † is an algebraic set. Note that the first of these equations is implied by the others (in characteristic 0!); given the dimension statement of Proposition 7.1 below, this means N n is a complete intersection of n C 1 hypersurfaces of bidegree .1;d 1/ in P P  that † . N The image † in P D is the set of singular hypersurfaces, called the discriminant . of D † The next proposition shows that ! D is a resolution of is a hypersurface, and that singularities: With notation as above, suppose that d  2 . Proposition 7.1. (a) The variety † is smooth and irreducible of dimension N 1 (that is, codimension 1 n N n are projective spaces . P C P over † ); in fact, the fibers of 1 n

260 246 Chapter 7 Singular elements of linear series The general singular hypersurface of degree d (b) has a unique singularity, which is an N is birational to its image P † . D ordinary double point. In particular,  N P . D (c) is an irreducible hypersurface in n n Let P x 2 ;:::;x be a point, and let be homogeneous coordinates on P p Proof: n 0 d 0 D / F.x ;:::;x p D =x .1;0;:::;0/ / D x . Let f.x such that =x be ;:::;x 0 n 0 n 0 1 0 F D 0 . For d  the affine equation of the hypersurface , the n C 1 coefficients of 1 the constant and linear terms f and f are equal to in the Taylor expansion of f at p 1 0 N † certain coefficients of p is a projective subspace of P F of over , so the fiber of C 1 . The first part of the proposition follows from this, and implies that n codimension D D  the discriminant .†/ is irreducible. 1 To prove the statements in the second part of the proposition, note that the fiber of n p 2 P † contains the hypersurface that is the union of d 2 hyperplanes over a point n 1 with vertex P not containing with a cone over a nonsingular quadric in p . This p hypersurface has an ordinary double point at p , and is generically reduced. By the previous argument, the hypersurfaces corresponding to points of the fiber of † over p form a linear system of hypersurfaces, with no base points other than p . Bertini’s theorem p . Thus the fiber of the shows that a general member of this system is smooth away from N over a general point of † ! D  P map W D consists of just one point, showing  1 that the map is birational onto its image. Since smoothness is an open condition on a quadratic form, the general member has only an ordinary double point at p . † , which has dimension N 1 , is birational to D The fact that D also shows that has dimension 1 , completing the proof. N N D P is difficult to write down explicitly, though of The defining equation of  course it can be computed in principle by elimination theory. There are determinantal formulas in a few cases: see for example Gelfand et al. [2008] and Eisenbud et al. n [2003]. Even in relatively simple cases such as 1 the discriminant locus has a lot of D interesting features, as a picture of the real points of the discriminant of a quartic f.a/ D 4 2 C ax x C bx C c in one variable suggests (see Figure 7.1). For a nice animation of the discriminant of a quartic polynomial, see http://youtu.be/MV2uVYqGiNc , created by Hans-Christian Graf v. Bothmer and Oliver Labs as part of their Geometrical Animations Advent Calendar. In view of Proposition 7.1, we can rephrase the first keynote question of this chapter N as asking for the number of points of intersection of a general line L  P with the D ; that is, the degree of hypersurface . How can we determine this if we cannot write D down the form? As we will see below, Chern classes provide a mechanism for doing exactly this. N There is an interpretation of the discriminant hypersurface in P that relates D to n an object previously encountered in Chapter 1. The d -th Veronese map embeds P  in d N  0 n .d// the dual P . O P H of the projective space , in such a way that the intersection P n N / , P of  with the hyperplane corresponding to a point F 2 P  is isomorphic, via . d d

261 Bundles of principal parts Section 7.1 247 Figure 7.1 Real points of the discriminant of a quartic polynomial. n to the corresponding hypersurface in P F . Thus the discriminant is the set of 0 D N  n  hyperplanes in . that have singular intersection with P / , or, equivalently, those P d n to . P / that contain a tangent plane to  dual variety . This is the definition of the d n P / , which we first encountered in Section 2.1.3. Proposition 7.1 shows that the dual .  d n / P  of is a hypersurface, and that the general tangent hyperplane is tangent at just . d one point, at which the intersection has an ordinary double point. 7.2 Bundles of principal parts We can simplify the problem of describing the discriminant by linearizing it. We do not ask “is the hypersurface X D V.F/ singular?”; rather, we ask for each point n 2 P p in turn the simpler question “is X singular at p ?”. This is very much analogous to our approach to lines on hypersurfaces, where instead of asking “does X contains “does contain L ?”. As in that context, this approach lines?” we asked for each line X L into a family of systems of converts a higher-degree equation in the coefficients of F linear equations, whose solution set we can then express as the vanishing of a section of a vector bundle. n 2 P For each point , we have an .n C 1/ -dimensional vector space p n p at f germs of sections of O g .d/ P D E : p  2 germs vanishing to order p g f at This space should be thought of as the vector space of first-order Taylor expansions of forms of degree d . We will see that the spaces E fit together to form a vector bundle, p 1 n .d// P . called the bundle of O first-order principal parts , which we will write as . P

262 248 Singular elements of linear series Chapter 7 of degree F  d of this vector bundle whose value at A form will give rise to a section F is the first-order Taylor expansion of locally at p , and whose vanishing p the point F 0 . locus is thus the set of singular points of the hypersurface F D has a naturally An important feature of the situation is that each vector space E p defined subspace: the space of germs vanishing at . These subspaces will, as we will p 1 see, glue together into a subbundle of P . Using the Whitney formula (Theorem 5.3), this will help evaluate the Chern classes of the bundle. C 1 ,” with We can generalize this in two ways: We can replace “2” by “ any m m d positive integer; and we can replace the forms of degree by the sections of a coherent (though in practice we will be working almost exclusively sheaf on an arbitrary variety X L be a quasi-coherent with line bundles on smooth varieties). To make this precise, let -scheme X , and write  X ; sheaf on a W k  X ! X for the projections onto the two 1 2 factors. Let I be the ideal of the diagonal in X  X . We set m 1 C m  L  . D /; L ̋ O . / I = P  2 X X  1 X which is a quasi-coherent sheaf on is smooth we call this the bundle of . When X . We will parse and explain this expression below, but first we list its very principal parts useful properties: m P have the following properties: . L / Theorem 7.2. The sheaves If 2 X is a closed point, then there is a canonical identification of the fiber p (a) m m L / ̋ .p/ of P P . L / at p with the sections of the restriction of L to the . -th-order neighborhood of p ; that is, m m C 1 m 0 ̋ .p/ D P H . L ̋ O / = m . L / X;p X;p O .p/ = m D D k . In other words, as vector spaces over X;p X;p g p at L germs of sections of f m D ̋ / L . .p/ P : m C 1 at p g f germs vanishing to order  0 0 m H If . L / is a global section, then there is a global section  // 2 H F . P 2 . L (b) F m C 1 0 H / . L whose value at O p = m is the class of F in . ̋ X;p X;p 0 (c) . L / D L P m > 0 there is a natural right exact sequence , and for each m m m 1 0; / ! P L . L / ! P ̋ Sym L . . / ! X  denotes the sheaf of k -linear differential forms on X . where X m If X is smooth and of finite type over k and L is a vector bundle on X , then P (d) . L / is a vector bundle on X , and the right exact sequences of part (c) are left exact as well.

263 Bundles of principal parts Section 7.2 249 Since the constructions all commute with restriction to open sets, we may Proof: D Spec R is affine. Thus also X  X D Spec S , where harmlessly suppose that X  L WD  L as coming from an R -module L , and then WD R S R . We may think of ̋ k 1 L ̋ R M on X  X forward by  . Pushing a (quasi-coherent) sheaf simply means  2 K -module as an R -module via the ring map R ! S considering the corresponding S to . ̋ sending r r 1 k I X In this setting, the sheaf of ideals in defining the diagonal embedding of X  I  S that is the kernel of the multiplication map X corresponds to the ideal D R ̋ ! R S R . If R is generated as a k -algebra by elements x is generated , then I i k S by the elements as an ideal of ̋ 1 1 ̋ x x . i i m can With this notation, we see that the . L / R P -module corresponding to the sheaf be written as m m C 1 ̋ R/; R/=I P .L .L/ D .L ̋ k k R 7! 1 ̋ r as above. f regarded as an -module by the action -rational point p corresponds to the maximal ideal Part (a) now follows: If the k  ' R m ! k D ; ' W x ; 7! a Ker i i then in R= m ̋ . S Š R the class of x a ̋ 1 1 ̋ x x D is x a ̋ ̋ 1 1 k i i k i k i i k i R Thus m 1 C m m C 1 L=. f x ̋ .L/ 1 1 ̋ ̋ x / g / L; R= m L D L=. f x g D a P i k k i i i R as required. Part (b) is similarly obvious from this point of view: The section can be taken to  F m C 1 .L be the image of the element F R/=I ̋ 1 in .L ̋ R/ . As the construction is ̋ k k k natural, these elements will glue to a global section when we are no longer in the affine case. Part (c) requires another important idea: The module of k -linear differentials  R= k 2 2 -module, to I=I is isomorphic, as an , which has a universal derivation ı W R ! I=I R given by ı.f / ̋ D 1 1 ̋ f f . This is plausible, since when X is smooth one can k k 2 .I=I , is ;R/ see geometrically that the normal bundle of the diagonal, which is Hom X Hom . ;R/ . See Eisenbud [1995, , which is isomorphic to the tangent bundle of R= k Section 16.8] for further discussion and a general proof. Given this fact, the obvious m m 1 2 m m C Š Sym Sym .I=I . / ! I yields the desired right exact =I surjection / k R= sequence. Finally, it is enough to prove part (d) locally at a point q 2 X X . If q is not on  the diagonal then, after localizing, I is the unit ideal, and the result is trivial, so we may L assume that D .p;p/ . Locally at p , the module q is free, so it suffices to prove the result when L D R .

264 250 Chapter 7 Singular elements of linear series X Å X X  X X Š T N X Å  X=X X  Å  X is isomorphic to the Figure 7.2 The normal bundle of the diagonal X X X . tangent bundle of W R !  Write d for the universal k -linear derivation of R . Since X is smooth, R= k  is locally free at p , and is generated there by elements d.x /;:::;d.x / , where 1 n R= k m ;:::;x / is a system of parameters at p , and thus x is the free module . Sym n 1 R= k I m / . Since R is a domain, d.x is a in the generated by the monomials of degree i prime ideal. m / is free, it suffices to show that the map . Because Sym R= k m 1 C m . ! Sym / S=I k R= m C 1 m I =I ) after localizing at the is a monomorphism (in fact, an isomorphism onto 2 2 prime ideal I Š  of the elements . Since is free on the classes mod I I=I k R= x , Nakayama’s lemma shows that, in the local ̋ / 1 1 ̋ d.x x that correspond to the i k i i k 1 ̋ ring is generated by the images of the x ̋ I 1 S x , themselves, and it follows k i k i I I 2 =I that these are a regular sequence. Thus the associated graded ring ̊ S ̊ =I I I I I I x x is a polynomial ring on the classes of the elements 1 1 ̋ , and in particular ̋ i k i k m C 1 m m =I in these elements freely generate I the monomials of degree . Consequently, I I 1 C m m m Sym S . is an isomorphism, as desired. ̋ / ! I the map =I S I k R= I I The name “bundle of principal parts,” first used by Grothendieck and Dieud- Remark. ́ e, was presumably suggested by the (conflicting) usage that the “principal part” of onn a meromorphic function of one variable at a point is the sum of the terms of negative degree in the Laurent expansion of the function around the point — a finite power series, m albeit in the inverse variables. It is not the only terminology in use: . L / would be P called the bundle of m -jets of sections of L by those studying singularities of mappings (see for example Golubitsky and Guillemin [1973, II.2]) and some algebraic geometers -jet terminology is in use in m (for example Perkinson [1996].) On the other hand, the m -jets” of a scheme X another conflicting sense in algebraic geometry: the “scheme of m C 1 Œxç=.x is used to denote the scheme parametrizing mappings from Spec k / into X . ́ So we have thought it best to stick to the Grothendieck–Dieudonn e usage.

265 Singular elements of a pencil Section 7.2 251 Example 7.3. We will not use this in any of the calculations below, but in the simplest n n O P m and L D 1 , X .d/ , it is possible to D D and most interesting case, where P 1 n P . .d// very explicitly: describe the bundle O P  n n 0; O ̊ D d  if P P 1 n . Š O P .d// P 1 n C n .d if d 1/ 0: ¤ O P This curious dichotomy is explained by the answer to a more refined question: By part (d), we have a short exact sequence 1 n n n .d/ ! !  . O 0; P .d// ! O ! 0 .d/ P P P and we can ask for its class in 1 1 1 n n n n n : Ext .d// Š Ext k D / . O O . ; . .d/; / D H n n P P P P P P P on a smooth variety X , the short exact sequence L More generally, for any line bundle in part (d) gives us a class in 1 1 / . ; . ̋ L L D H Ext /; X X X called the Atiyah class L and denoted at . L / (Atiyah [1957] and Illusie [1972]). The of 1 n .d// P O . follows at once from the more refined and more uniform formula for P result that n O D at .d// . d  ; P 1 n n ; is the class of the tautological sequence / . O 2 Ext where  n P P P / ;:::;x .x n 0 1 C n n n n . O ! 1/  ! 0 ! O 0: ! P P P m See Perkinson [1996, 2.II] and Re [2012] for an analysis of all the P . 7.3 Singular elements of a pencil 7.3.1 From pencils to degeneracy loci Using the bundle of principal parts, we can tackle a slightly more general version of Keynote Question (a): How many linear combinations of general polynomials F and G n d on P of degree have singular zero loci? By Proposition 7.1, none of the hypersurfaces X V.t F C t D G/ of the pencil will be singular at more than one point. Furthermore, 1 0 t no two elements of the pencil will be singular at the same point, since otherwise every member of the pencil would be singular there. Thus, the general form of the keynote n p 2 P question is equivalent to the question: For how many points is some element X t n of the pencil singular at p ? This, in turn, amounts to asking at how many points p 2 P

266 252 Chapter 7 Singular elements of linear series 1 n O .p/ in the fiber of P  . and are the values  .d// at p linearly dependent, .p/ G F P given that they are dependent at finitely many points? We can do this with Chern classes, provided that the degeneracy locus is reduced; we will establish this first. n and  To start, consider the behavior of the sections around a point p 2  P F G where they are dependent. At such a point, some linear combination F C t t G — which 1 0 we might as well take to be G were also zero at p , then the — vanishes to order 2. If F would have (at least) a double point at p . But Bertini’s theorem shows scheme V.F;G/ that a general complete intersection such as V.F;G/ is smooth, so this cannot happen; G.p/ thus we can assume that 0 . ¤ To show that V. ^  , we restrict our attention to an affine / is reduced at p G F where all our bundles are trivial. By Proposition 7.1, the hypersurface neighborhood of p V.F/ p , so if we work on an affine neighborhood where the bundle D has a node at C n , we is trivial, and take p to be the origin with respect to coordinates x O ;:::;x .d/ n 1 P may assume that the functions and G have Taylor expansions at p of the form F D . f C f terms of order > 2/; 2 g 1 C . terms of order  1/: D The sections  are then represented locally by the rows of the matrix and  G F    f @f=@x @f=@x 1 n :  g @g=@x @g=@x n 1 ^ The vanishing locus of   near p is, by definition, defined by the 2  2 minors G F of this matrix, and to prove that it is a reduced point we need to see that it contains n p ) with independent linear terms. Suppressing all the terms of functions (vanishing at the functions in the matrix that could not contribute to the linear terms of the minors, we get the matrix   =@x  @f 0 @f =@x n 2 1 2 : 1 0  0: 2  2 minors whose linear terms are @f =@x ;:::;@f =@x Thus there are , and these are 2 n 1 2 linearly independent because f D 0 is a smooth quadric and the characteristic is not 2. 2 As usual, if we assign multiplicities appropriately, we can extend the calculations to V. pencils whose degeneracy locus ^  is nonreduced. In Section 7.7.2 we will see / G F one way to calculate these multiplicities. 7.3.2 The Chern class of a bundle of principal parts n F;G be general forms of degree Once again, let on P . As we saw in the previous d section, the linear combinations t that are singular correspond exactly to points F C t G 1 0  are dependent. The degeneracy locus of and  and where the two sections  F G F 1 n .d// n -th Chern class of the rank- .n C 1/ bundle P , so we turn to the . O  is the G P

267 Singular elements of a pencil 253 Section 7.3 1 1 n P .d// to P . .d/ , but the computation of this class. For brevity, we will shorten O P 1 !” tensored with O .d/ P not reader should keep in mind that this is “a bundle n n 1 . Stated explicitly, if  2 denotes the class of a hyperplane in P A , we want to P / n in compute the coefficient of  1 n n C 1 / P Š .d// Œç=. 2 A. P /: c. Z Parts (c) and (d) of Theorem 7.2 give us a short exact sequence 1 n n .d/ P ! .d/ ! O 0  .d/ ! 0; ! P P 1 m n n c. O c. P .d//  c. D .d// .d// . (See Proposition 7.5 for the other P so .d/ .) P P n On the other hand,  fits into a short exact sequence P n C 1 n n n O 0 . 1/  ! O ! ! ! 0: P P P Tensoring with O .d/ , we get an exact sequence n C 1 n n n ! O ! 0 .d 1/ !  0 ! O .d/ .d/ P P P n 1 1 C 1 n . This does not mean that O P .d/ .d similar to the one involving 1/ .d/ P and P are isomorphic (they are not), but by the Whitney formula their Chern classes agree: 1 1 C n 1 C n n .d// .d 1/ c. O c. / D .1 C .d 1// D P : P Putting this formula together with the idea of the previous section, we deduce: The degree of the discriminant hypersurface in the space of forms of Proposition 7.4. n on P degree is d 1 n ; . P 1/ .d// D .n C 1/.d deg c n and this is the number of singular hypersurfaces in a general pencil of hypersurfaces of n d P . degree in In particular, this answers Keynote Question (a): A general pencil of plane curves of 2 will have 3.d degree 1/ d singular elements. It is pleasant to observe that the conclusion agrees with what we get from elementary geometry in the cases where it is easy to check, such as those of plane curves ( n D 2 ) with d D 1 or d D 2 . For d D 1 , the statement c simply means that there are no D 0 2 d D 2 corresponds to the number of singular elements in a pencil of lines. The case 2 f C , g of conics. To see that this is really 3.d 1/ singular conics in a general pencil D 3 t note that the pencil f C g consists of all conics passing through the four (distinct) base t points, and a singular element of the pencil will thus be the union of a line joining two of the points with the line joining the other two. There are indeed three such pairs of lines (see Figure 7.3).

268 254 Chapter 7 Singular elements of linear series A X C Y B are the singular elements of the pencil of plane conics containing A;B;C Figure 7.3 and Y . X We could also get the number 3 by viewing the pencil of conics as given by a 2 M of linear forms on P  whose entries vary linearly with a 3 3 symmetric matrix ; the determinant of M parameter t . t will then be a cubic polynomial in As the reader may have noticed, there is a simpler way to arrive at the formula of Proposition 7.4. We observed in Section 7.1 that the universal singularity N n p .Y;p/  P P † 2 2 Y Df g j sing N n 1 hypersurfaces of bidegree .1;d 1/ in P is a complete intersection of  P n . C N n N n and  P the pullbacks to of the hyperplane classes in P P  P ̨ , and Denoting by † has class this means that n C 1 .d 1// Œ†ç D . ̨ C     n C 1 1 C n 1 n n n n n C : CC .d ̨ 1/ ̨ C .d 1/ ̨  D 1 n N P , all the terms go to 0 except the last, from When we push this class forward to D is Œ D ç D which we can conclude that the class of the discriminant hypersurface n n 1/.d 1/ 1/.d ̨ ; that is, D D  . .†/ has degree .n C .n 1/ C 1 Why did we adopt the approach via principal parts, given this alternative? The answer is that, as we will see in Section 7.4.2, the principal parts approach can be applied to linear series on arbitrary smooth varieties; the alternative we have just given applies only to projective space. It is easy to extend the Chern class computation in Proposition 7.4 to all the m n P . .d// , and this will be useful in the rest of this chapter: O P  C n m m . / n n D .1 C .d O m// .d// P c Proposition 7.5. . : P

269 Singular elements of a pencil Section 7.3 255 Proof: We will again use the exact sequences of Theorem 7.2. With the Whitney formula, they immediately give m Y  m j n n P . c .d// O D Sym c. . /.d//: P P 0 j D To derive the formula we need, we apply Lemma 7.6 below to the exact sequence 1 C n n n n ! 1/ !  . O ! 0 0 ! O P P P n C 1 n n L U D O WD O . 1/ . To simplify the notation, we set and the line bundle .d/ . P P The lemma yields j j j 1 1 n Sym . . U /.d//  c. Sym c. Sym /.d// . U /.d// D c. P 0 n n for all . j . Combining this with the obvious equality /.d/ D O 1 Sym .d/ , we  P P  m n O . P .d// see that the product in the formula for is c P 2 1 Sym /.d// . U c. /.d// U . c. Sym n .d// O   c. ;  P 1 n O .d// c. . /.d// Sym c. U P which collapses to  m m n O /.d//: Sym .d// P D c. c . . U P But  m C n C 1 m m n . / n n n D c. Sym /.d// . c D O .d/ . . m/ O U /.d// . 1/ c. Sym P P  n C m / . n n O m/ c .d D P n C m / . n ; .d m// D C .1 yielding the formula of the proposition. Lemma 7.6. If ! A ! 0 ! C ! 0 B is a short exact sequence of vector bundles on a projective variety X with rank C D 1 , then, for any  1 , j 1 j 1 j j A L c. Sym ̋ . B / ̋ L /  c. Sym / D . . B / ̋ C ̋ L / Sym c. : / Proof: For any right exact sequence of coherent sheaves E ! F ! G ! 0; the universal property of the symmetric powers (see, for example, Eisenbud [1995,  1 there is a right exact sequence j Proposition A2.2.d]) shows that for each 1 j j j ̋ . F Sym ! Sym 0: . F / ! Sym E . G / ! /

270 256 Chapter 7 Singular elements of linear series A ; and C are vector bundles, the dual of the exact sequence in the hypothesis is Since B   D F D B A and ; exact, and we may apply the result on symmetric powers with G  . C D E D rank C D 1 , the sequence In this case, since rank E j 1 j j F . F / ! Sym Sym . ! / ! Sym E . G / ! 0 0 ̋ is left exact as well, as one sees by comparing the ranks of the three terms (this is a special case of a longer exact sequence, independent of the rank of , derived from the E Koszul complex). Since these are all bundles, dualizing preserves exactness, and we get an exact sequence   j 1  j    j . / Sym / ! ! Sym / A . . B Sym ! 0 ̋ C ! 0: B Of course the double dual of a bundle is the bundle itself, and the dual of the j -th j symmetric power is naturally isomorphic to the -th symmetric power of the dual, so all  the ’s cancel, and we can deduce the lemma from the Whitney formula. 7.3.3 Triple points of plane curves We can adapt the preceding ideas to compute the number of points of higher order in linear families of hypersurfaces. By way of example we consider the case of triple points of plane curves. N P be the projective space of all plane curves of degree d  3 , and let Let 2 0 N .C;p/ 2 Df  P † j mult P .C/  3 g : p 2 p 2 P The condition that a curve have a triple point at a given point C is six independent C , from which we see linear conditions on the coefficients of the defining equation of 2 0 ! † on the second factor are linear spaces that the fibers of the projection map P 6 N N 0 P , and hence that †  is irreducible of dimension N P 4 . It follows that the set of curves with a triple point is irreducible as well. An argument similar to that f 0 with a triple point has only for double points also shows that a general curve D N 0 ! P one. In particular, the projection map on the first factor is birational onto † N the locus ˆ P its image. It follows in turn that of curves possessing a point of  N 4 . We also see that multiplicity 3 or more is an irreducible variety of dimension C is a general curve with a triple point at p , then p is an ordinary triple point of C if ; that is, the projectivized tangent cone C T X is smooth or, equivalently, the cubic term p f of the Taylor expansion of f around p has three distinct linear factors. 3 We ask now for the degree of the variety of curves with a triple point, or, equivalently, the answer to Keynote Question (e): If ;:::;F are general polynomials of degree d F 4 0 2 (up to scalars) will on F D t F P CC t F , for how many linear combinations 0 4 t 4 0 2 have a triple point? the corresponding plane curve D V.F C /  P t t

271 Singular elements of a pencil Section 7.3 257 2  F in P If we write . O for the section defined by has a triple C , then .d// 2 F P  p vanishes at point at . An argument analogous to the one given in p if and only if F Section 7.3.1, together with the smoothness of the tangent cone at a general triple point, 5 2 minors of the map O shows that the 5  ! P 5 . O generate the maximal ideal .d// 2 2 P P F defines a curve with an locally at a general point where a linear combination of the i ordinary triple point. Thus the number of triple points in the family is the degree of the second Chern  2 . By Proposition 7.5, . O P .d// c class 2 2 P  2 2 6 2 c O . P C .d 2// : D 1 C .d// 2/ C 15.d D 4d C 4/ .1 6.d 2 P N If ˆ D ‰  P is the locus in the space of all curves of degree d Proposition 7.7. d;n 2 in of curves having a triple point, then for d  2 P 2 deg D 15.d 4d C 4/: .ˆ/ In case d D 1 , the number 15 computed is of course meaningless, because the expected dimension N of ˆ is negative — any five global sections  4 of the bundle F i 2 P .1// are everywhere-dependent. On the other hand, the number 0 computed in the . O D , which is 0, really does reflect the fact that no conics have a triple point. For d case 2 3 D d , the computation above gives 15, a number we already computed as the degree of the locus of “asterisks” in Section 2.2.3. 7.3.4 Cones As we remarked, the calculation in the preceding section is a generalization of ˆ parametrizing triples of the calculation in Section 2.2.3 of the degree of the locus 9 P parametrizing plane cubic curves. There concurrent lines (“asterisks”) in the space N is another generalization of this problem: We can ask for the degree, in the space P n in P cones , of the locus ‰ of parametrizing hypersurfaces of degree . We are now in d a position to answer that more general problem, which we will do here. We will not go through the steps in detail, since they are exactly analogous to the ‰ is the degree of the n last calculation; the upshot is that the degree of -th Chern class 1 d n P . O . By Proposition 7.5, of the bundle .d// P  1 n C d 1 d / . n n / O c P .d// C D .1 . ; P and so we have: N If ‰ D ‰ is the locus of cones in the space of all hyper-  P Proposition 7.8. d;n n d in P , then surfaces of degree    n C d 1 n .‰/ deg D : n

272 258 Chapter 7 Singular elements of linear series d Thus, for example, in case we see again that the locus of singular quadrics D 2 n 2 n C 1 , and in case d D 3 and n D P the locus of asterisks has degree 15. in is 14 P Likewise, in the space of quartic plane curves, the locus of concurrent 4-tuples of lines has degree    5   10 2 45: D D 2 2 Compare this to the calculation in Exercise 2.57! 7.4 Singular elements of linear series in general be a smooth projective variety of dimension n , and let W X . L ;W / be Let D X . We think of the elements of P W as divisors in a linear system on , and, as in X Section 7.1, we introduce the incidence correspondence X  .Y;p/ 2 P W † Df j p 2 Y g sing W with projection maps  . Also as in Section 7.1 we denote W † ! P W and  X W † ! 2 1 W D by .†/  P W the locus of singular elements of the linear series  , which we D 1 again call the discriminant . As mentioned in the introduction to this chapter, the techniques developed so far apply as well in this generality. What is missing is the analog of Proposition 7.1: We do † is irreducible of codimension n C 1 , we do not know that it not know in general that D maps birationally onto (as we will see more fully in Section 10.6, the discriminant may have dimension strictly smaller than that of ) and we do not know that the D † general singular element of W has one ordinary double point as its singularity. Thus enumerative formulas , in the sense the formulas we derive in this generality are only of Section 3.1: They apply subject to the hypothesis that the loci in question do indeed have the expected dimension, and even then only if multiplicities are taken into account. That said, we can still calculate the Chern classes of the bundle of principal parts 1 P . L / , and derive an enumerative formula for the number of singular elements of a pencil of divisors (that is, the degree of D  P W , in case D is indeed a hypersurface); we will do this in Section 7.4.1 below. in case the linear series P W is very ample. If D We note one interpretation of N P X is a smooth variety and  0 D . O j .1/;W / with W D H W . O .1// N X X P is the linear series of hyperplane sections of X , then a section in W is singular if and only if the corresponding hyperplane is tangent to X . Thus the set of points in P W

273 Singular elements of linear series in general Section 7.4 259 to corresponding to such sections is the X dual variety , and the number of singular elements in a general pencil of these sections is the degree of the dual variety. We will treat dual varieties more thoroughly in Section 10.6. 7.4.1 Number of singular elements of a pencil X W D . L ;W / a pencil of be a smooth projective variety of dimension n and Let (typically, a general pencil in a larger linear series). We can use the Chern X divisors on W . To simplify class machinery to compute the expected number of singular elements of 1 2 A the notation, we will denote the first Chern class of the line bundle .X/ , L by  ;:::;c of simply by c and the Chern classes of the cotangent bundle ;c  X . n 2 1 X From the exact sequence 1 0 ̋ L ! P  . L / ! L ! 0 ! X 1 . L / is the Chern class of P and Whitney’s formula, we see that the Chern class of  . ̋ ̊  / . Since L , we may apply the formula for . O ̊ O / D c c . / D c i i i X X X X X the Chern class of a tensor product of a line bundle (Proposition 5.17) to arrive at k   X C n i 1 1 i k . L // D . P c :  c i k i k 0 i D In particular, n X n 1 i // L . P c . D c 1 i/ C .n i n 0 D i n n 1 .n C C n c D CC 2c (7.1) 1/ : C c n 1 n 1 As remarked above, this represents only an enumerative formula for the number of singular elements of a pencil. But the calculations of Section 7.3.1 hold here as well: A singular element Y of a pencil corresponds to a reduced point of the relevant degeneracy locus if has just one ordinary double point as its singularity. Thus we have Y the following: . Let n . If W D be a smooth projective variety of dimension L ;W / is Proposition 7.9. X a pencil of divisors on having finitely many singular elements D such that ;:::;D X 1 ı (a) each D has just one singular point, i (b) that singular point is an ordinary double point, and (c) that singular point is not contained in the base locus of the pencil, ı of singular elements is the degree of the class then the number 1 n n 1 n . P 2c . L // D .n C 1/ . C n L / WD c c CC .X/; A 2 c C 1 n 1 n n where  D c / . . / and c . D c L i i 1 X

274 260 Chapter 7 Singular elements of linear series Naturally, there will be occasions when we want to apply this formula but may not be able to verify hypotheses (a)–(c) of Proposition 7.9 — for example, as we will see in Section 10.6, these hypotheses are not necessarily satisfied by a general pencil r  . It is worth asking, of hyperplane sections of a smooth projective variety P X accordingly, what can we conclude from the enumerative formula in the absence of these hypotheses. L / is nonzero, then we can conclude that the pencil W must First off, if the class . W have singular elements ; this applies to any pencil on any variety. Secondly, if is a D . V ;V / , we can form the universal general pencil in a very ample linear series L , as in Section 7.1: singular point † Df 2 P V  X j p 2 .D .v;p/ / ; g sing v D . As in the proof of  X is the divisor corresponding to the element v 2 where V P v Proposition 7.1, we see that the fibers of † over X are projective spaces of codimension n C 1 in P V , and hence that † has codimension n C 1 in P V  X ; it follows that the preim- age of a general pencil P  P V is † will be finite. We can conclude, therefore, that in W the degree of the class L / must be nonnegative, and if it is 0 then W will this situation . have no singular elements — in other words, the locus of singular elements of the linear V has codimension >1 in system V , and every singular element will have positive- P dimensional singular locus. We will see an example of a situation where this is the case in Exercise 7.28 below, and investigate the question in more detail in Section 10.6. Finally, we will see in Section 7.7.2 below a way of calculating multiplicities of the relevant degeneracy locus topologically, so that even in case the singular elements of W do not satisfy the hypothesis of having only one ordinary double point we can say something about the number of singular elements. (The conclusions of Section 7.7.2 are stated only for pencils of curves on a surface, but analogous statements hold in higher dimension as well.) 7.4.2 Pencils of curves on a surface By way of an example, we will apply the results of Proposition 7.9 to pencils of curves on surfaces. For the case d D 2 , see Exercises 7.22 and 7.23. 3 Suppose that P X is a smooth surface of degree d and that V is the linear series  D of intersections of e , so that L with surfaces of degree O . .e/ X X We claim that the three hypotheses of Proposition 7.9 are satisfied for a general pencil  V : W (a) The fact that a general singular element of W (equivalently, of V ) has only one singularity in the case e D 1 is somewhat subtle; it is equivalent to the statement that the Gauss map from the surface to its dual variety is birational. (This is sometimes false in characteristic p !) This statement is proven for all smooth hypersurfaces in Corollary 10.21.

275 Singular elements of linear series in general Section 7.4 261 e > 1 The case D 1 by using Bertini’s theorem and can be deduced from the case e Proposition 7.10. W is an ordinary double (b) The fact that the singularity of a general singular element of 1 e 1 , and when point is also tricky. Again, it follows for D D it from the case e > 1 e 3 P can be done for a general surface by an incidence correspondence/dimension X  , however, it requires the introduction X count argument (Exercise 7.42). For an arbitrary ; we will describe this in the following section and use it of the second fundamental form to prove the statement we want in Theorem 7.11. (c) Finally, the third hypothesis of Proposition 7.9 follows much as in the case of plane curves: By Bertini’s theorem, the base locus of a general pencil in a very ample linear series is smooth — in this case a set of reduced points — and a reduced point on a smooth surface cannot be the intersection of two divisors if one of them is singular. We have used: n Proposition 7.10. Ä  P be a finite subscheme with homogeneous coordinate Let S . ring Ä (a) imposes independent conditions on forms of degree deg Ä 1 . Ä Ä Ä 2 if and only if Ä fails to impose independent conditions on forms of degree (b) is contained in a line. Let l be a general linear form, and set R Ä WD S . In general, =.l/S imposes (c) Ä Ä Ä n e P is 0 in degree if and only if R in independent conditions on forms of degree e . Ä R is generated as an S -module in degree 0, the range of integers e such Since Ä Ä that .R is called the / r ¤ 0 is an interval in Z of the form Œ0;:::;rç , and the number e Ä R . See Eisenbud [2005, Chapter 4] for more on Castelnuovo–Mumford regularity of Ä this important notion. The condition that imposes independent conditions on forms of degree s is Ä Proof: . dim S equivalent to the statement that / . D deg Ä s Ä Using the exact sequences  l .S 0; / ! / .R 0 ! ! .S ! / t t t 1 Ä Ä Ä P s .R . / dim we see that dim .S D / k t s k Ä Ä 0 t D By Eisenbud [1995, Section 1.9], the dimension of R , is the degree of the scheme Ä Ä Ä is not contained in a line, proving Part (c). Part (a) is an immediate consequence. If then .R , proving Part (b). See Eisenbud and Harris / 0  2 , so .R D / 1 deg Ä 1 Ä Ä [1992].

276 262 Chapter 7 Singular elements of linear series 1 2 .X/ denote the restriction of the hyperplane class, we have c A . L / D  Letting 1 , and, as we have seen, e T c. / 3 P D T / c. X / c. N 3 P X= 4 C / .1 D d C 1 2 2 2 6 D .1 d C d C  4 / C /.1 2 2 .4 d/ C .d D 4d C 6/ 1 : C 2 2 4/ and c Thus D .d c D 4d C 6/ .d . From (7.1), above we see that 2 1 2 2 2 . L // D .3e P C 2.d 4/e C d . 4d C 6/ c : 2 2 deg . Finally, since / D d , the number of singular elements in the pencil of curves on a 3 e smooth surface of degree d cut by a general pencil of surfaces of degree P is X  2 2 . P . L // D d.3e deg C 2.d 4/e c d C 4d C 6/: (7.2) 2 As explained above, this will be the degree of the dual surface of the -th Veronese e  X .X/ of image . For example, when e D 1 this reduces to e  2 1/ D d.d deg X ; as calculated in Section 2.1.3. D 2 , we are computing the expected number of singular points in the When e 3 3 f Q g  P intersection of X , and with a general pencil P of quadric surfaces in 1 t P 2 t we find that it is equal to 3 C 2d: d The reader should check the case 1 directly! We invite the reader to work out some D d more examples, and to derive analogous formulas in higher (and lower!) dimensions, in Exercises 7.22–7.27. 7.4.3 The second fundamental form A useful tool in studying singularities of elements of linear series is the second n S . The notion was first considered of a smooth variety X  P fundamental form X in differential geometry, and is usually described using a metric, but we give a purely algebro-geometric treatment. We will explain the definition and an application; more information can be found, for example, in Griffiths and Harris [1979].

277 Singular elements of linear series in general Section 7.4 263 As we shall see at the end of this section, the second fundamental form is closely Gauss map W X ! G .k;n/ , which sends each point p 2 X to G related to the X n X P : The information S  carries is equivalent to that of its tangent plane T p X the differential  W d G G ! : T T X X G .k;n/ X (See Section 2.1.3 for the definition of the Gauss map for hypersurfaces, and below for the general case.) Since we will be dealing with both duals and pullbacks of vector bundles in this _  section, we will write the dual of a bundle E E as . E instead of our more usual n k in P X . Throughout this section will denote a smooth subvariety of dimension We will define D to be a map of sheaves on X S S X 2 2 _ I = I W S . ! T Sym /: X X X 2 _ . T Sym We regard / as the bundle of quadratic forms on the tangent spaces to X . Let X n be a function on an open subset of defined in a neighborhood of p 2 X and P f , so that is a local section of I . When restricted to the tangent space vanishing on X f X n  P T of X at p X f is singular at p , so the restriction f D f j , the function p T X p vanishes together with all its first derivatives at p . Because of this, the quadratic part f at p is independent of the choice of coordinates. Via the of the Taylor expansion of , we define T T , to be the X/ D T p X . S.f / at , the value of S.f / identification p p p p f at p . quadratic term in the expansion of 2 2 S 2 I S.f / We claim that , and thus that vanishes if defines a map I f = I ! X X X 2 _ Sym . / . We work in local coordinates z T at p . In these terms, S.f / is the quadratic i X form defined by the Hessian matrix   2 f @ : .p/ @z @z i j If D gh is the product of two functions vanishing on X , then the locally defined f @f=@z , so the Hessian matrix is D g.@h=@z function / C h.@g=@z X / vanishes on i i i . Since S is linear, this suffices to prove the claim. X identically 0 on Recall from Eisenbud [1995, Chapter 16] that there is an exact sequence 2 n n = I I 0 ! !  ! !  j 0: n X X X= P P X= P X is smooth this is a short exact sequence of vector bundles. Composing the Since 2 n n = I inclusion with the first map of the restriction of the Euler I !  j n X P X= P P X= sequence n C 1 n j ! ! O  0; 0 ! . 1/ ! O X X P X n C 1 2 n we get an inclusion of bundles = I W I  ! O X W ! G . The Gauss map n P X= X X= P  G.n k;n C 1/ G / . S may be defined as the unique map such that the pullback of the inclusion of the universal subbundle on G.n k;n C 1/ is  .

278 264 Chapter 7 Singular elements of linear series (The more usual definition of the Gauss map is dual to this one: Starting from the n ! T T derivative j of the inclusion map, one takes the pullback of the image under X X P the surjection 1 n C n O j ! T I X P X C 1;n C 1/ Š the two descriptions are related by the duality isomorphism G.k k;n 1/ .) G.n C to @ As explained in Section 3.2.5, the derivative of this map takes a derivation to the entries of a matrix representing  and then projecting to @ the result of applying n 1 C = E . We can put all these actions into the composite map O 2 n n I = ; . ̋ I  ! j T / !  n X X X P X= P P X= or equivalently the map 2 n n  ! :   j ̋  ! ̋ I I = n X X X X P X= P X= P x A local computation in coordinates on X shows that the image of the class of a i n f 2 I has the form function X= P X @f dx ̋ dx ; i j @x @x j i i;j 2 Sym and is thus a symmetric tensor, an element of /   ; in fact, it is the ̋  . X X X quadratic term of the Taylor expansion of f . We can now complete the proof of the result of Section 7.4.2, based on Corol- lary 10.21, which will be proven independently: n X Theorem 7.11. P Let be any smooth hypersurface of degree d > 1 . The set of  is an algebraic X  X where the rank of the quadratic form points i is at most S.f / p i subset of dimension at most i . In particular: (a) There are at most finitely many points where the tangent hyperplane section Y D X \ T . X has multiplicity 3 or more at p ; that is, S.f / 0 D p p If X 2 X is a general point, then the tangent hyperplane section Y D X \ T p (b) p has an ordinary double point at ; that is, for general p the rank of S.f / p is equal p to the dimension of X . Proof: Since X is a hypersurface, the ideal of X near p is generated by a single function , so we may regard S.f / as a map from X to the total space of a twist of f 2 _ T . the vector bundle Sym is thus the (reduced) preimage of the closed / . The locus X i X algebraic set of forms of rank  i .

279 r Inflection points of curves in Section 7.4 265 P X > i It suffices to prove the general statement. Suppose that had dimension i  for some p be a general (and in particular smooth) point of X 0 . Since i , and let i 0  T X , the null-space is general in T p of S.f / has constant dimension q i X;q X;q X  i for q in a neighborhood U  X dim of p , and these null-spaces form a i 0 j  T T . The tangent spaces to U also form a subbundle, and by our subbundle X U U 0 these two subbundles intersect in a subbundle hypothesis on the dimension of T X \ T i U X of rank .  1 We may assume that the ground field is the complex numbers. Integrating a local 0 , we obtain the germ of a curve in T T \ X along which analytic vector field inside U X the Gauss map has derivative 0. This contradicts the assertion of Corollary 10.21 that is a finite mapping from to its dual. G X X r P 7.5 Inflection points of curves in Bundles of principal parts are very useful for studying maps of curves to projective space. The connection with “singular elements of linear series” comes from the fact that a hyperplane in projective space is tangent to a nondegenerate curve if and only if its intersection with the curve — an element of the linear system corresponding to the embedding — is singular. If the plane meets the curve with a higher degree of tangency — think of the tangent line at a flex point of a plane curve — then that will be reflected in a higher-order singularity. Thus the technique we developed in Section 7.2 will allow us to solve the third of the keynote questions of this chapter: How to extend the notion of n P , and how to count them. flexes to curves in C  X is a reduced curve on a scheme X and D Recall that if X an effective  Cartier divisor on C , then for any closed point p 2 D \ C we defined the multiplicity of intersection of C with D at p to be the length (or the dimension over the ground field k is algebraically closed) of , which will be the same since we are supposing that k D = .D/  O O . Thus, for example, when p ... the multiplicity is 0, and the \ I C C;p C;p and D multiplicity is 1 if and only if p and meet transversely there. C are both smooth at For the purpose of this chapter it is convenient to expand this notion. Suppose that is a smooth curve, f W C ! X is a morphism and D is any subscheme of X such that C 1 .D/ is a finite scheme. We define the order of contact of D with C at p 2 C to be f 1  =f ord .D/ WD dim I O . .D//: f p C;p .p/ is smooth, the local ring O is a discrete valuation ring, Since we have assumed that C C;p 1  so ord .D/ is the minimum of the lengths of the algebras O g =f f .g/ , where p C;p ranges over the local sections of I .D/ at p , or over the generators of this ideal. r r is the inclusion map of a smooth curve C  X D P If , and D D ƒ  P f is a 1 linear subspace, then the order of contact f .ƒ/ is the minimum, over the set of ord p . H containing ƒ , of the intersection multiplicity m .C;H/ hyperplanes p

280 266 Chapter 7 Singular elements of linear series 2 p  P 2 is a smooth point of a plane curve and L  p is any For example, if C , then the order of contact of line through C at p is at least 1; L is tangent to L with p if and only if it is at least 2. The line L is called a flex tangent if the order is at p C at is called a flex of C . Carrying this further, we say that least 3, and in this case is a p p if the tangent line at p meets C with order  4 . We adopt similar definitions hyperflex L 2 W C ! P is a nonconstant morphism from a smooth curve. For in the situation where f a curve in 3-space we can consider both the orders of contact with lines and the orders of contact with hyperplanes. 7.5.1 Vanishing sequences and osculating planes W We will systematize these ideas by considering a linear system . L ;W / D . Given a point C W D r C 1 dim p 2 C and a section with on a smooth curve 2 W , the order of vanishing ord is defined to be the length of the  of  at p  p O -module L is algebraically closed =. O k / . Again, because the ground field p C;p C;p .p/ D k , so we have D ord dim L  =. O /: k p p C;p Given p 2 C , consider the collection of all orders of vanishing of sections  2 W at p . We define the vanishing sequence ;p/ of the linear system W at p to be the a. W of sections in , arranged in p sequence of integers that occur as orders of vanishing at W strictly increasing order: ;p/ WD .a a. . W ;p/ < a W . W ;p/ <  /: 0 1 Since sections vanishing to distinct orders are linearly independent, a. W ;p/ has at most W elements. On the other hand, we can find a basis for W consisting of sections dim p vanishing to distinct orders at (start with any basis; if two sections vanish to the same order replace one with a linear combination of the two vanishing to higher order, and a. W ;p/ is exactly dim : W D r C 1 repeat). It follows that the number of elements in k W ;p/ D .a a. . W ;p/ <  < a . W ;p//: r 0 We set ̨ D a , and call the associated weakly increasing sequence i i i W ;p/ D . ̨ ;p// . ̨. ;p/  ̨ W . W r 0 the ramification sequence of W at p . When the linear system W or the point p we are referring to is clear from context, we will drop it from the notation and write .p/ or a i just a ̨ in place of . , and similarly for . W ;p/ a i i i

281 r Inflection points of curves in 267 P Section 7.5 is a base point of p For example, .p/ D ̨ a .p/ > 0 , and more W if and only if 0 0 .p/ is the multiplicity with which p generally W . If p is a appears in the base locus of 0 then, since C is a smooth curve, we may remove it; that is, W is in W a base point of 0 0 L . the image of the monomorphism a H ! H . . L / , and we may thus consider p// 0 0 W WD . L . a W p/;W / . In this way most questions about as defining a linear series 0 linear systems on smooth curves can be reduced to the base point free case. r W , so that W defines a morphism f When C ! P is not a base point of in a W p p , we have a D .p/ D 1 if and only if f is an embedding near p . If r neighborhood of 2 1 and f is an embedding, we thus have ̨ W .p/ D ̨ is very ample, so that .p/ D 0 0 1 ̨ p .p/ > 0 and p if and only if there is a line meeting for all for some particular 2 at p ; that is, p is an inflection point of the the embedded curve with multiplicity > 2 embedded curve. The geometric meaning of the vanishing sequence is given in general by the next result: W D . L ;W / Let C , and Proposition 7.12. be a linear series on a smooth curve 0 0 C . If p is not a base point of W , we let W let D W ; in general, let W p D 2  ;p/p/;W L . W a . . 0 0 ;p/ . W (a) a D a . . W ;p/ a ;p/ . W 0 i i 0 Choose  H ;:::; , and let 2 W such that  ;p/ vanishes at p to order a (b) . W j j 0 j r  P / corresponding to  be the hyperplane in . The plane .W j \\ D L H H 1 r C i i with highest order of contact with C is the unique linear subspace of dimension i . , and that order is p . ;p/ a W at 1 C i L . We always have are called the osculating planes to The planes at p f.C/ i L D . If f.C/ is smooth at f.p/ then p L is the tangent line, and in general it is the 1 0 reduced tangent cone to the branch of f.C/ that is the image of an analytic neighborhood p 2 of . C Proof: L . dp/ that vanishes to order (a) A section of as a section of L . dp/ will m vanish to order m C d at p as a section of L . 0 (b) Writing p of the morphism defined by W , it follows from the f for the germ at 0 -plane L with higher order of contact, D a definitions that L ord . If there were an i 1 p i C i and we wrote 0 0 0 D L \\ H H r 1 i C 0 0 , then each strictly for some hyperplanes H H p would have order of contact with C at r r , and taking a greater than . But these would correspond to independent sections in W 1 C i linear combinations of these sections we would get r i sections with vanishing orders at p strictly greater than a i r . This contradicts the assumption that the highest 1 i C elements of the vanishing sequence are a . ;:::;a 1 C r i

282 268 Chapter 7 Singular elements of linear series ucker formula 7.5.2 Total inflection: the Pl ̈ of dimension is an . L ;W / for a linear system r if p inflection point We say that , ;:::; ̨ the ramification sequence / is not .0;:::;0/ , or, equivalently, if ̨ . ̨ > 0 r 0 r r > r . When W arises from a morphism f W C ! a which is the same as that is P r an embedding near p , p is an inflection point of W if and only if some hyperplane has  r C 1 at p . contact weight We define the p 2 C with respect to W to be of r X : W w. WD ̨ ;p/ i D i 0 W p . We This number is a measure of what might be called the “total inflection” of at P can compute the sum as a Chern class of the bundle of principal parts ;p/ w. W C p 2 L of . ̈ W ucker formula) . If Theorem 7.13 is a linear system of degree (Pl and dimension r d on a smooth projective curve C of genus g , then X 1/: 1/r.g w.W;p/ D .r C 1/d C .r C C 2 p This is our answer to Keynote Question (c). Note that it is only an enumerative r or more C formula, in the sense that each hyperplane having contact of order 1 p at a point w. W ;p/ . We might expect C with has to be counted with multiplicity is a suitably general curve — say, one corresponding to a general point on a that if C component of the open subset of the Hilbert scheme parametrizing smooth, irreducible, would have weight 1, but this is nondegenerate curves — then all inflection points of C actually false (see Exercises 7.40–7.41). It can be verified in some cases, such as plane curves (see Exercise 7.32), and it is true also for complete intersections with sufficiently high multidegree (see Exercise 7.39 for a step forward in that direction); it remains an open problem to say when it holds in general. Proof: The key observation is that both sides of the desired formula are equal to the r degree of the first Chern class of the bundle . L / . We can compute the class of this P bundle from Theorem 7.2 as r Y r j c. D c. . L /: L P Sym // . ̋ / C 0 D j j j j is a line bundle we have Sym Since . / / D   L , and thus c.. Sym D . ̋ / C C C C jc . . It follows that 1 C c C . L / / 1 1 C   r C 1 r L // D .r C 1/c c . L / C . . P /: c . 1 1 1 C 2

283 r Inflection points of curves in 269 P Section 7.5  2 , the degree of this class is is Since the degree of 2g C r .r . . L // D deg C 1/d C P C 1/r.g 1/; c .r 1 ̈ ucker formula. the right-hand side of the Pl 1 C r r ! P . L / by choosing any basis O '  We may define a map ;:::; W of W r 0 C r C 1 r i and sending the to the section  corresponding O of P -th basis element of . L /  i C  C . We will complete the proof by showing that for any point p 2 to the determinant i of the map p to order exactly w.W;p/ , and that there are only finitely ' vanishes at where the determinant is 0. many points w.W;p/ p 2 C . Since the determinant of ' depends on the choice To this end, fix a point only up to scalars, we may choose the basis ;:::; of basis   so that the order of i r 0 . / D a at p vanishing a ord .W;p/ . Trivializing L in a neighborhood of p , we is p i i i  locally as a function, and ' is represented by the matrix may think of the section i 0 1     0 1 r B C 0 0 0     B C r 0 1 B C ; : : : B C : : : : : : @ A .r/ .r/ .r/     r 0 1 .r/ 0 vanishes to order   where the r -th derivative. Because  denotes the derivative and i i i  i at p , the matrix evaluated at p is lower-triangular, and the entries on the diagonal are a is not an inflection D i for each i ; that is, if and only if p all nonzero if and only if i point for W . ' det We can compute the exact order of vanishing of at an inflection point as the .r C 1/ -vector . follows: Denote by v.z/ , so that the determinant .z// .z/;:::; r 0 of ' is the wedge product 0 .r/ v ^ v det ^^ v .'/ : D n det .'/ is then a linear combination of -th derivative of Applying the product rule, the terms of the form .ˇ .ˇ C 1/ .ˇ r/ C / r 1 0 v ^^ ^ v ; v P .ˇ / / .ˇ .ˇ / 0 1 0 .p/ D 0 unless with .p/  . Now, v ; similarly, v ˇ 0 D .p/ ^ v ̨ ˇ n D 0 0 i unless ˇ of C ˇ .'/  ̨ det C ̨ any derivative of , and so on. We conclude that 1 0 0 1 P w w ̨ -th derivative of vanishes at order less than , and the expression for the D p i det .'/ has exactly one term nonzero at p , namely r/ . ̨ C / . ̨ 1/ . ̨ C r 1 0 ^^ v ^ v : v Since this term appears with nonzero coefficient, we conclude that det .'/ vanishes to order exactly w at p .

284 270 Chapter 7 Singular elements of linear series C It remains to show that not every point of can be an inflection point for W — is not identically zero. To prove this, suppose that .'/ does vanish ' det det that is, that identically, that is, that 0 .k/ v v  0 (7.3) v ^ ^^  r . Suppose in addition that k is the smallest such integer, so that at a for some k 2 we have general point p C 1/ 0 .k I v .p/ v .p/ ¤ 0 ^^ v.p/ ^ 1/ .k .k/ .p/ are linearly independent, but in other words, v.p/;:::;v .p/ lies in their v span . Again using the product rule to differentiate the expression (7.3), we see that ƒ d 0 .k 1/ .k/ 0 .k 1/ 1/ .k C / D v ^ v v ^^ v 0;  ^ ^ v v ^^ .v ^ v dz C 1/ .k .k 1/ so that v .p/ also lies in the span of .p/ . Similarly, taking the v.p/;:::;v second derivative of (7.3), we see that 2 d 2/ C .k 1/ 0 .k 0 .k 1/ .k/ v ^ D v ^ v v ^^ v ^^ v ^ ^ v .v / 0;  2 dz C 1/ -fold .k where are all the other terms in the derivative are zero because they are -dimensional space. Continuing in this way, we see k wedge products of vectors lying in a .m/ ƒ .p/ 2 ƒ for all m ; it follows by integration that v.z/ 2 that for all z . This implies v W 1 k < r C that the linear system , contradicting our assumptions. has dimension Flexes of plane curves Theorem 7.13 gives the answer to Keynote Question (c). We do not even need to C is smooth; if C is singular, as long as it is reduced and irreducible we view it assume r z W C ! P as the image of the map from its normalization. For example, when r D  , 2 ̈ if we apply the Pl ucker formula to the linear system corresponding to this map, we see that C has .r C 1/d C r.r C 1/.g 1/ D 3d C 6g 6 z flexes, where C , that is, geometric genus of C . If the curve C is indeed is the genus of g 2 D d.d 3/ , and so this yields smooth, then 2g C 6g 6 D 3d C 3d.d 3/ D 3d.d 2/: 3d 2 z C To be explicit, this formula counts points p such that, for some line L  P 2 ,  the multiplicity of the pullback divisor L at p is at least 3. In particular:  (a) It does not necessarily count nodes of C , even though at a node p of C there will be lines having intersection multiplicity 3 or more with C at p . (b) It does count singularities where the differential d vanishes, for example cusps.

285 r Inflection points of curves in 271 P Section 7.5 ̈ ucker formula appear in Exercises 7.35–7.37. Some applications of the general Pl We mention that there is an alternative notion of a flex point of a (possibly singular) 2 2  p 2 C curve L P P  through p , we have C : a point such that, for some line .C  m  3: L/ p p C is a flex point, since the tangent lines to the In this sense, a node of a plane curve C branches of the curve at the node will have intersection multiplicity at least 3 with at Cartesian . When we want to talk about flexes in this sense, we will refer to them as p 2 C P rather than its flexes, since they are defined in terms of the defining equation of  parametrization by a smooth curve. There is a classical way to calculate the number of flexes of a plane curve that does C count Cartesian flexes. Briefly, if is the zero locus of a homogeneous polynomial F.X;Y;Z/ , we define the Hessian of C be the zero locus of the polynomial ˇ ˇ 2 2 2 ˇ ˇ F F @ @ F @ ˇ ˇ 2 ˇ ˇ @X@Z @X@Y @X ˇ ˇ 2 2 2 ˇ ˇ F @ @ F F @ H : D ˇ ˇ 2 ˇ ˇ @X@Y @Y@Z @Y ˇ ˇ 2 2 2 ˇ ˇ @ F F @ @ F ˇ ˇ 2 @X@Z @Y@Z @Z For a smooth plane curve C , the Cartesian flexes are exactly the points of intersection C with its Hessian (it is even true that on a smooth curve C the weight of a flex p is of C with its Hessian at p ). In Exercise 7.33, we equal to the intersection multiplicity of will explore what happens to the flexes on a smooth plane curve when it acquires a node. Hyperflexes First, the bad news: We are not going to answer Keynote Question (d) here. The 2 C  P question itself is well-posed: We know that a general plane curve of degree d has only ordinary flexes, and it is not hard to see that the locus of those curves 4  N that do have a hyperflex is a hypersurface in the space P of all such curves (see Exercise 7.38). Surely the techniques we have employed in this chapter will enable us to calculate the degree of that hypersurface? Unfortunately, they do not, and indeed the reason we included Keynote Question (d) is so that we could point out the problem. Very much by analogy with the analysis of lines on surfaces and singular points on curves, we would like to determine the class of the “universal hyperflex:” that is, in the universal curve N 2 .C;p/ 2 P  ˆ P Df j p 2 C g ; the locus : Df .C;p/ 2 ˆ Ä p is a hyperflex of C g j

286 272 Chapter 7 Singular elements of linear series Moreover, it seems as if this would be amenable to a Chern class approach: We would define a vector bundle ˆ whose fiber at a point .C;p/ 2 ˆ would be the vector space E on at germs of sections of O .1/ p g f C E : D .C;p/ germs vanishing to order at p g f 4  from the trivial bundle with fiber ˆ We would then have a map of vector bundles on 0 . O , and the degeneracy locus of this map would be the universal hyper- E H to .1// 2 P flex Ä . Since this is the locus where three sections of a bundle of rank 4 are linearly dependent, we could conclude that D c ŒÄç . E /: 2 As we indicated, though, there is a problem with this approach. The description E makes sense as long as p is a smooth point of above of the fibers of , but not C otherwise E by taking Å  ˆ  . Reflecting this fact, if we were to try to define the ˆ N P diagonal and setting 4   ̋ I . E = O O D .1/ /; 2  1  ˆ ˆ 2 N Å P P of U  ˆ would have fiber as desired over the open set .C;p/ with C the sheaf E p , but would not even be locally free on the complement. The fact that bundles smooth at of principal parts do not behave well in families (except, of course, smooth families) is a real obstruction to carrying out this sort of calculation. There is a way around this problem: Ziv Ran [2005a; 2005b] showed that — at least 1 N ˆ of a general line P over the preimage  C  — the vector bundle E j extends P \ C U to a locally free sheaf on a blow-up of C , realized as a subscheme of the relative Hilbert 1 scheme of P C . This approach does yield an answer to Keynote Question (d), and over indeed applies far more broadly, albeit at the expense of a level of difficulty that places it outside the range of this text. And now, the good news: there is another way to approach Keynote Question (d), and we will explain it in Section 11.3.1. 7.5.3 The situation in higher dimension ̈ ucker formula for linear series on varieties of dimension Is there an analog of the Pl r X  P , we might greater than 1? Assuming that the linear series yields an embedding ask, for a start, what sort of singularities we should expect the intersection X \ ƒ of X r with linear spaces  P of a given dimension to have at a point p , and ask for the ƒ locus of points that are “exceptional” in this sense. We do not know satisfying answers to these questions in general. One issue is that, while the singularities of subschemes of a smooth curve are simply classified by their multiplicity, there is already a tremendous variety of singularities of subschemes of sur- faces. (If we have a particular class of singularities in mind, such as the A -singularities n

287 Nets of plane curves Section 7.5 273 described in Section 11.4.1, then these questions do become well-posed; see for ex- A ample the beautiful analysis of elements of a linear system having an -singularity n in Russell [2003].) Another problem is that the analog of the final step in the proof of r C P can be an Theorem 7.13 — showing that not every point on a smooth curve  inflection point — may not hold. For example, a dimension count might lead us to expect 5  P p no hyperplane on a smooth, nondegenerate surface that for a general point S 5 \ intersects S in a curve C D H H S  p , but there are such P with a triple point at surfaces for which this is false, and we do not know a classification of such surfaces. We will revisit this question in Chapter 11, where we will describe the behavior of 3  . plane sections of a general surface P S 7.6 Nets of plane curves We now want to consider larger-dimensional families of plane curves, and in partic- ular to answer the second keynote question of this chapter. A key step will be to compute D f .C;p/ j the class of the universal singular point 2 C † g as a subvariety of p sing N 2 N 0 , where P  D P H P . O P . .d// 2 P 7.6.1 Class of the universal singular point 0 n .d// P . H Let D , so that O W is the projective space of hypersurfaces of W P n in P m , and consider the universal degree -fold point d n P † Df .X;p/ 2 P W  † D j mult g .X/  m ; p n;d;m and let  2 n n - P P  P W  1 ? W P n 2 A. P W be the projection maps. We can express the class P Œ†ç / in terms of  Chern classes: Proposition 7.14. † is the zero locus of a section of the vector bundle n;d;m m m   n O .d//;  .1/ ̋  WD O P P . W P P 1 2 which has Chern class n C m m / . n .d c. m/ ; C  P / / D .1 C n W n  W and  respectively. are the pullbacks of the hyperplane classes on P where and P n W

288 274 Chapter 7 Singular elements of linear series  C n m n / is the sum of the terms of total degree A. P  P in † W Thus the class of n;d;m n 2;m D 1 this is in this expression. For example, in the case n D 2 3 2 2 2 3 1/ D   C 3.d Œ†ç 1/ C  3.d P  A /: . P W  2 2 W 2 W W The computation is similar to the one used in the calculation of the class of the Proof: 2 W defines a section  universal line in Section 6.6. Since every polynomial of F F m n , we have a map O .d// . P P m n n W ̋ P O . O .d// ! P P n . Likewise, we have the tautological inclusion of vector bundles on P . 1/ ! O ̋ O W P W W P 2 . We pull these maps back to the product P W  P on and compose them to obtain P W a map   m n  . 1/ !  O O P .d//; . P W P 2 1 n m †  or, equivalently, a section of the bundle W  P P , P . The zero locus of this map is n m in A. P W  P † / is the class of a section of P so the class of , as claimed. n;d;m m P To compute the Chern class of , we follow the argument of Proposition 7.5, pulling back the sequences 1 i i i n n n 0 .d// ! ! P . O .d// O . P ! /.d/ Sym 0 . ! P P P  and tensoring with the line bundle  O .1/ to get P W 1 m Y m  j n D / P c. Sym //; .  .d C O c. / ̋  n W P 2 0 D j   n . C  .d/ / as shorthand for the line bundle  ̋ where we write O O O .1/ .d  n W W P P 1 2 Using the exact sequences i i i 1 n n n 0 . O . 0 . ! 1// ! Sym Sym / ! . O ! Sym . 1// P P P we get a collapsing product as before, yielding the desired formula for the Chern class m of . To deduce the special case at the end of the proposition, it suffices to remember P 3 0 is the pullback from a two-dimensional variety we have   that since D . 2 2 7.6.2 The discriminant of a net of plane curves We return to the case of a net of plane curves of degree d . Throughout this section we fix a general net of plane curves of degree d , that is, the family of curves associated 0 .d//  H to a general linear subspace . O of dimension 3, parametrized by W 2 P 2 . B D P W Š P

289 Nets of plane curves 275 Section 7.6 B  of the net B . Let be the set of singular curves, called the D discriminant curve B P W , its degree is is the intersection of D Since with the discriminant hypersurface in 2 2 1/ D by Proposition 7.4. Next, let Ä  P deg be the plane curve traced out 3.d D by the singular points of members of the net, so that if we set 2 † \ . B † P  /; WD B  † restrict to surjections then the projection maps on i  j B 2 - Ä † B  j 1 B ? D Since is smooth of codimension 3, Bertini’s theorem shows that † is a smooth curve † B 2 P in . Since the generic singular plane curve is singular at only one point, the B  2 map ! Ä is birational. Since the fiber of † over a given point p 2 P † is a linear B space of dimension 3 , the general 2-plane B containing a curve singular at p will N D is also birational, and † is the † ! contain a unique such curve. Thus the map B B and D . In particular the geometric genus of normalization of each of and that of Ä Ä D † . are the same as the genus of B From the previous section, we know that † is the zero locus of a section of the B 2 1 B . This makes it easy to compute the degree and genus P P on j  rank-3 bundle 2 P  B † of , and we will derive the degree and genus of Ä , answering Keynote Question (b): B Proposition 7.15. With notation as above, the map † ! Ä is an isomorphism, so both B  3d 4 3 , and thus has genus Ä 3d curves are smooth. The curve . When has degree 2 2 D is singular. , the curve  d arise, what they look like and how many We will see how the singularities of D there are in Chapter 11. Ä , the number of points of intersection of Ä with a We begin with the degree of Proof: 2 ! L . Since † P  Ä is birational, this is the same as the degree of the product line B 2 4 .) 2 A Œ† . B  P ç /: (More formally,  ç Œ† Œ† D ç D Ä and  çŒLç Œ† 2  B  B 2 2 2 B B N for the restriction of Write  , to , the pullback of the hyperplane section from P  B W 2 2 2 2 . The degree of a class in B  P  is the coefficient of  P in its expression in  B 2 B 2 3 3 / D Z Œ  ; B ç=. P /: ; A. 2 B 2 B 3 D Since  0 , the last formula in Proposition 7.14 gives B  2 2 2  3.d 1/ .Ä/ deg  C 3.d 1/ deg  D  2 B 2 2 B 2 2 D deg 3.d 1/  2 B D 3d 3:

290 276 Chapter 7 Singular elements of linear series  3d 4 Ä is a plane curve, the arithmetic genus of the curve Since . Ä is 2 † of the smooth curve † . The normal bundle of g Next we compute the genus B B † B 2 2 1 P B , and the canonical divisor on  is the restriction of the rank-3 bundle  B P in P 3 has class 3 , so by the general adjunction formula (Part (c) of Proposition 6.15) 2 B the degree of the canonical class of is the degree of the line bundle obtained by † B V 3 2 1 tensoring the canonical bundle of B  P and restricting the result to † with . P B This is the degree of the class  2 1 2 1 2 c  . P //Œ† 3 ç D . 3 1/ 3 C c 3.d P 3 //  . 3.d 1/ C  C . :  2 2 1 1 2 B B B B 2 B 1 / P c / D 3..d 1/ Substituting the value C  from Proposition 7.14 and taking . 1 2 W 2 , this becomes  account of the fact that   D B W B  2 2 2 2 2 1/   .3d C 3.d 1/ 3.d 6/  3/  D .3d 6/.3d  2 2 B 2 2 B B with degree 2g , and we see that 6/ 2 D .3d 3/.3d † B   4/.3d 5/ .3d 4 3d D : D g.† / B 2 2 Since this coincides with the arithmetic genus of computed above, we see that Ä † Ä ! Ä is an isomorphism. On the other hand the degree is smooth and the map B 2 for all of D is different from that of Ä 3.d d  2 , so in these cases the arithmetic 1/ and geometric genera of differ, and D must be singular, completing the proof. D Here is a different method for computing the degree of : The net B of curves, Ä having no base points, defines a regular map 2 ' P ! ƒ; W B 2 P where is the projective plane dual to the plane parametrizing the curves in ƒ Š 2 2 the net B as a d . This map expresses -sheeted branched cover of ƒ , and the curve P 2 Ä  P is the ramification divisor of this map. By definition,  .1/ O I ' D O .d/ 2 ƒ P   , we have '  ƒ the hyperplane class on D d . so that, if we denote by ƒ ƒ 2 W P ! Pulling back a 2-form via the map ƒ ' we see that  C D ' K K Ä; 2 ƒ P and since K D 3 , this yields ƒ ƒ 3 D 3d C ŒÄç or ŒÄç .3d 3/ . D W These ideas work for a net . L ;W / on an arbitrary smooth projective surface S , D as long as we know the classes c and can evaluate the . / /;c L . . / and  D c 1 2 1 S S degrees of the relevant products in A.S/ . See Exercise 7.31 for an example.

291 The topological Hurwitz formula Section 7.6 277 7.7 The topological Hurwitz formula In this section we will work explicitly over the complex numbers, so that we can use the topological Euler characteristic. Using this tool, we will give a different approach to questions of singular elements of linear series. It sheds additional light on the formula of Proposition 7.4, and is applicable in many circumstances in which Proposition 7.4 cannot be used. In addition, it will allow us to describe the local structure of the discriminant hypersurface, such as its tangent planes and tangent cones. By the Lefschetz principle (see for example Harris [1995, Chapter 15]), moreover, the purely algebro-geometric consequences of this analysis, such as Propositions 7.19 and 7.20, hold more generally over an arbitrary algebraically closed field of characteristic 0. (There are also alternative ways of defining an Euler characteristic with the desired properties algebraically.) This approach is based on the following simple observation: Proposition 7.16. be a smooth projective variety over C Let Y  X a divisor. X , and  the topological Euler characteristic (in the classical, or analytic, If we denote by top topology), then Y/: .X/ D  n .Y/ C  .X  top top top This will follow from the Mayer–Vietoris sequence applied to the covering of X Proof: V D n Y by X of Y . U and a small open neighborhood L D Let . Introducing .Y/ , and let  be the section of L vanishing on Y O X Hermitian metrics on X and the line bundle L , we can use the gradient of the absolute 1  to define a C Y map V ! Y expressing V as a fiber bundle over value of with 2 fiber a disc , and simultaneously expressing V \ U as a bundle over Y with fiber a D punctured disc. It follows that  .V / D  0; .Y/ and  D .V \ U/ top top top and we deduce the desired relation.  applies .X/ D  It is a surprising fact that the formula .Y/ C  Y/ .X n top top top much more generally to an arbitrary subvariety Y of an arbitrary X ; see for example Fulton [1993, pp. 93–95, 142]. Now let be a smooth projective variety, and let f W X ! X be a map to a smooth B curve B of genus g . This being characteristic 0, there are only a finite number of points p is singular. We can apply the relation on Euler ;:::;p X 2 B over which the fiber 1 p ı i characteristics to the divisor ı [ D Y X  X: p i i D 1

292 278 Chapter 7 Singular elements of linear series P is a fiber .Y/  Naturally, .X D Y / , and on the other hand the open set X  n top top p i p nf g , so that B bundle over the complement ;:::;p 1 ı .X n Y/ D   .X /; / .X .B nf p ı/ ;:::;p g / D .2 2g top top   1 top top ı is a general point of B . Combining these, we have where again  ı X D .2  2g ı/ / .X .X / C .X/  top  p top top i D 1 i ı X D .B/ .X / C  .X //:  / . .X top top   top top p i i D 1 In this form, we can extend the last summation over all points q 2 B . We have proven: (Topological Hurwitz formula) . Let f W X ! B be a morphism from a Theorem 7.17 smooth projective variety to a smooth projective curve; let  2 B be a general point. Then X //: .X/ D  .X .B/  .X / C  / .X . top q  top top top  top B q 2 X is what it would be if were a fiber bundle In English: The Euler characteristic of X — that is, the product of the Euler characteristics of and the general fiber X B — B over  with a “correction term” coming from each singular fiber, equal to the difference between its Euler characteristic and the Euler characteristic of the general fiber. To see why Theorem 7.17 is a generalization of the classical Riemann–Hurwitz formula (see for example Hartshorne [1977, Section IV.2]), consider the case where X h and f W X ! C a branched cover of degree is a smooth curve of genus . For each d point 2 C , we write the fiber X p as a divisor: p X  D f q:  m .p/ q 1 .p/ f 2 q We call the integer m ramification 1 the ramification index of f at q ; we define the q divisor of f to be the sum R X q; D  1/ .m R q 2 X q branch divisor B and we define the f to be the image of R (as a divisor, not as a of scheme!) — that is, X X B m 1: b D  p; where b D p p q 1 C 2 p 2 f .p/ q

293 The topological Hurwitz formula Section 7.7 279 1 , for each D .p/ of f X d f p 2 C Now, since the degree of any fiber is equal to p 1 .p/ will be d f b , so its contribution to the topological Hurwitz the cardinality of p . The formula then yields b formula is p 2h D d.2 2g/ deg .B/; 2 the classical Riemann–Hurwitz formula. 7.7.1 Pencils of curves on a surface, revisited To apply the topological Hurwitz formula to Keynote Question (a), suppose that 2 P D V.t . Since the F C t d G/  f C g is a general pencil of plane curves of degree 1 t 0 G and Ä D V.F;G/ of the pencil will consist F are general, the base locus polynomials 2 reduced points, and the total space of the pencil — that is, the graph d of 2 1 P j p 2 C  g 2 P Df X .t;p/ t 1 2 2 - P . In particular, P — is the blow-up of P ŒF;Gç along Ä W of the rational map 1 P W X ! f is smooth, so Theorem 7.17 can be applied to the map that is the X projection on the first factor. 2 2 is the blow-up of P at d Since points, we have X 2 2 2 P C / C d  D d . D 3: .X/  top top of the map f is a smooth plane curve of degree d ; C Next, we know that a general fiber   1 d and hence as we saw in Example 2.17, its genus is 2 2 / D d C 3d:  .C top  C appearing in a general pencil We know from Proposition 7.1 that each singular fiber of plane curves has a single node as singularity. By the calculation in Section 2.4.6, then,  1 d z will be a curve of genus and hence Euler characteristic C its normalization 1 2 2 z 3d C 2 . Since C is obtained from C C d by identifying two points, we have 2  .C/ D d C 3d C 1; top f to the topological Hurwitz formula is so the contribution of each singular fiber of exactly 1. It follows that the number of singular fibers is 1 D  / .X/  .C . P ı / top  top top 2 2 d C 3 2. D d C 3d/ 2 D 3d 6d C 3; as we saw before.

294 280 Chapter 7 Singular elements of linear series S This same analysis can be applied to a pencil of curves on any smooth surface . 1 be a line bundle on with first Chern class c L . L / D  2 A .S/ Let , and let S 1 0 Dh ; be a two-dimensional vector space of sections with i H  . L / W 1 0 C  D V.t S  f C t /  g 1 0 0 t 1 1 P 2 t the corresponding pencil of curves. We make — for the time being — two assumptions: The base locus D V. f  g Ä (a) of the pencil is reduced; that is, it consists of / W 2  2 / deg points. . (b) Each of the finitely many singular elements of the pencil has just one node as singularity. S . D c c . We also denote by / the Chern classes of the cotangent bundle to i i S X be the blow-up of Given this, the calculation proceeds as before: we let along Ä , S 1  and apply the topological Hurwitz formula to the natural map W W f Š P P . To X ! start, we have 2 .X/ D   .S/ C # .Ä/ D c  C 2 top top (we omit the “ deg ” here for simplicity). Next, by the adjunction formula, the Euler C of the pencil is given by characteristic of a smooth member  2  .C / D deg .! ; c / D .c  C /   D  top 1 1 C  and, by Section 2.4.6, as in the plane curve case the Euler characteristic of each singular element of the pencil is 1 greater than the Euler characteristic of the general element. In sum, then, the number of singular fibers is 1  .X/ ı  / . P D / .C top  top top 2 2 C D c / 2.   c 1 2 2 3 c C 2c ; C D 2 1 agreeing with our previous calculation. We will see how this may be applied in higher dimensions in Exercises 7.43–7.44. 7.7.2 Multiplicities of the discriminant hypersurface One striking thing about this derivation of the formula for the number of singular elements in a pencil is that it gives a description of the multiplicities with which a given singular element counts that allows us to determine these multiplicities at a glance.

295 The topological Hurwitz formula Section 7.7 281 In the derivation of the formula, we assumed that the singular elements of the pencil had only nodes as singularities. But what if an element C of the pencil has a cusp? In that case the calculation of Section 2.4.6 says that the geometric genus of the curve — z C the genus of its normalization — is again 1 less than the genus of the smooth fiber, but z we are just “crimping” the curve at one C this time instead of identifying two points of z point. (In the analytic topology, and C are homeomorphic.) Thus, C z C D .C/ .  C/ D  2; .C  /  top top top “counts with multiplicity 2,” in the sense that its contribution to the C and the fiber sum in the right-hand side of Theorem 7.17 is 2. Similarly, if has a tacnode, we have C z z z g.C to / 2 , so that  C . g. C/ D  .C C/ , but we identify two points of D C 4 /  top  top C , so in all form .C/ D  C .C 3; /   top top and the contribution of the fiber C to the rightmost term in Theorem 7.17 is thus 3. If C has an ordinary triple point — consisting of three smooth branches meeting at a point — z z C/ D g.C then / g. 3 , but we identify three points of C to form C , so   .C/ D  4; .C / C  top top C and the contribution of the fiber is 4. Moreover, if a fiber has more than one isolated singularity, the same analysis shows that the multiplicity with which it appears in the formula above is just the sum of the contributions coming from the individual singularities. In addition to giving us a way of determining the contribution of a given singular fiber to the expected number, this approach tells us something about the geometry of N N the discriminant locus in the space P D of plane curves of degree d . To see  P 2  is any plane curve of degree d with isolated singularities. C this, suppose that P be a general plane curve of the same degree, and consider the pencil B of plane Let D N B  a general line through the point curves they span — in other words, take P N P 2 . By what we have said, the number of singular elements of the pencil B other C 2 will be 3.d 1/ than C is a smooth plane curve C , where .C/ .C  // .  top top  d ; it follows that the intersection multiplicity is of degree . B ; D / of B and D at C m p   .C / . .C/ top top  2 Proposition 7.18. C  P be any plane curve of degree d with isolated singularities. Let Then mult /; . D / D  .C .C/  top  top C where C is a smooth plane curve of degree d  Thus a plane curve with a cusp (and no other singularities) corresponds to a double point of D , a plane curve with a tacnode is a triple point, and so on. A curve C with one node and no other singularities is necessarily a smooth point of D .

296 282 Chapter 7 Singular elements of linear series 7.7.3 Tangent cones of the discriminant hypersurface We can use the ideas above to describe the tangent spaces and tangent cones to the N P discriminant hypersurface . To do this, we have to remove the first assumption  D in our application of the topological Hurwitz formula to pencils of curves, and deal with pencils whose base loci are not reduced. 2  P To consider the simplest such situation, suppose that are a point and a p L 2 d such that V.F/ and V.G/ F;G line in the plane, and that are general forms of degree and are tangent to L at p . Let Ä D pass through be the base locus of the p V.F;G/ 2 2 V.t pencil F C t 2 G/ , so that will be a scheme of degree d C consisting of d D Ä t 1 0 p . Since being singular at p reduced points and one scheme of degree 2 supported at is one linear condition on the elements of the pencil, exactly one member of the pencil (which we may take to be C after re-parametrizing the pencil) will be singular at p . 0 We could arrive at such a pencil by taking F to be the equation of a general curve and a general polynomial vanishing at p ; thus for the general pencil p G with a node p of the pencil will have a node at above, the singular element , with neither branch C 0 L , while all the others elements are smooth at p and have a common tangent tangent to T . .C L / D line at p p t X Let be the minimal smooth blow-up resolving the indeterminacy of the rational 2 1 P associated to the pencil — that is, to P map ' from is obtained by blowing up X 2 0 P at Ä and then blowing up the resulting surface at the point p on the exceptional red divisor corresponding to the common tangent line L to the smooth members of the pencil at p the blow-up of S along the scheme Ä , which is singular! See . (This is not 2 P for example Eisenbud and Harris [2000, IV.2.3].) Note that we are blowing up a total 2 2  d .X/ times, so that the Euler characteristic 3 C d of , just as in the is equal to top general case. 1 ! P over What is different is the fiber of the map t D 0 : Rather than being X C a copy of the curve V.F/ , it is the union of the proper transform of C D and the 0 0 proper transform of the first exceptional divisor, that is, the union of the normalization E 1 z z C C E of P lying over the node , meeting at the two points of C p and a copy of 0 0 0 (See Figure 7.4). t D 0 is In sum, the Euler characteristic of the fiber over z z C / C  .E/  2 D  . . C / D  2; .C / C top 0 top  0 top top and the fiber counts with multiplicity 2. We can use this to analyze the tangent planes to at its simplest points: D Proposition 7.19. Let C be a plane curve with a node at p and no other singularities. N T of curves containing the D  P The tangent plane is the hyperplane H D  p C point p .

297 Section 7.7 The topological Hurwitz formula 283 1 X X D blow-up of 1 P z C  1 z  f D ./ C  f z z C C 0 0 1 Q 0 C E f .0/ D C 0 E 2 at three points D X P blow-up of 1 2 P C 0 C  p L a pencil of conics tangent to L at p 1 W ! P f coming from the pencil of conics tangent to Figure 7.4 The morphism X p . L at 2 Proof:  P is a plane curve with one node If p and no other singularities, then, by C C D . It thus suffices to show that H is contained is a smooth point of Proposition 7.18, p N at C in the tangent space to . But, as we have seen, if B  P D is a general pencil 0 2 C and having p as a base point, B will meet D in exactly 3.d 1/ including 1 N points — in other words, a general line B through C and lying in H  will be P p D tangent to somewhere. is spanned by a curve F D 0 with a node at p As above, we may suppose that B D 0 that passes through p . and no other singularities and a smooth curve G To complete the argument — to show that such a line is indeed tangent to D specifi- C , and not somewhere else — we have to do two things: We have to relate the cally at B pencil to nearby general pencils, and we have to localize the Euler characteristic. For 0 , and consider the family of pencils f the first, choose a general polynomial G g with B s 0 ; and B B the pencil spanned by F and a linear combination G D G C sG D B 0 s s that is, 0 1 V.F C t.G C sG // B j t 2 P Df g : s 1 s , we let X the map be the total space of the pencil B For each and f P W X ! s s s s 0 ŒF;G C sG ç .

298 284 Singular elements of linear series Chapter 7 , the pencil will be a general pencil of curves of degree d ; in B For general s s 0 < j s j <  the pencil B particular, if will  > 0 is sufficiently small, then for any s 2 transversely in exactly 3.d 1/ points. Moreover, by our description of D intersect 2 D , we know that as B approaches s , two of these 3.d 1/ \ points will approach a 0 0 particular element C 2 B — the point of tangency of B with D — and the remaining t 0 2 C 2 will remain distinct from each other and from 3.d . 1/ t 0 For the second component of the argument (localizing the Euler characteristic), we 1 -line P a disc around the point cover the by a pair of open sets: U D . j t j < / t t D 0 , t 1 and V n . j t j =2/ the complement of a smaller closed disc. We can choose  P D B other than small enough so that no singular fiber of lies in U ; in particular, no C 0 singular fiber lies in \ V . It follows that, for some  > 0 , the same is true for all U lie in the overlap with j s j <  : none of the singular fibers of B . For any B U \ V s s s j <  , accordingly, the number of singular fibers of j 0 < in U is the intersection B s m B , which we claim is 2. . multiplicity ; D / of B and D at C 0 C 0 Now consider the total space of our family of pencils: 1 2 0 2 Å  P  ˆ P Df j F.p/ C t.G.p/ C sG .s;t;p/ .p// D 0 g : V V V Å ˆ . Since the fiber ˆ Let V of ˆ in over each s 2 be the preimage of is ˆ s V V Å and, in particular, all the ˆ ˆ is a fiber bundle over have the same Euler smooth, s V 2 characteristic. We know that has exactly 3.d ˆ 1/ 2 singular fibers, each a curve 0 with a single node, so that by Theorem 7.17 2 V 2 .ˆ / D d.d 3/ C 3.d 1/  I top 0 V ¤ 0 the since for ˆ s have the same Euler characteristic, the same logic tells us that s 2 V 1/ singular fibers over 2 they also have exactly V . It follows that ˆ 3.d has two s singular fibers for s j <  , completing the argument. 0 < j is a plane curve with a unique C This argument shows that, more generally, if , the tangent cone to D at singular point will be a multiple of the hyperplane H p , C p C p ;:::;p , the tangent cone has isolated singularities and, more generally still, if 1 ı . D is supported on the union of the planes H T p C i There is also a sort of converse to Proposition 7.18: Proposition 7.20. D consists exactly of those curves with a single The smooth locus of node and no other singularity. Proof: Proposition 7.18 gives one inclusion: if has a node and no other singularity, it C is a smooth point of D . Moreover, if C has more than one (isolated) singular point, then the projection map † ! D is finite but not one-to-one over C ; it is intuitively clear (and follows from Zariski’s main theorem) that is analytically reducible and hence singular D at C . Moreover, we observe that if d  3 any curve with multiple components is a limit of curves with isolated singularities and at least three nodes — just deform each multiple

299 Exercises Section 7.7 285 mC to a union of m general translates of C — so these must also lie component C  0 0 . in the singular locus of D It remains to see that if C is a singular curve having a singularity p other than a is singular at C . This follows from an analysis of plane curve singularities: node, then D has isolated singularities including a point of multiplicity k  3 , then, as we saw p C If z is at most in Section 2.4.6, the genus of the normalization C   k.k 1/ d 1 Q  D/ g. 2 2 k p are identified in C , and, since at most points of the normalization lying over 2 z  2 2g.  C/ C k 1  d.d 3/ C : 1/ .C/ .k top p other than a node, we have already done the case of a cusp; As for double points other double points will drop the genus of the normalization by 2 or more, and since p , we must have  .C/ we have at most two points of the normalization lying over  top d.d 3/ C 3 . Finally, note that the techniques of this section can be applied in exactly the same way in one dimension lower! d 0 . P H Let D O P d .d// be the space of polynomials of degree Proposition 7.21. 1 P 1 d D P P on the discriminant hypersurface, that is, the locus of polynomials , and  F 2 D is a point corresponding to a polynomial with exactly with a repeated root. If p d one double root 2 simple roots, then D is smooth at F with tangent space the and p . space of polynomials vanishing at We leave the proof via the topological Hurwitz formula as an exercise; for an algebraic proof, see Proposition 8.6. We add that there are many, many problems having to do with the local geometry of D and its stratification by singularity type, only a small fraction of which we know how to answer. The statements above barely scratch the surface; for more, see for example ̈ orrer [1986] or Teissier [1977]. Brieskorn and Kn 7.8 Exercises 1 1 Exercise 7.22.  P  P S , and let f Let be a general pencil of curves D S g C 1 t t 2 P of type .a;b/ on S , where a;b > 0 . What is the expected number of curves C that are t D .1;1/ !) .a;b/ singular? (Make sure your answer agrees with (7.2) in the case Prove that the number found in the previous exercise is the actual number Exercise 7.23. of singular elements; that is, prove the three hypotheses of Proposition 7.9 in the case of 1 1 O .a;b/  P . and the line bundle P S D

300 286 Singular elements of linear series Chapter 7 3 S Let be a smooth cubic surface and L  S a line. Let f C  g Exercise 7.24. P 1 t P 2 t 3 H be the pencil of conics on  S cut out by the pencil of planes g containing L . f P t are singular? Use this to answer the question of how many How many of the conics C t S L . other lines on meet 2 2 P p be a point, and let f C Exercise 7.25.  P Let g be a general pencil of 2 1 t P 2 t — in other words, let p and G be two general polynomials plane curves singular at F p , and take C F D V.t C vanishing to order 2 at C t . How many of the curves G/ 1 t 0 t will be singular somewhere else as well? 4 Exercise 7.26.  \ X S D P X be a smooth complete intersection of hyper- Let 2 1 4 f surfaces of degrees f H e  P and g is a general pencil of hyperplanes in . If 1 t 2 P t 4 P S \ H , find the expected number of singular hyperplane sections . (Equivalently: if t 2 4  P ƒ is a general 2-plane, how many tangent planes to P intersect ƒ in a line?) Š S 4 X  P Exercise 7.27. be a smooth hypersurface of degree d . Using formula (7.1) , Let X in a pencil. Again, find the expected number of singular hyperplane sections of compare your answer to the result of Section 2.1.3. 1 2 5 Š P P  P ,  Let X be the Segre threefold. Using formula (7.1) Exercise 7.28. find the number of singular hyperplane sections of in a pencil. X 4 Let X S \ X D  P Exercise 7.29. be a smooth complete intersection of hyper- 2 1 surfaces of degrees f . What is the expected number of hyperplane sections of S and e e f D 2 !) having a triple point? (Check this in the case D n S  P d be a smooth surface of degree Let whose general hyperplane Exercise 7.30. 2 ; let e and f be the degrees of the classes c 2 . T / / section is a curve of genus ;c g . T 2 1 S S 2 in terms of . Find the class of the cycle T .S/ S  G .1;n/ of lines tangent to A .S/ 1 d;e;f and g ; from Exercise 4.21, we need only the intersection number ŒT . .S/ç   3 1 Consider instead the variety of tangent planes .S/  G .2;n/ , and find the T Hint: 2 2 minus the intersection with . /  intersection with  . as the intersection with 1;1 2 1 3 Exercise 7.31.  P Let S d and B a general net of be a general surface of degree plane sections of S (that is, intersections of X with planes containing a general point 3  p ). What are the degree and genus of the curve Ä P S traced out by singular 2 points of this net? What are the degree and genus of the discriminant curve? Use this to 2  . W S ! P p given by projection from describe the geometry of the finite map p 2 Exercise 7.32. C  P the number of degree d Verify that for a general curve 3d.d 2/ is the actual number of flexes of C , that is, that all inflection points of C have weight 1.

301 Exercises Section 7.8 287 2 Let f  P Exercise 7.33. g C ; 3  be a general pencil of plane curves of degree d 1 t 2 t P is a singular element of suppose C (so that in particular by Proposition 7.1 C C will 0 0 have just one node as singularity). By our formula, will have six fewer flexes than C 0 C the general member of the pencil. Where do the other six flexes go? If we consider t the incidence correspondence 1 2 2 P ;  P ˆ j C Df is smooth and p is a flex of C .t;p/ g t t ˆ near t D 0 ? Bonus question: Describe the what is the geometry of the closure of geometry of 1 2 2  z a flex of  P Df  P .t;p;L/ 2 j C g smooth, p P C C and ˆ D T L p t t t t 0 . D near 1 P Find the points on , if any, that are ramification points for the maps Exercise 7.34. 1 3 P ! P given by 3 4 3 3 3 2 4 2 3 3 / 2 ;t and .s;t/ 7! .s t;st ;s ;s t;st P ;t : / 2 P .s;t/ 7! .s r Exercise 7.35. C P Show that the only smooth, irreducible and nondegenerate curve  with no inflection points is the rational normal curve. elliptic normal curve to be a smooth irreducible nondegen- Exercise 7.36. We define an r C 1 in P erate curve of genus 1 and degree . Observe that for an elliptic normal curve r 2 ̈ the Pl E C 1/ .r of inflection points. Show that these ucker formula yields the number r , 1 on E are exactly the images of any one under the group of translations of order C each having weight 1. 2 C be a smooth curve of genus g  Let . A point p 2 C is called a Exercise 7.37. Weierstrass point if there exists a nonconstant rational function on C with a pole of order g or less at and regular on C nf p g . p Show that the Weierstrass points of are exactly the inflection points of the C (a) g 1 . ! C W canonical map P ' (b) Use this to count the number of Weierstrass points on C . N Let P 4 be the space of all plane curves of degree d  Exercise 7.38. , and let N H  be the closure of the locus of smooth curves with a hyperflex. Show that H is P a hypersurface. (We will be able to calculate the degree of this hypersurface once we have developed the techniques of Chapter 11.) 3 To prove that a general complete intersection C  P does not have Exercise 7.39. weight-2 inflection points, we need to prove that it does not have flex lines (lines with multiplicity-3 intersection with the curve) or planes with a point of contact of order 5. S and S Prove the first statement: that a general complete intersection of two surfaces 1 2 does not have a flex line. d > 1  d of degrees 2 1

302 288 Chapter 7 Singular elements of linear series The following two exercises show how to construct an example of a component of the Hilbert scheme whose general member is a smooth, irreducible, nondegenerate curve > 1 having inflection points of weight . n n 1 P E   P D be a cone over an elliptic normal curve S p;E Exercise 7.40. Let (that is, a smooth curve of genus 1 embedded by a complete linear system of degree n ), and ;:::;L  S be lines of the ruling. Show that, for n > 9 L m  0 : and let n 1 1 n S with a general hypersurface X  P C of m (a) The residual intersection of degree L ;:::;L containing is a smooth, irreducible and nondegenerate curve. 1 1 n Every deformation of C also lies on a cone over an elliptic normal curve. (The (b) is necessary to ensure that the surface has itself no deformations n > 9 condition S other than cones. This follows from the classification of del Pezzo surfaces ; see for example Beauville [1996].) constructed in this fashion Thus the smooth, irreducible and nondegenerate curves C n form an open subset of the Hilbert scheme of curves in P . n C  S  P be a curve as constructed in the preceding problem. Exercise 7.41. Let C Show that (look at points where C is tangent to a > 1 has inflection points of weight ). S line of the ruling of 3 S  P Exercise 7.42. be a general surface of degree d  2 , p 2 S a general Let 3 H D T . Show by an elementary S  P point and the tangent plane to S at p p dimension count (not using the second fundamental form or quoting Theorem 7.11) that H S has an ordinary double point at p . the intersection \ 1 1 Exercise 7.43. P   P Let , and let f C S D S g be a general pencil of curves 1 t 2 t P .a;b/ on of type . Use the topological Hurwitz formula to say how many of the curves S C are singular. (Compare this with your answer to Exercise 7.22.) t 2 2 p 2 P g be a point, and let f C be a general pencil of  P Exercise 7.44. Let 1 t t 2 P d p , as in Exercise 7.25. Use the topological Hurwitz plane curves of degree singular at formula to count the number of curves in the pencil singular somewhere else. 5 5 P be the space of conic plane curves and D  P Let the discriminant Exercise 7.45. hypersurface. Let C 2 D be a point corresponding to a double line. What is the multiplicity of at C , and what is the tangent cone? D 14 14 P P be the space of quartic plane curves and D  Exercise 7.46. Now, let the discriminant hypersurface. Let C 2 D be a point corresponding to a double conic. What is the multiplicity of D at C , and what is the tangent cone?

303 Chapter 8 Compactifying parameter spaces Keynote Questions 2 (a) ;:::;C (The five conic problem) Given five general plane conics C P  , how 1 5 2  P many smooth conics are tangent to all five? (Answer on page 308.) C 2 (b) Given general points p ;:::;p 11 2 P in the plane, how many rational quartic 1 11 2 C  P curves contain them all? (Answer on page 321.) All the applications of intersection theory to enumerative geometry exploit the fact that interesting classes of algebraic varieties — lines, hypersurfaces and so on — are parameter space , themselves parametrized by the points of an algebraic variety, the and our efforts have all been toward counting intersections on these spaces. But to use intersection theory to count something, the parameter space must be projective (or at least proper) so that we have a degree map, as defined in Chapter 1. In the first case we treated in this book, that of the family of planes of a certain dimension in projective space, the natural parameter space was the Grassmannian, and the fact that it is projective is what makes the Schubert calculus so useful for enumeration. When we studied the questions about linear spaces on hypersurfaces, we were similarly concerned with parameter spaces that were projective — the Grassmannian G .k;n/ and, in connection with questions involving families of hypersurfaces, the projective space N of hypersurfaces itself. These spaces have an additional feature of importance: a P universal family of the geometric objects we are studying, or (amounting to the same thing) the property of representing a functor we understand. This property is useful in many ways, first of all for understanding tangent spaces, and thus transversality questions. In many interesting cases, however, the “natural” parameter space for a problem is not projective. To use the tools of intersection theory to count something, we must add points to the parameter space to complete it to a projective (or at least proper) variety. It is customary to call these new points the boundary , although this is not a topological

304 290 Chapter 8 Compactifying parameter spaces boundary in any ordinary sense — the boundary points may look like any other point of of the space — and (more reasonably) to call the enlarged space a compactification the original space. If we are lucky, the boundary points of the compactification still parametrize some sort of geometric object we understand. In such cases we can use this structure to solve geometric problems. But as we shall see, the boundary can also get in the way, even when it seems quite natural. In such cases, we might look for a “better” compactification. . . but just how to do so is a matter of art rather than of science. Perhaps the first problem in enumerative geometry where this tension became clear is the five conic problem, which was solved in a naive way, not taking the difficulty into account (and therefore getting the wrong answer) by Steiner [1848], and again, with the necessary subtlety (and correct answer!) by Chasles [1864]. In this case there is a very beautiful and classical construction of a good parameter space, the space of complete conics . In this chapter we will explore the construction, and briefly discuss two more general constructions: Hilbert schemes and Kontsevich spaces. 8.1 Approaches to the five conic problem To reiterate the problem: Given five general plane conics ;:::;C , how many C 1 5 smooth conics are tangent to all five? Here is a naive approach: 5 (a) . The locus of conics tangent to each The set of plane conics is parametrized by P 5 is an irreducible hypersurface given Z , as one sees by considering the  P C i i incidence correspondence 5 2 P at  C f j C a conic tangent to C g .C;p/ p i i  2 C Z i i 5 of dimension 3. (Here, are linear subspaces of P  and noting that the fibers of 2 2 “tangent to at p ” means m of the .C  C C /  , that is, the restriction to C i i p i C defining equation of p .) vanishes to order at least 2 at 5 (b) Z — that is, is 6. To see this, we intersect Z P with a general line in The degree of i i C . The conic we take a general pencil of conics and count how many are tangent to i 1 2 C by the complete linear system in P may be thought of as the embedding of P i of degree 2. Thus a general pencil of conics cuts out a general linear series on C of i is the number of divisors in this family with fewer Z degree 4, and the degree of i 1 than four distinct points. The linear series defines a general map ! P of degree C i 4 with distinct branch points, and by the Riemann–Hurwitz theorem (Section 7.7) the number of branch points of this map is six.

305 Approaches to the five conic problem Section 8.1 291 Thus the number of points of intersection of Z ;:::;Z (c) , assuming they intersect 5 1 5 D 7776 . will be transversely, 6 not Alas, 7776 is the answer to the question we posed. The problem is not hard to spot: far from being transverse, the hypersurfaces Z do not even meet in a finite set! i 5 P To be sure, the part of the intersection within the open set U  of smooth conics is (which is what we wanted to count) finite, and even transverse, as we will verify below. The trouble is with the compactification: we used the space of all (possibly singular) Z takes place along the boundary. conics, and “excess” intersection of the i 5 is the In more detail: the hypersurface in P Z of the locus of smooth closure i C tangent to C exactly when the defining . A smooth conic C conics C is tangent to i i 1 1 P C of Š P F and viewed as a quartic polynomial on C , restricted to , has a equation i multiple root. When we extend this characterization to arbitrary conics C we see that 5 Z  ;:::;Z a double line is tangent to every conic . Thus the five hypersurfaces P 5 1 5 5 S P of double lines, which is a Veronese surface in the P will all contain the locus  T of conics. As we shall see, the intersection Z is the union of S and the finite set of i smooth conics tangent to the five C . The presence of this extra component S means that i Q 5 5 the number we seek has little to do with the intersection product 2 A ŒZ . P ç / . i There are at least three successful approaches to dealing with this issue: Blowing up the excess locus Suppose we are interested in intersections inside some quasi-projective variety U and we have a compactification V of ; in the example above, U is the space of smooth U V the space of all conics. We could blow up some locus in the boundary V n U conics and to obtain a new compactification. This is the classical way of separating subvarieties of a given variety that we do not want to meet. In the five conic problem, we would blow 5 z P S and consider the proper transforms in Z up the surface of the hypersurfaces Z i i 5 X D Bl in the blow-up P . If we are lucky (and in this case we are), we will have S z Z eliminated the excess intersection — that is, the will not intersect anywhere in the i  X of the blow-up. (If this were not the case we would have to exceptional divisor E T z z E \ Z .) In our case, the Z intersect blow up again, along the common intersection i i U . To finish the argument, we could determine the Chow transversely, and only inside 1 z of the blow-up, find the class  2 A ring .X/ of the A.X/ Z (as members of a family i 5 P , they all have the same class) and evaluate the parametrized by an open subset of 5 5 A 2 product  .X/ . Readers who want to carry this out themselves can find a description of the Chow ring of a blow-up in Section 13.6; there is also a complete account of this approach in Griffiths and Harris [1994, Section 6.1]. This approach has the virtue of being universally applicable, at least in theory: Any component of any intersection of cycles can be eliminated by blowing up repeatedly. But often we cannot recognize the blow-up as the parameter space of any nice geometric

306 292 Chapter 8 Compactifying parameter spaces objects, and this makes the computations less intuitive and sometimes unwieldy. For example, this approach to the problem of counting cubics satisfying nine tangency condi- tions (solved heuristically by Maillard and Zeuthen in the 19th century and rigorously in Aluffi [1990] and Kleiman and Speiser [1991]) requires multiple blow-ups of the space 9 P of cubics and complex calculations. Excess intersection formulas Excess intersection problems were already considered by Salmon in 1847, and were generalized greatly by Cayley around 1868. The excess intersection formula of Fulton and MacPherson (see Fulton [1984, Chapter 9]) subsumes them all: It is a general T Z formula that assigns to every connected component of an intersection  X a class i in the appropriate dimension, in such a way that the sum of these classes (viewed as classes on the ambient variety via the inclusion) equals the product of the classes of X the intersecting cycles. This applies whenever all but at most one of the subvarieties ; in our case all are hypersurfaces. We will are locally complete intersections in Z X i give an exposition of the formula in Chapter 13, and show in Section 13.3.5 how it may be applied to the five conic problem, as was originally carried out in Fulton and MacPherson [1978]. As a general method, excess intersection formulas are often an improvement on blowing up. But, as with the blow-up approach, they require some knowledge of the normal bundles (or, more generally, normal cones) of the various loci involved. Changing the parameter space To understand what sort of compactification is “right” for a given problem is, as we have said, an art. In the case of the five conic problem, we can take a hint from the fact that the problem is about tangencies. The set of lines tangent to a nonsingular conic is again a conic in the dual space (we will identify it explicitly below). But when a conic degenerates to the union of two lines or a double line, the dual conic seems to disappear — the dual of a line is only a point! This leads us to ask for a compactification of the space of smooth conics that keeps track of information about limiting positions of tangents. There are at least two ways to make a compactification that encodes the necessary information. One is to use the Kontsevich space . It parametrizes not subschemes of 2 2 , but rather maps P W C ! P f with C a nodal curve of arithmetic genus 0. This is an important construction, which generalizes to a parametrization of curves of any degree and genus in any variety. We will discuss it informally in the second half of this chapter. But proving even the existence of Kontsevich spaces requires a considerable development, and we will not take this route; the reader will find an exposition in Fulton and Pandharipande [1997]. The other way to describe a compactification of the space of smooth conics that preserves the tangency information is through the idea of complete conics . The space of

307 Complete conics 293 Section 8.1 complete conics is very well-behaved, and we will spend the first half of this chapter on this beautiful construction. It turns out that the space we will construct is isomorphic 5 to the Kontsevich space for conics (and, for that matter, to the blow-up of Bl P S 5 along the surface of double lines), but generalizes in a different direction: There P are analogs for quadric hypersurfaces of any dimension, for linear transformations (“complete colineations”) and, more generally, for symmetric spaces (see De Concini and Procesi [1983; 1985], De Concini et al. [1988] and Bifet et al. [1990]), but not for curves of higher degree or genus. (There is an analogous construction but, as we will remark at the end of Section 8.2.2, in general the space constructed is highly singular and not well-understood.) 8.2 Complete conics We begin with an informal discussion. Later in this section we will provide a rigorous 2 C is the P of a smooth conic dual foundation for what we describe. Recall that the  2    , regarded as a curve . As we shall see,  C C P is also a C set of lines tangent to smooth conic (this would not be true in characteristic 2!). 8.2.1 Informal description Degenerating the dual Consider what happens to the dual conic as a smooth conic degenerates to a singular conic — either two distinct lines or a double line. That is, let C ! B be a one-parameter t , with C ¤ smooth for t family of conics with parameter 0 . Associating to each curve t 2   to  P C C we get a regular map from the punctured disc B nf the dual conic g 0 t t 2  5  2  2  P the space of conics in P V and P P P D P V D , the space of conics on . (If 2 2  Sym P and P Sym V V — in particular, they are naturally dual each are respectively 5 5  to one another, so if we write the former as P P it makes sense to write the latter as .) 2  5  Since the space of all conics in is proper, this extends to a regular map on all P P   lim C D . However, as B of C — in other words, there is a well-defined conic t ! 0 t 0 and not just on the curve C we will see, this limit depends in general on the family : C 0  C is not determined by the limit of the curves C in other words, the limit of the duals . t t U of smooth conics that captures this To provide a compactification of the space 5 5  phenomenon, we realize  P U as a locally closed subset of : As we will see in the P  C 7! C following section, the map is regular on smooth conics, so U is isomorphic to 5  the graph of the map ! P sending a smooth conic C to its dual. That is, we set U  2   2  5 5 its dual  P 2 j C a smooth conic in P / and C .C;C  P Df U P g : The desired compactification, the , is the closure variety of complete conics 5 5   : X  P D U P

308 294 Chapter 8 Compactifying parameter spaces M L p C t C 0 D Figure 8.1 Conics specializing to a conic M of rank 2. C L [ 0 The dual of the dual of a smooth conic is the original conic, as we shall soon see (in fact, the same statement holds for varieties much more generally, and will be proven in 5 5  P is symmetric under exchanging Section 10.6), so the set . It follows that X U and P  .C;C / X and is symmetric too. (As one consequence of this symmetry, note that if 2 5  or is smooth, then the other is too.) The set U  P of smooth conics is by C either C X , and it follows that is irreducible and of dimension 5 as well. definition dense in X  C when C becomes singular? Let us first consider the case of a What happens to f C is g of smooth conics approaching a conic C of rank 2, that is, family C M D L [ 0 0 t the union of a pair of distinct lines; for example, the family given (in affine coordinates 2 on P ) as 2 2 2 D t y j ; g x 2 Df P  B C .t;x;y/ as shown in Figure 8.1. The picture makes it easy to guess what happens: Any collection L through the point g of lines with f L tangent to C approaches a line for t ¤ 0 L t 0 t t through D \ M p L tangent to L p is a limit of lines L , and conversely any line t 0 0 . (Actually, the second statement follows from the first, given that the limit C D C t 0 0  C is one-dimensional.) Since C is by definition a conic, it must be the double lim t ! 0 t 0 2  of the line in p , irrespective of the family f C P g used to construct it dual to the point t or of the positions of the lines L and M . Things are much more interesting when we consider a family of smooth conics 2  C P C g specializing to a double line C f D 2L , and ask what the limit lim of  t 0 ! t 0 t 2   the dual conics C  P may be. One way to realize such a family of conics is as the t 1 2 . Such a family of maps is given by a triple W P images of a family of maps ! P ' t / of polynomials .x/;g , whose .x/;h .f .x// , homogeneous of degree 2 in x D .x ;x 0 t t 1 t coefficients are regular functions in t . In our present circumstances, our hypotheses t are linearly independent (and so span 0 the polynomials f ;g and h say that for ¤ t t t 0 H O  W .2// ), but for . D 0 they span only a two-dimensional vector space t 1 P 0 H O .2// . . For now, we will make the additional assumption that the linear system 1 P is base point free; the case where it is not will be dealt with below. .2/;W / . O D W 1 P

309 Complete conics Section 8.2 295 r p t q t q p 2 2 Figure 8.2 The family of conics 2y/ . D t.x y 1  u;v 2 P will be in this situation, let C To see what the limit of the dual conics t 1 1 P be the ramification points of the map ! P ' associated to W (note that the W W 1 ' P map as the is just the composition of this map with the inclusion of the target 0 2 q L ), and let p D ' P .u/ and  D ' line .v/ 2 L be their images. We claim that 0 0  2     of the dual conics is the conic in this case the limit D p C C q C  P lim ! t 0 t 0 and lines through q . p consisting of lines through 2 P be any point not in To prove this, let L and not in any curve C r , and let 2 t 2 1 1 ! L  r to L . The composition  ı ' W W P P ! L Š P be the projection from t 1 L ;v 2 2 P has degree 2; let be the ramification points of this map and p u ;q t t t t  the corresponding branch points. Suppose that ' is the map associated to the ı t 0 W . As D . O .2// O .2/;W . / for a two-dimensional vector space W  pencil H 1 1 t t t P P approaches the linear system W t 0 W ; correspondingly, the , the linear system ! t v C divisor . In other words, the approaches u C v and p q C q C approaches p u t t t t C r passing through tangent lines to — which are exactly the lines r;' and .u r;p / D t t t t independently of .v . Thus every line / D r;q r — approach the lines r;p and r;q , r;' t t t through p or q is a limit of tangent lines to C , and conversely. t C is the union It is important to note that in this situation, unlike in the case where 0  C of two distinct lines, the limit of the dual conics is not determined by the conic t C . As we will see in Section 8.2.2, the points p and q may be any pair of points of L , 0 depending on the path along which approaches C . C t 0 p of the maps ;q L The remaining case to consider is when the branch points 2 t t ı '  p 2 L . (Typically, this corresponds to the case where approach the same point t W has a base point: When W has a base point u , the ramification of W is concen- trated at this point, which must then be the limit as of both the ramification t ! 0 W and v points of u .) In this case, the same logic shows that the limit of the dual t t t 2     dual to the image point of the line p C  P will be the double 2p conics t p D ' .u/ . 0

310 296 Chapter 8 Compactifying parameter spaces Types of complete conics 0 In conclusion, there are four types of complete conics, that is, points 2 X : .C;C / 0  0 0 C and C C are both smooth and . We will call these ; that is, D C 2 U .C;C (a) / smooth complete conics. 2  0   , where C D 2p [ L p M  P is of rank 2 and C is the line dual to (b) D D L \ p . M 2  0   (c) D p D [ q 2L is the union of the lines in P is of rank 1, and C dual to two C p;q L . points 2 2  0  C (d) D 2p C is the double of the line in D 2L is of rank 1, and dual to a point P p L . 2 0 C Note that the description is exactly the same if we reverse the roles of C , and except that the second and third types are exchanged. Note also that the points of each type form a locally closed subset of X , with the first open and the last closed, and all 5 5  four are orbits of the action of PGL  P . on P 3 As we have already explained, the locus of complete conics of type (a) is isomorphic U to ; in particular, it has dimension 5. It is easy to see that those of type (b) are , and thus form a set of dimension 4. By symmetry determined by the pair of lines L;M (or inspection) the same is true for type (c). Finally, those of type (d) are determined by L the line p 2 L ; thus these form a set of dimension 3, which is in fact the and the point intersection of the closures of the sets of points described in (b) and (c). 8.2.2 Rigorous description Let us now verify all these statements, using the equations defining the locus  5 5 P  P . We could do this explicitly in coordinates, but it will save a great deal of  X ink if we use a little multilinear algebra. The reader to whom this is new will find more than enough background in Appendix 2 of Eisenbud [1995]. The multilinear algebra allows us to treat some basic properties in all dimensions with no extra effort, so we begin with some general results about duality for quadrics. Duals of quadrics Let V be a vector space. Recall that since we are assuming the characteristic of the ground field k is not 2 the following three notions are equivalent:   ' W V ! V A symmetric linear map .  A quadratic map W V ! k . q 2 0   Sym An element .V 2 / . q  '.x/;x ' ! V Explicitly, if we start with a symmetric map then we take q.x/ Dh V i , W 2 0   Sym q .V 2 / comes about from the identification of Sym .V and the element / with the ring of polynomial functions on V .

311 Complete conics Section 8.2 297 Q Any one of these objects, if nonzero, defines a quadric hypersurface ,  P V V.q/ q , or equivalently the locus D of defined as the zero locus Q P V jh '.v/;v iD 0 g f v 2 : (Here, and in the remainder of this discussion, we will abuse notation and use the same v 2 V and the corresponding point in to denote both a nonzero vector V .) symbol v P  P V will be smooth if and only if ' is an isomorphism; more generally, The quadric Q will be the (projectivization of the) kernel of Q rank of Q is the singular locus of ' . The , or dim .Q / (where we ' defined to be, equivalently, the rank of the linear map n sing . ¿ / D 1 ); another way to adopt, for the present purposes only, the convention that dim is the cone with vertex a linear space k characterize it is to say that a quadric of rank n k n k 1 P . over a smooth quadric hypersurface Q  P Q P  Š sing n n  X P  is defined to be the closure in of the locus Now, the dual of any variety P (that is, containing the tangent space X of hyperplanes tangent to X at a smooth point T p 2 X p ). (We will describe this construction in far more detail in Section 10.6.) Given the description in the last paragraph of a quadric Q of rank k as a cone, we see that the k has dimension dual of a quadric of rank 2 . That said, we ask: what, in these terms, k is the dual to ? Q ' V ! W is any map of vector spaces of dimension To state the result, recall that if W c 1 , then there is a n ' C W W ! V , represented by a matrix whose entries cofactor map c c  n minors of ' , satisfying ' ı ' ' D det .'/ Id . and ' are signed ı n D det .'/ Id V W c ' In invariant terms, is the composite V n  ' V V n n   Š W W ! V Š V; V V n n   and Š are defined by choices of nonzero V where the identifications Š V W W V V 1 1 C n C n   W V and respectively. Note that vectors in the one-dimensional spaces c ' when the rank of the map ' < n is zero. is n D  P .V / Proposition 8.1. P Let be the quadric corresponding to the symmetric Q  ' W V ! V be a nonzero vector such that , and let v 2 V map h '.v/;v iD 0 , so that v 2 . The tangent hyperplane to Q at v is Q iD P Df w 2 T .V / jh '.v/;w Q 0 g : v The dual of Q is thus   Df '.v/ 2 P .V Q / j v 2 Q and '.v/ ¤ 0 g :  ), then ' is n C 1 is nonsingular (that is, if the rank of Q In particular, if is the Q   of Q under the induced map ' W P V ! P V image , and Q '.Q/ is the quadric c corresponding to the cofactor map ' .

312 298 Chapter 8 Compactifying parameter spaces c Q , and Q is is the quadric corresponding to On the other hand, if the rank of n c c  is the unique double hyperplane containing Q the cofactor map ; that is, , then Q ' c is the hyperplane corresponding to the annihilator of the singular the support of Q point of Q . w 2 V , the line v;w  P V spanned by v and w Proof: Q at v if For any is tangent to and only if 2 w/;v C w iD 0 mod . h /: '.v C Expanding this out, we get '.w/;v iCh '.v/;w iD 0; h and, by the symmetry of ' and the assumption that we are not in characteristic 2, this is the case if and only if h '.v/;w 0; iD  Q '.Q/ . proving the first statement and identifying the dual variety as D c ' or n Q 1 . Let n is be the matrix of cofactors of ' , so Suppose the rank of C c is the identity map. Since ' D det ' ı I , where I that rank ' D rank '  n , the Q c c  Q map is by definition the set of all w 2 V is nonzero. The quadric such that ' c v .w/ D 0 . If w;' 2 Q then h c '.v/;' iD '.v/ iD . det '/ h '.v/;v h 0; c so Q '.Q/ . is contained in  rank n C 1 , so that ' is an isomorphism, then Q D D '.Q/ is again a quadric If ' c c  Q hypersurface, and we must have . If rank ' D n , then since ' Q ' D 0 D the c rank of is 1, and the associated quadric is a double plane. On the other hand, Q ' 1 n  P , and Q is the dual of that quadric is the cone over a nonsingular quadric in n   spans the plane inside a hyperplane (corresponding to the vertex of . Thus Q P ) in Q c contained in Q . The following easy consequence will be useful for the five conic problem: 0 0 Q and If are smooth quadrics, then Q and Q Q have the same tangent Corollary 8.2. 0  0 Q v 2 Q \ l D if and only if Q 0 and Q hyperplane at some point of intersection 0  D 0 l 2 Q v \ Q have the common tangent hyperplane . at the point of intersection D is a smooth plane conic then the divisor Z  X , In particular, it follows that if D 0 .C;C C / such that which is the closure of the set of complete conics is smooth and  tangent to , is equal to the divisor defined similarly starting from the dual conic D , D 2  0 0 such that that is, the closure of the set of .C;C  P C / is smooth and tangent to the  dual conic D .

313 Complete conics Section 8.2 299 0 Suppose that Q Q correspond to symmetric maps ' and . Since the Proof: and  0 \ 2 Q are the same, Proposition 8.1 shows that D Q .v/ . tangent planes at v '.v/  1 1 1 0 D  D .'.v// .'.v// , we see that Q . .v// and Q v are in fact Since ' '.v/ . (In addition to the fact that the duality interchanges points and planes, tangent at 0  is we are really proving that the dual of , and similarly for Q Q . Such a thing is Q actually true for any nondegenerate variety, as we will see in Section 10.6.) Equations for the variety of complete conics 2 P , and suppose that V is three-dimensional. We now return to the case of conics in Proposition 8.3. The variety 5 5 2  2  X V Sym P . Sym . V / D P P  P  /  5 5  of complete conics is smooth and irreducible. Thinking of  .'; / 2 P as coming P   W V ! V ! and W V from a pair of symmetric matrices ' V , the scheme X is defined by the ideal I generated by the eight bilinear equations specifying that the ı ' product has its diagonal entries equal to one another (two equations) and its off-diagonal entries equal to zero (six equations). has codimension 5, I (For the experts: it follows from the proposition that the ideal and that its saturation, in the bihomogeneous sense, is prime. Computation shows that the polynomial ring modulo I is Cohen–Macaulay. With the proposition, this implies that I is preserved under the interchange of factors ' and , I is prime. In particular, which does not seem evident from the form given.) Proof: Let Y be the subscheme defined by the given equations. We first show that agrees set-theoretically with X on at least the locus of those points .'; / where Y ' 2 or rank  2 , that is, where ' or  corresponds to a smooth conic or the rank ' has rank 3 and .'; / 2 Y union of two distinct lines. On the locus of smooth conics, 1 D ' if and only if up to scalars, so Proposition 8.1 shows that the dual conic is defined by ' is 2 and .'; / 2 Y , then we see from the . Moreover, if the rank of c equations that D 0 . Up to scalars, ı D ' is the unique possibility, and again ' 0 C C is the dual of Proposition 8.1 shows that the corresponding conic . To see the uniqueness (up to scalars) in terms of matrices, note that in suitable bases 0 1 1 0 0 A @ ' D 0 1 0 0 0 0 and the symmetric matrices annihilating the image have the form 0 1 0 0 0 c @ A D : a' D 0 0 0 0 0 a

314 300 Chapter 8 Compactifying parameter spaces c , ' D The same arguments show that when and again they correspond  rank 2 c c 1 on this locus we do not have ' D there.) rank to dual conics. (Note that since D 5 5   P is defined as the closure in X of the locus U of pairs .C;C P / with Since  X Y Y . We will show next that smooth, we see now in particular that is smooth of C  locally at any point 2 5 where both ' and .'; / have rank 1. We will dimension Y Y is everywhere smooth of dimension 5. use this to show that .'; / 2 Y and that both ' and have rank 1. The tangent To this end, suppose that Y space to may be described as the locus of pairs of symmetric at the point .'; /   W V V ; ˇ W V ! ! ̨ such that V matrices 2 mod .' C  ̨/ ı . ˇ/ / C . ' and have rank 1, has equal diagonal entries and zero off-diagonal entries. Since both ̨ C ˇ ı ' is at most 2, so this is equivalent to saying that ı the rank of ı ̨ C ˇ ı ' D 0: . ̨;ˇ/ is equivalent We must show that this linear condition on the entries of the pair to five independent linear conditions. In suitable coordinates the maps '; will be represented by the matrices 0 1 0 1 1 0 0 0 0 0 @ A @ A D ' D and : 0 0 0 0 1 0 0 0 0 0 0 0 Multiplying out, we see that 1 0 1 0 0 0 0 0 0 ˇ 1;1 @ A @ A ı D and ˇ ı ' D ̨ : ̨ ˇ 0 0 ̨ ̨ 2;1 2;3 2;2 2;1 0 0 0 0 0 ˇ 3;1 ̨ C ˇ' Thus the equation 0 is equivalent to the equations ̨ and C ˇ 0 D D 2;1 2;1 D ˇ ̨ : five independent linear conditions, as required. D ̨ 0 D ˇ D 3;1 1;1 2;3 2;2 To complete the proof of smoothness, note that Y is preserved scheme-theoretically by the action of the orthogonal group G .'; / 2 Y and ̨ is orthogonal, . (Proof: If    ̨ / 2 then since ̨ ; ̨ ̨ . ̨' ̨ D 1 .) Any closed point on Y where rank '  2 Y degenerates under the action of G to a point where rank ' D 1 . (Proof: If ̨ is orthogonal,     ̨ ̨ ' is diagonal if and only if ̨' ̨ that is, D 1 D ̨' ̨ , then the matrix ̨ ̨ ' is diagonal, stretching one of the coordinates is diagonal. Thus, in a basis for which c ' approach zero, and D ' will make the corresponding entry of moves at the same time; a similar argument works when rank  2 .) Consequently, if the singular locus of Y were not empty it would have to intersect the locus of pairs of matrices of rank 1, and we have seen that this is not the case.

315 Complete conics Section 8.2 301 is equal to Y X Finally, to see scheme-theoretically it is enough to show that the open Y Y . We use the fact that each point .'; / of Y corresponds subset U of is dense in 0 . When ' has rank 2 , Q corresponds to a pair of / to a unique pair of quadrics .Q;Q 0 is uniquely determined. Thus this set is four-dimensional. The Q distinct lines, and same goes for the case where ' and have has rank 2. On the other hand, when both 0 L and Q Q is the double of a line corresponding to one rank 1, is the double of a line ; thus this set is only three-dimensional. Since is everywhere smooth L Y of the points of must intersect the set where ' and of dimension 5, any component of have rank 3, Y as required. The classification of the points of X into the four types on page 296 follows from Proposition 8.3: If .'; / 2 X , then one of the following holds: Corollary 8.4. ' is of rank 3, then must be its inverse. (a) (Smooth complete conics) If ' X is symmetric) the products ı ' and ' ı If must is of rank 2, then (since (b)  both be zero; it follows that ! V is the unique (up to scalars) symmetric map V and whose image is the kernel of ' . ' whose kernel is the image of If ' is of rank 1, may have rank 1 or 2; in the latter case, it may be any symmetric (c)  ! V whose kernel is the image of map and whose image is the kernel of ' . V ' If ' and both have rank 1, they simply have to satisfy the condition that the kernel (d) ' and vice versa. of contains the image of 5 P Relations with the blow-up of We mentioned at the beginning of this chapter that another approach to the problem of excess intersection in the five conic problem would be to blow up the excess compo- 5 5 5 Bl of P S of P nent — that is, to pass to the blow-up along the surface Z  P D S double lines. It is natural to ask: what is the relation of the space of complete conics X Z ? to the blow-up The answer is that they are in fact the same space. To see this, it is helpful to recall the characterization of blow-ups given in Eisenbud and Harris [2000, Proposition IV-22]: and subscheme A  Y with ideal .f For an affine scheme ;:::;f Y / , the blow-up 1 k k 1 Y of Y along A is the closure in Y Bl P ! Y ! of the graph of the map Y n A  A k 1 given by Œf a ;:::;f P ç . We can globalize this: Let Y be any scheme and A  Y 1 k 0 L is a line bundle on and  Y ;:::;s sections generating the 2 H subscheme. If . L / 1 k k 1 L , then the closure of the graph of the map Y subsheaf A ! P I given by ̋ n A=Y Œf . ;:::;f A ç is the blow-up Bl along Y ! Y of Y 1 A k This is exactly what we have in the construction of the space X of complete conics. 5 of conics as the space of symmetric Again, we think of the space 3  3 matrices M , P 5 S  P and the Veronese surface of double lines as the locus of matrices of rank 1. The .2/ .2/ six minors of M are then sections of O I  , so that the blow-up generating 5 5 i P S= P

316 302 Chapter 8 Compactifying parameter spaces 5 5 5 P ;:::; . But as we ç W P Œ n S ! P is the closure of the graph of the map Bl 6 1 S have just seen this is the map sending a conic to its dual, so the closure of the graph is X of complete conics. the variety N One note: We could construct an analogous compactification of the space  P U 2 C  P d its dual of smooth plane curves of any degree by associating to each smooth M M is the space of plane curves of P ! , where U P curve. This defines a regular map N M , and we can compactify U by taking the closure in degree d.d  P of the 1/ P d D 3 , graph of this map. The resulting spaces are highly singular — already in the case 9 Aluffi [1990] showed it takes five blow-ups of to resolve the singularities — so in P general this is not a useful approach. 8.2.3 Solution to the five conic problem of complete conics is smooth and Now that we have established that the space X projective, we will show how to solve the five conic problem. To any smooth conic 2 we associate a divisor Z  D P  X , which we define to be the closure D Z D  .C;C in / 2 X with C smooth and tangent to D , and let X of the locus of pairs 1 ŒZ  ç 2 A D .X/ be its class. We will address each of the following issues: D 5  of the space P We have to show that in passing from the “naive” compactification U X , we have in fact of smooth conics to the more sensitive compactification eliminated the problem of extraneous intersection; in other words, we have to show C X the corresponding divisors Z  intersect only that for five general conics i C i 0  0 in points X with C and C .C;C D C / smooth. 2  We have to show that the five divisors Z are transverse at each point where they C i intersect.  We have to determine the Chow ring of the space X , or at least the structure of a A.X/ containing the class  of the hypersurfaces Z subring of we wish to intersect. C i   in this ring and find the degree of the fifth power We have to identify the class 5 5 A .X/ .  2 Complete conics tangent to five general conics are smooth X is symmetric under the operation of interchanging the We begin by recalling that 5 5  P and P . factors 0 .C;C of type (b) lies in the / Let us start by showing that no complete conic intersection of the divisors associated to five general conics. The first thing we need to 0 do is to describe the points / of type (b) lying in Z .C;C for a smooth conic D . This D is a conic of rank 2 which is a limit of smooth conics D L [ M is straightforward: If C tangent to D , then C also must have a point of intersection multiplicity 2 or more with D ; thus either L or M is a tangent line to D , or the point p D L \ M lies on D . (Note that by symmetry a similar description holds for the points of type (c): the complete   .) C q .2L;p / will lie on Z D conic L is tangent to D , or p or q lie on only if D

317 Complete conics Section 8.2 303 0 .C;C is a complete conic of type (b) lying in the intersection of Now, suppose that / C the divisors Z associated to five general conics C . Write D D L [ M , and set Z i i C i are general, no three are concurrent; thus . We note that since the C p \ p can D M L i . We will proceed by considering three cases in turn: lie on at most two of the conics C i  C . This is the most immediate case: Since the conics lies on none of the conics p i  are also general, it is likewise the case that no three of them are concurrent. C i In other words, no line in the plane is tangent to more than two of the , and C i .L M;p/ 2 Z correspondingly [ for at most four of the C . i C i  p C , say C are general with and lies on two of the conics C . Since C and C ;C 4 2 5 1 i 3 C and C , none of the finitely many lines tangent to two of them passes respect to 2 1 C can each be tangent to at most one of \ C ; thus through a point of L and M 1 2 M;p/ C C , and again we see that .L [ and 2 Z the conics ;C for at most four 3 4 5 C i of the . C i  C p , say C is general with respect . Now, since C lies on exactly one of the conics 1 1 i and ;C to ;C C , it will not contain any of the finitely many points of pairwise C 5 2 3 4 L intersection of lines tangent to two of them. Thus M cannot each be tangent and to two of the conics C ;:::;C for , and once more we see that .L [ M;p/ 2 Z 2 5 C i . C at most four of the i Z ; by symmetry, Thus no complete conic of type (b) can lie in the intersection of the C i no complete conic of type (c) can either. 0 It remains to verify that no complete conic / of type (d) can lie in the inter- .C;C T section , and again we have to start by characterizing the intersection of a cycle Z C i Z Z with the locus of complete conics of type (d). D D  .2M;2q M / , with q 2 To do this, write an arbitrary complete conic of type (d) as .  0 Z .2M;2q , then there is a one-parameter family .C C ;C If 2 / 2 Z with / t t D D t 0 0   .2M;2q C for t ¤ 0 and .C D ;C be the point of D / D \ C / ; let p C 2 smooth, t t 0 t t 0 C tangency of with D , and set p D lim T . The tangent line M p 2 C D T D t ! t p t 0 p t t t  C q at p and will have as limit the tangent line L to D at p , so L 2 q to . Thus both p t t D L . If p D q then in particular q 2 M . On the other hand, if p ¤ q , are in both and  D p;q D L , so then we must have 2 D M . We conclude, therefore, that a complete M   / of type (d) can lie in Z . only if either q .2M;2q D or M 2 D conic 2 D Given this, we see that the first condition ( q 2 C ) can be satisfied for at most two i  C , and the latter ( M 2 C of the ) likewise for at most two; thus no complete conic i i  for all / Z . .2M;2q of type (d) can lie in i D 1;:::;5 C i Transversality In order to prove that the cycles  X intersect transversely when the conics Z C i C at points ;:::;C Z are general, we need a description of the tangent spaces to the 5 1 C i T Z of . We have just shown that such points are represented by smooth conics, and C i the open subscheme parametrizing smooth conics is the same whether we are working

318 304 Compactifying parameter spaces Chapter 8 M M q D D q  .2M;2q . / Figure 8.3 The two types of complete conics of type (d) tangent to D 5 P in or in the space of complete conics, so we may express the answer in terms of the 5 . P geometry of 5 2 ı P Let  be a smooth conic curve and  P D the variety of smooth Z Lemma 8.5. D C tangent to D . plane conics C has a point p of simple tangency with D and is otherwise transverse to D , then (a) If ı is smooth at ŒCç . Z D ı ı is the hyperplane Z In this case, the projective tangent plane ŒCç to Z T at (b) ŒCç D D 5 P H p .  of conics containing the point p 1 with P Proof: , and consider the restriction map First, identify D 0 0 0 .4//: O .2// ! H . . O . .2// D H O H 1 2 D P P This map is surjective, with kernel the one-dimensional subspace spanned by the section D itself. In terms of projective spaces, the restriction induces a rational map representing 5 0 0 4  D P H W . O P .2// H ! . O P P .4// D 2 1 D P P 5 4 5 P P P to from the point D ). The closure 2 (this is just the linear projection of 5 4 5 ı D is thus the cone with vertex 2 P Z over the hypersurface D  P P of in D j j singular divisors in the linear system ; Lemma 8.5 will follow directly from the .4/ O 1 P next result: d 0 P H D P Proposition 8.6. Let . O on d .d// be the space of polynomials of degree 1 P 1 d D the discriminant hypersurface, that is, the locus of polynomials with P P and  F 2 D is a point corresponding to a polynomial with exactly one a repeated root. If double root p and d 2 simple roots, then D is smooth at F with tangent space the space of polynomials vanishing at . p Proof: Note that we have already seen this statement: it is the content of Proposition 7.21 (stated in Section 7.7.3 as a consequence of the topological Hurwitz formula). For another proof, this time in local coordinates, we can introduce the incidence correspondence 1 d j .F;p/ P Df  P ‰ 2 ord g .F/  2 ; p

319 Complete conics Section 8.2 305 d 1 .a;x/  P in : ‰ is the zero locus and write down its equations in local coordinates P of the polynomials d d 1 a a R.a;t/ x x D C a CC a x C 0 1 1 d d and d 1 d 2 .d 1/a D da S.a;t/ x C CC 2a : x C a x 1 2 d 1 d .a;x/ where a , all the partial derivatives D a 0 D x Evaluated at a general point D 0 1 of R vanish except S and 0 1 @R @R @R   C B @a @x @a 0 1 0 1 0 B C D : A @ 1 0 2a @S @S @S 2 @x @a @a 1 0 2  2 minor is nonzero assures us that ‰ is smooth at the point, The fact that the first a ¤ 0 and the characteristic is not 2 assures us that the differential and the fact that 2 d d is injective, with image the ‰ ! T d P W of the projection  W D ! P T a .a;0/ plane a is one-to-one at such a point tells us the image D 0 . Finally, the fact that  0 D .‰/ is smooth at the image point. D 2 C  is a conic with a point p of Getting back to the statement of Lemma 8.5, if P and is otherwise transverse to , then, by Proposition 8.6, D is D simple tangency with D 4 smooth at the image point in P , with tangent space the space of polynomials vanishing Z . Since at is the cone over D it follows that p ; the tangent space is smooth at C Z D D statement follows as well. In order to apply Lemma 8.5, we need to establish some more facts about a conic tangent to five general conics: 2 2 be general conics, and Lemma 8.7.  P Let any smooth conic C  P ;:::;C C 1 5 C tangent to all five. Each conic is simply tangent to C at a point p and is otherwise i i transverse to C , and the points p 2 C are distinct. i Proof: Let U be the set of smooth conics, and consider incidence correspondences 5  g ;:::;C C I C/ 2 .U ˆ .C U/ j each C is tangent to Df i 5 1 5 5 0  Df ;:::;C .C I C/ 2 .. P ˆ /  : U/ j each C g is tangent to C i 1 5 0 ˆ is an open subset of the set ˆ The set . Since U is irreducible of dimension 5 and the 0 5 / ! U on the last factor has irreducible fibers .Z of dimension 20, ˆ projection map C 0 we see that ˆ , and thus also ˆ , is irreducible of dimension 25.

320 306 Compactifying parameter spaces Chapter 8 satisfied : ˆ There are certainly points in where the conditions of the lemma are simply choose a conic tangent to it. Thus the set of C and five general conics C i .C I C/ 2 ˆ ;:::;C where the conditions of the lemma are not satisfied is a proper 1 5 5 closed subset, and as such it can have dimension at most 24, and cannot dominate U under the projection to the first factor. This proves the lemma. T 2 ŒCç be a point corre- To complete the argument for transversality, let Z C i 2 P sponding to the conic . By Lemma 8.7 the points p C of tangency of C with the  i C . Since C is the unique conic through these five points, the are distinct points on C i at ŒCç Z intersection of the tangent spaces to C i \ \ Z T D H g Df ŒCç p C ŒCç i i is zero-dimensional, proving transversality. 8.2.4 Chow ring of the space of complete conics T Having confirmed that the intersection Z indeed behaves well, let us turn now C i to computing the intersection number. We start by describing the relevant subgroup of . the Chow group A.X/ 5 5  1 .X/ be the pullbacks to X  P First, let  ̨;ˇ 2 A of the hyperplane classes P 5 5  P P . These are respectively represented by the divisors on and  / j p 2 C g Df A .C;C p 2 p 2 P (for any point ) and   2 / j L C g B .C;C Df L 2  2 P ). L (for any point 4 2 A Ä .X/ be the classes of the curves Also, let and ˆ that are the pullbacks ;' 5 5  and P P X . These are, respectively, the classes of the loci of of general lines in to  complete conics / such that C contains four general points in the plane, and such .C;C 2  2  L ). 2 P that C (that is, C is tangent to four lines in P contains four points i 1 A .X/ of divisor classes on X has rank 2, and is generated The group Lemma 8.8. ̨ . The intersection number of these classes with ˇ over the rationals by and ' and are given by the table ̨ ˇ ! 1 2 ' 2 1 1 Proof: A U .X/ is at most 2. The open subset We first show that the rank of  X of   pairs / with C and C smooth is isomorphic to the complement of a hypersurface .C;C 5 P , and hence has torsion Picard group: Any line bundle on U extends to a line bundle in 5 5 n U P , a power of which is represented by a divisor supported on the complement P on .

321 Complete conics Section 8.2 307 L is any line bundle on , a power of L is trivial on U and so is represented Thus, if X n X U in X has U by a divisor supported on the complement . But the complement of D of the loci of complete conics D just two irreducible components: the closures and 2 3 of type (b) and (c). Any divisor class on X is thus a rational linear combination of the X and D classes of , from which we see that the rank of the Picard group of D is 3 2 at most 2. Since passing through a point is one linear condition on a quadric, we have D 1 and dually deg .ˇ'/ D 1 . Similarly, since a general pencil of conics deg . ̨ / 2  P a pencil of degree 2, which will have two branch points, will cut out on a line L .ˇ / . ̨'/ deg and again by duality deg D D 2 . Since the matrix of in- we see that 2 ̨;ˇ and tersections between is nonsingular, we conclude that ̨ and ˇ generate ;' Pic ̋ Q . .X/ 1 ̨ generate A and .X/ over Z as well, as we could see from the description In fact, ˇ 5 as a blow-up of P . of X The class of the divisor of complete conics tangent to C 1 D p ̨ C qˇ 2 A It follows from Lemma 8.8 that we can write .X/  Q for some ̋ p;q Q . To compute p and q , we use the fact that, restricted to the open set U  X , 2 C the divisor deg  D p is a sextic hypersurface; it follows that 2q D 6 , and since  Z is symmetric in ̨ and ˇ we get deg ' D q C 2p D 6 as well. Thus 1  C 2ˇ 2 A D .X/ ̋ Q : 2 ̨ 5 5 / D 32 deg . ̨ C ˇ/ deg , and it suffices to evaluate the From this we see that . i i 5 5 D 2 A degree of the class .X/ for i ̨ 0;:::;5 . By symmetry, it is enough to do ˇ i 0 this for , 1 and 2. D  5 5  P X P onto the first factor is To do this, observe first that the projection of  5  0 2 .C;C / such that rank C  of pairs (the map U ! P U an isomorphism on the set 1 sending a smooth conic C to its dual in fact extends to a regular map on U sending a 1 5   [ M of rank 2 to the double line 2p L 2 P D L , where p D C \ M ). Since conic  , the intersections 2 all conics passing through three given general points have rank needed will occur only in U . Since the degree of a zero-dimensional intersection is equal 1 to the degree of the intersection scheme, this implies that we can make the computations 5 ́ X instead of on on . For this we will use B P ezout’s theorem:  0 : Passing through a point is a linear condition on quadrics. There is a unique D i 5 quadric through five general points, and the intersection of five hyperplanes in P 5 has degree 1, so deg / D 1 . . ̨ 5  : The quadrics tangent to a given line form a quadric hypersurface in P . Since D i 1 not all conics in the one-dimensional linear space of conics through four general 4 . ̨ . ˇ/ D 2 points will be tangent to a general line, deg

322 308 Chapter 8 Compactifying parameter spaces  i D 2 : Similarly, we see that the conics passing through three given general points 5 U  P . Not all these conics and tangent to a general line form a conic curve in 1 are tangent to another given general line. (For example, after fixing coordinates we p 1 may think of circles as the conics passing through the points ̇ on the line . Certainly there are circles through a given point and tangent to a given line 1 at 3 2 . ̨ ˇ that are not tangent to another given line.) It follows that / is the degree of deg the zero-dimensional intersection of a plane with two quadrics, that is, 4. Thus             5 5 5 5 5 5 5 2 C D C 4 / C .. ̨ 4 C 2 ˇ/ deg C C 4 2 5 0 1 3 1 C 10 C 40 C 40 C 10 C 1 D D 102 and, correspondingly, 5 5 3264:   102 D 2 D This proves: Theorem 8.9. There are 3264 plane conics tangent to five general plane conics. Of course, the fact that we are imposing the condition of being tangent to a conic is arbitrary; we can use the space of complete conics to count conics tangent to five general plane curves of any degree, as Exercises 8.11 and 8.12 show, and indeed we can extend this to the condition of tangency with singular curves, as Exercises 8.14–8.16 indicate. See Fulton et al. [1983] for a general formula enumerating members of a k -dimensional families of varieties tangent to k given varieties. Other divisor classes on the space of complete conics We will take a moment here to describe as well the classes of two other important of complete conics: the closures E and G of the strata of X divisors on the space complete conics of types (b) and (c). As we mentioned in the initial section of this 5  5 5 ! P  P P , X can also be realized, via the projection map X chapter, the space  5 5 P S  P of double lines, or dually as a as the blow-up of along the Veronese surface 5  P ; in these descriptions of X , the divisors blow-up of and E are the exceptional G divisors of the blow-up maps. We can describe the classes  and  of E and G by the same method we used to determine the class of Z , that is, by calculating their intersection numbers with d Ä and ˆ the curves E , we see that a general pencil of plane conics will have . For 5 . / D 3 (that is, the image of E in P three singular elements, so that is a cubic deg 5  in P hypersurface), while the image of E has codimension 3, and so will not meet a 5  general line in P at all, so that deg .'/ D 0 ; solving, we obtain  ̨: 2 ̨ ˇ; and dually  D 2ˇ D

323 Complete quadrics Section 8.2 309 5 P B Alternatively, we can argue that in the space of the locus of conics the closure L 2 L is a quadric hypersurface containing the  of smooth conics tangent to a given line P (the singular Veronese surface of double lines. It necessarily has multiplicity 1 along S n n locus of a quadric hypersurface in is contained in a proper linear subspace of P ), so P D  , and the relations above follow. ˇ that its proper transform has class 2 ̨ 8.3 Complete quadrics There are beautiful generalizations of the construction of the space of complete n and more general bilinear forms or homomorphisms. P conics to the case of quadrics in The paper Laksov [1987] gives an excellent account and many references. Here is a sketch of the beginning of the story. As usual we restrict ourselves to characteristic 0, though 2 . the description holds more generally as long as the ground field has characteristic ¤ n P V by a symmetric P D As in the case of conics, we represent a quadric in 2   transformation ' , or equivalently a symmetric bilinear form in Sym W V V . ! V To this transformation we associate the sequence of symmetric transformations V V V i i i   W ' / D . ! 1;:::;n: D V for i V / .V i V V i i   is canonical — see for example Eisenbud Here the identification .V / . D V / [1995, Section A2.3]. V i 2  Sym ' as an element of . V We think of / , and we define the variety of complete i n quadrics in , which we will denote by ˆ , to be the closure in P n Y V i 2  . // P . Sym V D 1 i of the image of the set of smooth quadrics under the map ' 7! .' . ;:::;' / n 1 V i  ' . in which the quadric corresponding to P lies is the ambient The space / V i n D space of the Grassmannian .i 1;n/ of .i 1/ -planes in P G , and in fact an G i n 1/ -plane ƒ  P .i is tangent to Q if the point Œƒç 2 G lies in this quadric. i ˆ U isomorphic to the open set has an open set From the definition we see that n corresponding to quadrics in the projective space of quadratic forms on . As with the P case of complete conics, there is a beautiful description of the points that are not in U . n P of subspaces of D P V and consider a flag V To start, let V of arbitrary length r and dimensions k k Df <  < k : g r 1 V V V:  V 0    k k k r 2 1 Now consider the variety F D of pairs . V ;Q/ , where V is a flag as above and Q k .Q ;:::;Q are smooth quadric hypersurfaces in the projective space Q , where the / 1 1 C i r P .V ; this is an open subset of a product of projective bundles over the variety =V / k k i 1 i of flags V . We then have:

324 310 Chapter 8 Compactifying parameter spaces There is a stratification of ˆ F Proposition 8.10. , where whose strata are the varieties k ranges over all strictly increasing sequences < r <  < k 0 < k k . 1 r D 2 U One can also describe the limit of a family of smooth quadrics F when q ¿ t q , as in of rank the family approaches a quadric C 1 k n 0 ! 0 t  I k : ' WD t 0 I k C n 1 The limit lies in the stratum F , where the flag consists of one intermediate space f k g 0  V /  V ; the k -plane V .V=V will be the kernel of ' P , the quadric Q on 2 0 k k k P Q Q will be the quadric on on the quotient, and V will be the quadric induced by 0 1 k associated to the limit j ' t V k lim : 0 t ! t F F lies in the closure of In general, the stratum ; the exactly when l  k l k specialization relations can be defined inductively, using the above example. 8.4 Parameter spaces of curves So far in this chapter we have been studying compactifications of parameter spaces 5 P of smooth conics. The most obvious is perhaps , which we can identify as the space 2 of all subschemes of having pure dimension 1 and degree 2 (and thus arithmetic P genus 0), and we have shown how the compactification by complete conics was more useful for dealing with problems involving tangencies. Here we have used the fact that the dual of a smooth conic is again a smooth conic. It would have been a different story 2 3 P P if the problem had involved twisted cubics in — if we had rather than conics in asked, for example, for the number of twisted cubic curves meeting each of 12 lines, or tangent to each of 12 planes, or, as in one classical example, the number of twisted cubic curves tangent to each of 12 quadrics. In that case it is not so clear how to make any parameter space and compactification at all! In this section, we will discuss two general approaches to constructing parameter spaces for curves in general: the Hilbert scheme of curves and the Kontsevich space of stable maps. (In specific cases, other approaches may be possible as well; for exam- ple, see Cavazzani [2016] in the case of twisted cubics.) The Hilbert scheme and the Kontsevich space each have advantages and disadvantages, as we will see. 8.4.1 Hilbert schemes n / is a parameter space for . P Recall from Section 6.3 that the Hilbert scheme H P n subschemes of P with Hilbert polynomial P ; in the case of curves (one-dimensional

325 Parameter spaces of curves Section 8.4 311 subschemes), this means all subschemes with fixed degree and arithmetic genus. We start 3 2 P ; P by describing the Hilbert schemes parametrizing conic and cubic curves in and when we come to Kontsevich spaces, we will describe these cases in that setting for contrast. 2 P The Hilbert schemes of conics and cubics in 5 9 P As we have observed, these are just the projective spaces associated to the and P 2 P ; they parametrize vector spaces of homogeneous quadratic and cubic polynomials on 2 2m with Hilbert polynomial subschemes of C 1 and 3m respectively. P 3 P The Hilbert scheme of plane conics in We will discuss this space at much greater length in the following chapter (where, in particular, we will prove all the assertions made here!). Briefly, the story is this: Any 3 subscheme of P 2m C 1 is necessarily the complete intersection with Hilbert polynomial of a plane and a quadric surface; the plane, naturally, is unique. This means that the 3  P ; the fiber over a point Hilbert scheme admits a map to the dual projective space 3  5 2 5 is the 2 P of conics in H Š P P . (This P H -bundle structure is what allows us to calculate its Chow ring; we will use this information to solve the enumerative problem of 3 .) In any event, the Hilbert P counting the conics meeting each of eight general lines in 3 is irreducible and smooth of dimension 8. P scheme of plane conics in The Hilbert scheme of twisted cubics 3 In the case of the Hilbert scheme parametrizing twisted cubic curves in P (that 3 is, parametrizing subschemes of with Hilbert polynomial 3m C 1 ) we begin to see P some of the pathologies that affect Hilbert schemes in general. It has one component of dimension 12 whose general point corresponds to a twisted cubic curve. But it also has a second component, whose general point corresponds to the union of a plane cubic 2 3 3 C  P  and a point p 2 P P . Moreover, this second component has dimension 15 (the choice of plane has three degrees of freedom, the cubic inside the plane nine more, and the point gives an additional three). These two components meet along the 11-dimensional subscheme of singular plane cubics C with an embedded point at the singularity, not contained in the plane spanned by C (see Piene and Schlessinger [1985]). 8.4.2 Report card for the Hilbert scheme The Hilbert scheme is from some points of view the most natural parameter space that is generally available for projective schemes. Among its advantages: As shown in Section 6.3, it represents a functor that is easy to understand. There is a useful cohomological description of the tangent spaces to the Hilbert scheme, and, beyond that, a deformation theory that in some cases can describe its local structure. It was shown

326 312 Chapter 8 Compactifying parameter spaces to be connected in characteristic 0 by Hartshorne [1966] and in finite characteristic by Pardue [1996] (see Peeva and Stillman [2005] for another proof). Reeves [1995] showed that the radius of the graph of its irreducible components is at most one more than the dimension of the varieties being parametrized. And, of course, associated to a point on the Hilbert scheme is all the rich structure of a homogeneous ideal in the ring k ;:::;x Œx ç and its free resolution. n 0 However, as a compactification of the space of smooth curves, the Hilbert scheme has drawbacks that sometimes make it difficult to use: (a) It has extraneous components, often of differing dimensions. We see this phe- nomenon already in the case of twisted cubics, above. Of course we could take just ı the closure in the Hilbert scheme of the locus of smooth curves, but we would H lose some of the nice properties, like the description of the tangent space. (Thus while it is relatively easy to describe the singular locus of , we do not know in H ı H along the locus where it intersects general how to describe the singular locus of other components; in the case of twisted cubics it was not known until Piene and ı Schlessinger [1985] that H is smooth.) In fact, we do not know for curves of higher degree how many such extraneous components there are, nor their dimensions: For r  3 and large d , the Hilbert r scheme of zero-dimensional subschemes of degree P will have an unknown in d number of extraneous components of unknown dimensions, and this creates even more extraneous components in the Hilbert schemes of curves. (b) . If we do choose No one knows what is in the closure of the locus of smooth curves ı H of the locus of smooth curves rather than the whole to deal with the closure Hilbert scheme — as it seems we must — we face another problem: except in a few special cases, we cannot tell if a given point in the Hilbert scheme is in this closure. That is, we may not know how to tell whether a given singular one-dimensional r C  P scheme is smoothable. (c) . The singularities of the Hilbert scheme are, in a precise It has many singularities sense, arbitrarily bad: Vakil [2006b] has shown that the completion of every affine local k -algebra appears (up to adding variables) as the completion of a local ring on a Hilbert scheme of curves. 8.4.3 The Kontsevich space In the case of curves in a variety, the Kontsevich space is an alternative compactifi- cation. A precise treatment of this object is given in Fulton and Pandharipande [1997]; here we will treat it informally, sketch some of its properties, and indicate how it is used, with the hope that this will give the interested reader a taste of what to expect.

327 Parameter spaces of curves 313 Section 8.4 r . ;d/ parametrizes what are called stable maps of The Kontsevich space M P g;0 r g P . These are morphisms and genus degree d to r P f ; W C ! g having only nodes as singularities, such C with a connected curve of arithmetic genus ŒCç that the image of the fundamental class of f is equal to d times the class of C  r . P a line in / , and satisfying the one additional condition that the automorphism A 1 f ' of C group of the map f ı ' D — that is, automorphisms — is finite. f such that f is finite; it is relevant only (This last condition is automatically satisfied if the map for maps that are constant on an irreducible component of C , and amounts to saying C f of C on which that any smooth, rational component is constant must intersect 0 r W C ! P the rest of the curve f C in at least three points.) Two such maps and r 0 0 0 P are said to be the same if there exists an isomorphism W ̨ W C ! C C f ! 0 ı ̨ D f . There is an analogous notion of a family of stable maps, and f with r . the Kontsevich space P M ;d/ is a coarse moduli space for the functor of fam- g;0 ilies of stable maps. Note that we are taking the quotient by automorphisms of the r . P ;d/ M shares with the Hilbert scheme source, but not of the target, so that g;0 r r H P  . P of de- / a common open subset parametrizing smooth curves C 1 g C dm d and genus g . gree r M W . P There are natural variants of this: the space ;d/ parametrizes maps f g;n r with C a nodal curve having n marked distinct smooth points p ;:::;p 2 C P ! n 1 C is an automorphism of C fixing the points p f and . (Here an automorphism of i commuting with f ; the condition of stability is thus that any smooth, rational component of on which f is constant must have at least three distinguished points, counting C C 0 C .) More both marked points and points of intersection with the rest of the curve X ˇ 2 Num .X/ , generally, for any projective variety and numerical equivalence class 1 M ! .X;ˇ/ parametrizing maps f W C we have a space X with fundamental class g;n nodal and ŒCç D ˇ , again with C f f having finite automorphism group.  It is a remarkable fact that the Kontsevich space is proper: In other words, if r r D  P C is a flat family of subschemes of P  parametrized by a smooth, one- dimensional base D C , and the fiber is a smooth curve for t ¤ 0 , then no matter what t r z C ! P the singularities of C which is the are there is a unique stable map f W 0 0 r r z W ! P  . Note that this limiting stable map f , C C ! P limit of the inclusions W t 0 t C depends on the family, not just on the scheme ; the import of this in practice is that the 0 Kontsevich space is often locally a blow-up of the Hilbert scheme along loci of curves with singularities worse than nodes. (This is not to say we have in general a regular map from the Kontsevich space to the Hilbert scheme; as we will see in the examples r z W ! C either.) below, the limiting stable map P f does not determine the flat limit C 0 0 We will see how this plays out in the four relatively simple cases discussed above in connection with the Hilbert scheme:

328 314 Chapter 8 Compactifying parameter spaces 1 1 1 P C [ C Š Š P P f f L L p D q q p Figure 8.4 Stable maps of degree 2 with image a line. Plane conics One indication of how useful the Kontsevich space can be is that, in the case of 2 P ;2/ (that is, plane conics), the Kontsevich space is actually equal to the space of . M 0 complete conics. 2 P To begin with, if is a conic of rank 2 or 3 — that is, anything but a double C  2 5 C , ! P  is a stable map; thus the open set W  P line — then the inclusion map of W 2 M . P such conics is likewise an open subset of the Kontsevich space ;2/ . 0 2  D  P But when a one-parameter family of conics specializes to a double C W C D 2L , the limiting stable map is a finite, degree-2 map f line C ! L , with 0 1 1 P isomorphic to either C meeting at a point. Such a map is or two copies of P B L , a characterized, up to automorphisms of the source curve, by its branch divisor  1 . (If B consists of two distinct points, then C Š P B , while if divisor of degree D 2p 2 p L , the curve C is reducible.) Thus the data of the limiting stable map is 2 for some equivalent to specifying the limiting dual curve. This suggests what is in fact the case: The identification of the common open subset 2 2 5 P D / ;2/ of the Kontsevich space and the Hilbert scheme H M . . P P W 1 C 2m 0 2 . M ;2/ with the extends to a regular morphism, and to a biregular isomorphism of P 0 5 of complete conics, commuting with the projection X ! P space : X Š 2 - P M ;2/ X . 0 -  5 2 . P / H D P 1 C 2m We will not verify these assertions, but they are not hard to prove given the analysis of limits of conics and their duals in Section 8.2.1.

329 Parameter spaces of curves 315 Section 8.4 3 Plane conics in P By contrast, there is not a natural regular map in either direction between the Hilbert 3 P M . Of course there is a . scheme of conics in space and the Kontsevich space ;2/ 0 common open set : its points correspond to reduced connected curves of degree 2 U 3 (such a curve is either a smooth conic in a plane or the union of two P embedded in coplanar lines). To see that the identification of this open set does not extend to a regular 3 D  P map in either direction, note first that, as before, if is a family of conics C  , then the limiting stable map is a finite, degree-2 cover specializing to a double line C 0 C ! L , and this cover is not determined by the flat limit C f of the schemes W 0 3  P C . Thus the identity map on U does not extend to a regular map from the Hilbert t scheme to the Kontsevich space. On the other hand, the scheme is contained in a C 0 C . Since it has degree 2, the plane plane — the limit of the unique planes containing the t . Thus containing it is unique. But this plane is not determined by the data of the map f does not extend to a regular map from the Kontsevich space to the identity map on U the Hilbert scheme either. The birational equivalence between the Hilbert scheme and the Kontsevich space is of a type that appears often in higher-dimensional birational geometry: the Kontsevich H by blowing up the locus of double lines, space is obtained from the Hilbert scheme and then blowing down the exceptional divisor along another ruling. (The blow-up 3 along the double line locus is isomorphic to the blow-up of M of . P H ;2/ along 0 the locus of stable maps of degree 2 onto a line; both can be described as the space 3   .C;C , where // of pairs H  P .H is a plane and .C;C I / a complete conic in 2 H Š .) The birational isomorphism between the Hilbert scheme and Kontsevich P flip in higher-dimensional space in this case is an example of what is known as a birational geometry. Plane cubics 2 . Here, we do have a regular map from the Kontsevich space M ;3/ to the Hilbert P 1 2 9 scheme P H / Š P , and it does some interesting things: It blows up the locus of . 3m triple lines, much as in the example of plane conics, and the locus of cubics consisting of a double line and a line as well. But it also blows up the locus of cubics with a cusp, and cubics consisting of a conic and a tangent line, and these are trickier: The blow-up along the locus of cuspidal cubics, for example, can be obtained either by three blow-ups with smooth centers or by one blow-up along a nonreduced scheme supported on this locus. 2 M P But what we really want to illustrate here is that the Kontsevich space ;3/ . 1;0 is not irreducible — in fact, it is not even nine-dimensional! For example, maps of the 2 W C ! P form with C consisting of the union of an elliptic curve E and a copy of f 1 1 , where P maps P f to a nodal plane cubic C , and maps E to a smooth point of C 0 0 form a 10-dimensional family of stable maps; in fact, these form an open subset of a 2 , as illustrated in Figure 8.5. ;3/ . P M second irreducible component of 1

330 316 Chapter 8 Compactifying parameter spaces E D f.E/ p f 1 D C P / f. 1 P 2 . P M ;3/ . Figure 8.5 A typical point in the 10-dimensional component of 1;0 1 P 1 f. D C P / 0 f E p D f.E/ 1 D f. P / C 1 1 P 2 M P Figure 8.6 General member of a third component of ;3/ . . 1;0 And there is also a third component, whose general member is depicted in Figure 8.6. Twisted cubics H D H has, as we Here the shoe is on the other foot. The Hilbert scheme 3m C 1 saw, a second irreducible component besides the closure H of the locus of actual 0 twisted cubics, and the presence of this component makes it difficult to work with. For is smooth, since we have no simple H example, it takes quite a bit of analysis to see that 0 way of describing its tangent space; see Piene and Schlessinger [1985] for details. By contrast, the Kontsevich space is irreducible, and has only relatively mild (finite quotient) singularities. 8.4.4 Report card for the Kontsevich space As with the Hilbert scheme, there are difficulties in using the Kontsevich space: (a) . These arise in a completely different way from the It has extraneous components extraneous components of the Hilbert scheme, but they are there. A typical example r of an extraneous component of the Kontsevich space . P M ;d/ consists of maps g r 1 C ! P was the union of a rational curve in which C f C , mapping to a Š P W 0 r in P rational curve of degree , and d an arbitrary curve of genus g meeting C C 1 0 in one point and on which was constant; if the curve C does not itself admit a f 1 r , this map cannot be smoothed. to P d nondegenerate map of degree

331 How the Kontsevich space is used: rational plane curves Section 8.4 317 So, using the Kontsevich space rather than the Hilbert scheme does not solve this problem, but it does provide a frequently useful alternative: There are situations where the Kontsevich space has extraneous components and the Hilbert scheme does not — like the case of plane cubics described above — and also situations where the reverse is true, such as the case of twisted cubics. (b) . This, unfortu- No one knows what is in the closure of the locus of smooth curves nately, remains an issue with the Kontsevich space. Even in the case of the space 2 P parametrizing plane curves, where it might be hoped that the Kontsevich ;d/ M . g space would provide a better compactification of the Severi variety parametrizing reduced and irreducible plane curves of degree and geometric genus g than simply d N , the fact that we do not of all plane curves of degree d its closure in the space P know which stable maps are smoothable represents a real obstacle to its use. It has points corresponding to highly singular schemes, and these tend to be in (c) r M . P ;d/ . Still true, but in this respect, at least, it turn highly singular points of g might be said that the Kontsevich space represents an improvement over the Hilbert r of a stable map f scheme: Even when the image C ! P f.C/ is highly singular, W the fact that the source of the map is at worst nodal makes the deformation theory of the map relatively tractable. Finally, we mention one other virtue of the Kontsevich space: It allows us to work with tangency conditions, without modifying the space and without excess intersection. r r  P . is a smooth hypersurface, the closure Z ;d/ in M The reason is simple: If X P g X is contained in the locus of maps f W X of the locus of embedded curves tangent to r 1 f ! P .X/ is nonreduced or positive-dimensional. Thus, such that the preimage C 2 for example, a point in . P M ;d/ corresponding to a multiple curve — that is, a map g 2 f W C ! P that is multiple-to-one onto its image — is not necessarily in Z . X 8.5 How the Kontsevich space is used: rational plane curves g 0 . One case in which the Kontsevich space is truly well-behaved is the case D r . Here the space P M ;d/ is irreducible — it has no extraneous components — and, 0 moreover, its singularities are at worst finite quotient singularities (in fact, it is the coarse moduli space of a smooth Deligne–Mumford stack). Indeed, the use of the Kontsevich space has been phenomenally successful in answering enumerative questions about rational curves in projective space. We will close out this chapter with an example of this; specifically, we will answer the second keynote question, and, more generally, the 2 d C P question of how many rational curves of degree  are there passing through 3d 1 general points in the plane.

332 318 Chapter 8 Compactifying parameter spaces C 1 2 p P 1;t p 1;0 p 2;t f.p / 2;t p 2;t p 2;0 / f.p 1;t p 1;t f t f.p / p p 3;t 3;t 3;t p 4;t f.p / p 4;t 4;t C 2 C Figure 8.7 A family of maps that blows down . 1 Since we have not even defined the Kontsevich space, this analysis will be far from complete. The paper Fulton and Pandharipande [1997] provides enough background to complete it. Before starting the calculation, let us check that we do in fact expect a finite 2 1 P ŒF;G;Hç P from are given by triples d of homogeneous number. Maps of degree to 1 on P with no common zeros; since the vector space of polynomials of degree d 1 on P , the space has dimension d C 1 polynomials of degree d of all such triples has U N N 3 . Now look at the map U ! P dimension from U to the space P C of plane 3d d , sending such a triple to the image (as divisor) of the corresponding curves of degree 1 2 P G . This has four-dimensional fibers (we can multiply F , map ! H by a P and 1 P ), so we conclude that common scalar, or compose the map with an automorphism of 3d 1 . In particular, we see that there are no rational curves the image has dimension 2 d 3d of degree P general points of , and we expect a finite number passing through 3d 1 . We will denote the number by N.d/ . (possibly 0) through 2 . WD M of stable maps from curves with We will work on the space P M ;d/ 0;4 d four marked points. This is convenient, since on M we have a rational function ' , given d by the cross-ratio : at a point of M ! corresponding to a map f W .C I p / ;p ;p ;p 3 2 4 1 d 1 1 2 irreducible, it is the cross-ratio of the points P ;p ;p C ;p p 2 P Š ; that with P 3 2 1 4 1 P is, in terms of an affine coordinate z on , / .z z z /.z 3 4 1 2 ; ' D /.z z / .z z 4 2 3 1 and z z.p only when two of the / . The cross-ratio takes on the values 0;1 D 1 where i i points coincide, which in our setting corresponds to when the curve C is reducible: For C , then C example, if and C , with p ;p 2 C and p ;p 2 C has two components 2 3 2 4 1 1 2 1 by blowing down the curve in the total space of the source family, we can realize C 1 with ;:::;p irreducible / as a limit of pointed curves .C C ;p .C;p .t/;:::;p .t// 1 4 t t 4 1 lim lim and .t/ D p p .t/ (see Figure 8.7). Thus ' has a zero at such a point. ! 0 t 2 ! 0 1 t will be equal Similarly, if three of the p , or all four, lie on one component of C , then ' i 1 , and so will not be 0, 1 or 1 . to the cross-ratio of four distinct points on P

333 How the Kontsevich space is used: rational plane curves Section 8.5 319 B  We now introduce a curve on which we will make the calculation. Fix a M d 2 2 p  P p passing through and two lines ; fix two more general points L;M 2 P point 2 2 Ä  P q;r of 2 4 general points. We consider the locus P 3d and a collection ( ) ˇ r; q; f.p D D / f.p / 1 2 ˇ 2 ˇ .C I p B ;p D ;p / ;p f W ! P  : M M; f.p L; f.p / / 2 2 1 3 2 4 4 3 d ˇ f.C/ Ä  2 in P d 3d 1 , Since, as we said, the space of rational curves of degree has dimension 3d and we are requiring the curves in our family to pass through points (the points q 2 and , and the 3d 4 points of Ä ), our locus B will be a curve. r B for which the source of the corresponding map There may be points in C 2 is reducible. But in these cases p f ;p will have no more than ;p C ;p I / W P .C ! 4 2 1 3 C has components . To see this, note that if the image of ;:::;D of two components D 1 k d can contain at most ;:::;d 2 , by the above the curve D degrees 3d 3d 1 of the i 1 i k points q;r g . Thus Ä [f k X 2  3d .3d k; 3d 1/ D i 1 i D k whence 2 . As a consequence, we see that the map f cannot be constant on any  component: By the stability condition, if f were constant on a component C C , then 0 0 f would have to meet at least three other components — but can be nonconstant on only two, and it follows that the stability condition is violated on some component. B This argument also shows that there are only finitely many points in for which the 2 D D D D [ source , C P is reducible: If , with D d a rational curve of degree  i i 2 1 Ä [f q;r g D , then by the above D 2 must contain exactly 3d and 1 of the 3d i i Ä [f q;r g . The number of such plane curves D is thus points   3d 2 /N.d N.d /: 1 2 1 3d 1 2 there are d with Moreover, for each such plane curve D stable maps f W C ! P d 2 1 image D : We can take C the normalization of D at all but any one of the points of intersection D will intersect transversely.) \ D D . (By Exercise 8.18, D and 1 2 1 2 On with the calculation! We equate the number of zeros and the number of poles of 2 ;p . To begin with, we consider points f W .C I p C ' with on B ;p of / ! P B ;p 4 3 2 1 irreducible. Since f.p , / D q and f.p M / D r are fixed and lie off the lines L and 2 1 the only way any two of the points p can coincide on such a curve is if i f.p : / D f.p g / D p; where L \ M Df p 4 3 Such points are zeros of ' ; the number of these zeros is the number of rational plane curves of degree d through the 3d 1 points p , q , r and Ä , that is to say, N.d/ . (Of course, to make a rigorous argument we would have to determine the multiplicities

334 320 Chapter 8 Compactifying parameter spaces of these zeros; since we are just sketching the calculation, we will omit the verification that all multiplicities are 1, here and in the following.) coming from points What about zeros and poles of ' 2 .C W ;p f ;p p ;p I / ! P 4 1 2 3 C D C reducible? As we have observed, we get a zero of [ C B with ' at such in 1 2 and p p lie on one component of C and p a point if and only if the points p and 1 4 3 2 lie on the other. How many such points are there? Well, letting d be the degree of 1 containing of C the component p C and p the degree of the , and d d D d 2 1 2 1 1 and , we see that other component / must contain q , r f.C 3d C 3 of the points of 1 1 2 C Ä , while 3d 4 .3d . For 3/ D 3d Ä 1 points of contains the remaining 2 1 2 3d any subset of 3 points of Ä , the number of such plane curves is N.d , /N.d / 2 1 1 1 d choices of the point 2 C .L/ \ f and for each such plane curve there are p 2 2 3 1 p d 2 C choices of the point \ and f choices of the point .M/ , as well as d d 1 2 2 2 4 . We thus have a total of \ C f.C / 2 f.C / / \ f.C 2 2 1 1 1 d   X 4 3d 3 d d /N.d N.d / 1 1 2 2 3 3d 1 D 1 d 1 zeros of arising in this way. ' ' are counted similarly. These can occur only at points The poles of 2 ;p I .C f W p ;p / ! P ;p 1 4 3 2 in B with C D C lying on [ C p reducible, specifically with the points p and 1 2 1 3 C and one component of and p p on the other. Again letting d be the degree of the 1 2 4 D of C containing p the degree of the other and p component , and d d C d 2 1 1 1 3 2 , we see that / must contain q C 3d f.C points of Ä , and f.C component / and 2 2 1 1 3d the remaining 4 .3d 2/ D 3d 2 2 points of Ä , plus r . For any subset of 3d 1 1 2 Ä , the number of such plane curves is N.d points of /N.d , and for each such plane / 2 1 1 choices of the point p d 2 C and curve there are f d choices of the point .L/ \ 2 3 2 1 1 2 C \ f . p .M/ , as well as d / d f.C choices of the point f.C \ \ C / / 2 f.C 1 1 2 2 1 4 2 2 We thus have a total of 1 d   X 3d 4 2 2 d / d /N.d N.d 2 1 2 1 2 3d 1 1 D d 1 poles of ' arising in this way. Now, adding up the poles and zeros, we conclude that 1 d   i  h X 4 3d 3d 4 2 N.d /N.d /; d d d d N.d/ D d 1 2 2 1 2 1 2 3d 2 3d 3 1 1 1 d D 1 0 0 . N.d a recursive formula that allows us to determine / for d N.d/ < d if we know

335 Exercises 321 Section 8.5 To see how this works, we start with the fact that there is a unique line through two 1 D / N.d D N.d . Next, if points, and a unique conic through five general points, so / 2 1 3 we see that we take d D h i       i h 5 5 5 5 2 D N.3/ 4 C 2 2 12: 2 D 3 0 4 1 In fact, we have already seen this in Proposition 7.4: The set of all cubics containing 2 ;:::;p P is a general pencil, and we are counting the number p eight general points 2 8 1 of singular elements of that pencil. D 4 , we have Continuing to d   h    i h   h  i i  8 8 8 8 8 8 12 D 4 9 3 N.4/ 620: C 3  12 D 3 C 4 3 4  7 1 4 0 6 3 Always ignoring the question of multiplicity, this answers Keynote Question (b): There 2 P are 620 rational quartic curves through 11 general points of . Exercises 8.19 and 8.20 suggest some additional problems that can be solved using spaces of stable maps. 8.6 Exercises 2 D  P Exercise 8.11. be a smooth curve of degree d , and let Z Let  X be the D closure, in the space X of complete conics, of the locus of smooth conics tangent to D . 1 ŒZ . ç 2 A Find the class .X/ of the cycle Z D D 2 d D  P Now let be general curves of degrees ;:::;D . ;:::;d Exercise 8.12. 5 1 1 5 Show that the corresponding cycles Z  X intersect transversely, and that the D i intersection is contained in the open set U of smooth conics. Exercise 8.13. Combining the preceding two exercises, find the number of smooth 2 conics tangent to each of five general curves  P D . i 2 D  P Exercise 8.14. be a curve of degree d with ı nodes and  Let ordinary cusps (for a definition of cusps, see Section 11.4.1), and smooth otherwise. Let Z  X be D the closure, in the space X of complete conics, of the locus of smooth conics tangent to 1 at a smooth point of D . Find the class ŒZ . ç 2 A .X/ D of the cycle Z D D , with Let Exercise 8.15. g be a family of plane curves of degree d D D smooth for f t t ¤ 0 and t ? having a node at a point p . What is the limit of the cycles Z 0 ! as t D 0 D t Exercise 8.16. Here is a very 19th century way of deriving the result of Exercise 8.11 above. Let f D t g be a pencil of plane curves of degree d , with D smooth for general t t D consisting of the union of d general lines in the plane. Using the description of and 0 in the preceding exercise, find the class of the cycle 0 the limit of the cycles as t ! Z D t Z for t general. D t

336 322 Chapter 8 Compactifying parameter spaces True or false: There are only finitely many Exercise 8.17. -orbits in the Kontsevich PGL 4 3 M . P ;3/ . space 0 2 3d and Ä Exercise 8.18. be collections of Let , 1 and 3d Ä 1 general points in P 2 1 1 2 2 P D and any of the finitely many rational curves of degree d passing through Ä .  i i i D and D intersect transversely. Show that 2 1 2 2 be general points and ;:::;p a general line. How 2 P Exercise 8.19. Let L  P p 7 1 many rational cubics pass through p ;:::;p and are tangent to L ? 1 7 2 (a) Let D M M . P Exercise 8.20. ;d/ be the Kontsevich space of rational plane 0 1 2 curves of degree  M be the open set of immersions f W P , and let ! P U d 2 ı  U D Z P  that are birational onto their images. For a smooth curve, let D 1 2 1 P / ! P be the locus of maps such that f. P f W is tangent to D at a smooth 1 M point of / , and Z f.  P its closure. Show that Z is contained in the locus D D r 1 C ! P f such that the preimage of maps f W .D/ is nonreduced or positive- dimensional. T Given this, show that for D is ;:::;D Z general curves the intersection (b) 1 D 1 3d i contained in U .

337 Chapter 9 Projective bundles and their Chow rings Keynote Questions 3 3 ;:::;L Given eight general lines  P L (a) P , how many plane conic curves in 1 8 meet all eight? (Answer on page 354.) 1 Can a ruled surface (that is, a P -bundle over a curve) contain more than one curve (b) of negative self-intersection? (Answer on page 341.) Many interesting varieties, such as scrolls, blow-ups of linear subspaces of projective spaces, and some natural parameter spaces for enumerative problems can be described as projective bundles over simpler varieties. In this chapter we will investigate such varieties and compute their Chow rings. This is a tremendously useful tool, and in particular will allow us to answer the first of the keynote questions above. It will also help us to describe the Chow ring of a blow-up, which we will do in Chapter 13. 9.1 Projective bundles and the tautological divisor class A projective bundle over a scheme Definition 9.1. is a map  W Y ! X such that X for any point p 2 X there is a Zariski open neighborhood U  X of p in X with r 1 as  .U/ Š U  P Y WD U -schemes; that is, there are commuting maps U Š 1 r -  .U/  P U  1  -  U r ! is projection on the first factor. U U  P W  where 1

338 324 Chapter 9 Projective bundles and their Chow rings r C 1 E Š One can make projective bundles from vector bundles as follows: First, if O X is a trivial vector bundle, then r  ç/ . O X Œx D ;:::;x /; P D Proj . Sym E Proj  r 0 X r  Sym E and the structure map corresponds to the projection  W X  O P ! X ! . X becomes trivial on an open cover of X , so P E WD By definition, any vector bundle E  Sym E / ! X is a projective bundle, called the projectivization of . . In fact, every Proj E E for some vector bundle E . Before we can prove projective bundle can be written as P 1 this, we need to know a little more about projectivizations of vector bundles. From the local description of E as a product, it follows that the points of P E P .x;/ with x 2 X and  correspond to pairs   E of the a one-dimensional subspace x  E E . The bundle  fiber E on P E thus comes equipped with a tautological subbundle of x E P E is the subspace   2 of rank 1, whose fiber at a point of the fiber E .x;/ x x  2  E P . This subbundle is denoted by O . E . 1/   corresponding to the point E x P r 1 where E becomes trivial, so that  D On an open set U U U  P  , the bundle X r . 1/ 1/ is the pullback of O D O . .1/ from the second factor. We write O P E E P P for the dual bundle. Dualizing the inclusion of the tautological om . O 1/; / . H O P E P E   ! O bundle, we get a surjection E .1/ .  E P E E from the case where We can get an idea of the relation between is a P and E 0 is locally X  P line bundle. In this case , so the projection  W P E ! X is an P E  E with X via  isomorphism. Identifying  P . E / D E , and moreover , we see that O . . 1/ D E E P E and O From this example we see that the bundles are not determined by . 1/ E P P E or even by the map  W the scheme E ! X — rather, the bundle E is an additional P piece of data that determines the bundle O . We shall soon see that, in general, the 1/ . E P E X alone determines E up to tensor product with a line bundle projective bundle P ! . completely (Proposition 9.4), and that the line bundle O 1/ on P E determines E E P (Proposition 9.3). 9.1.1 Example: rational normal scrolls Before continuing with the general theory we pause to work out the case of projective 1 1 bundles over . As we saw in Theorem 6.29, vector bundles on P are particularly P L r O .a / . simple: Each one is a direct sum of line bundles 1 i D 0 i P 1 P Write D P V , where V is a vector space of dimension 2, with homogeneous  the V coordinates . Recall that for a  1 2 rational normal curve of degree a is the s;t 1 There is a conflicting definition that is also in use. Some sources, following Grothendieck, define the projectivization  E to be what we would call the projectivization of E of , that is,  W Proj . Sym E / ! X: Its points correspond to 1-quotients of fibers of E . We are following the classical tradition, which is also the convention adopted in Fulton [1984]. Grothendieck’s convention is better adapted to the generalization from vector bundles to arbitrary coherent sheaves, which we will not use.

339 Projective bundles and the tautological divisor class 325 Section 9.1 image of the morphism 1 a a 1 a a /: ;s ! 7! t;:::;t ; .s;t/ .s P P ' W 0 1 ' W P P ! When a to be the constant map. More invariantly, for a  0 , D 0 we take a  ! P P D P W V given by the complete linear series as the map ' we can think of  0 jWD j O : O .a/;H .a/ . O .a// 1 1 1 P P P L C nonnegative integers a Fix a sequence of ;:::;a . , and let E D O 1 r . a / 1 r 0 i i P N E by mapping it to a projective space P using P We will analyze the projective bundle O .1/ . the line bundle E P L i 0  0  , and write W .a D // D Sym H V Set and W D W . . E O / D H 1 i i i P X X D dim N 1 D W C 1/ 1 D r C .a a : i i E , we consider the r C 1 rational curves P Inside 1 P . O D C : . a P // Š 1 i i P There are natural maps 0  0   0 / ! D H . .1//; E . / ! H E . O W H E P and from the commutative diagrams L 0 - .1// . O H W W D E P i C restriction to projection i ? ? Š 0 0 - .a W // H D . O H O . .1// 1 i i / O P a . 1 i P P 0 ! H we see that . O is the C .1// is a monomorphism and that its restriction to W E P i 1   W .a W / j . Let ' P W P be the corresponding complete linear series P O j  ! 1 i i P i C morphism, which embeds as the rational normal curve of degree a as above. i i 1 p 2 P . , the restriction of the linear series W WD For each O to the fiber .1/;W / E P r 1 r . Since the image contains the is a subseries of j O P D .1/ .p/  r C 1 linearly j P independent points ' .p/ , it is the complete linear series, and this fiber is mapped i r ' that is the linear span of the points isomorphically to the P .p/ . Thus the linear series i N . P E ! P W ' W is base point free, and defines a morphism rational normal scroll We define the   M N P  / W ;:::;a D P S.a 0 i r to be the image P E / of this morphism. It is the union of the r -dimensional planes '. 1 ' : .p/;:::;' spanned by .p/ as p runs over P r 0 [ WD .p/: .p/;:::;' ;:::;a ' / D S S.a r 0 0 r 1 2 P p

340 326 Chapter 9 Projective bundles and their Chow rings 0 1 L L 0 1 0 1 , the union of lines joining corresponding points on the parametrized S.1;1/ Figure 9.1 3 and L L P . skew lines , is a nonsingular quadric in 1 0 O ' .p/ Since each , and the distinct W is embedded by the restriction of P / . a 1 i i P 1 p is set-theoretically , it is already clear that ' are linearly independent for every P 2 an injection. W j O is the complete linear series In the next section, we will show that .1/ j , and P E a . The ideal of forms that when all the map ' induces an isomorphism P E Š S > 0 i vanishing on a rational normal scroll is also easy to describe (Exercises 9.27–9.29). We will also show that     r r M M P / O Š P .a .b O / 1 1 i i P P D i 0 0 D i if and only if there is an integer b such that (after possibly reordering the indices) ; thus the description above can also be applied to describe the D b i C b for all a i i  L r P are negative. a even when some of the O / bundles .a 1 i i D 0 i P r 0 , we have Some examples of this construction are already familiar. In the case D is the rational normal curve of degree / already noted that a S.a (or a point, if a ). D 0 0 0 0 3  P above as the union of lines joining corresponding From the construction of S.1;1/ points on two given disjoint lines, the images of ' and ' is the , we see that S.1;1/ 1 0 3 : the lines in the union are the lines in one of the two rulings, P nonsingular quadric in while the images of and ' are two of the lines in the other ruling (see Figure 9.1). ' 0 1 5 Another instance is the scroll S.1;1;1/  P , which is the Segre threefold , that is, the 5 2 1  P ! P image of the Segre embedding . P a D 0 , then from the construction we see that S.a ;:::;a If / is a cone over r 0 r / S.a . This remark allows us to reduce most a , and similarly for the other ;:::;a 1 i r 0 questions about scrolls to the case where all a . For example, the quadric in > 0 for all i i 2 3 with an isolated singularity, that is, the cone over a nonsingular conic in P , can be P S.2;0/ or S.0;2/ . described as 4 S.1;2/  P To describe the first example beyond these, the scroll , we choose an . isomorphism between a line and a nonsingular conic C lying in a plane disjoint from L L The scroll is then the union of the lines joining the points of L to the corresponding points of C .

341 Maps to a projective bundle Section 9.1 327 There is much more to say about the geometry of rational normal scrolls, some of which will be deduced from the more general situation of projective bundles in the next sections, some in Exercises 9.27–9.29. For more information see Eisenbud and Harris [1987] or Harris [1995]. 9.2 Maps to a projective bundle Y ! X is the  One of our goals is to show that every projective bundle W on , as stated above. In fact, we will construct the E projectivization of a vector bundle X bundle E from the geometry of  , as the dual of the direct image of a suitably chosen on Y line bundle Y ! P E , we will use the following L . To construct the isomorphism universal property, which generalizes the one for projective spaces: Proj ) . Given a vector bundle E (Universal property of X , Proposition 9.2 on a scheme commutative diagrams of maps of schemes ' - P E Y p -   X  L p E .  are in natural one-to-one correspondence with line subbundles  ' , we pull back the inclusion O E ' . 1/   Proof: Given via and get E P     . 1/  ' : O E E D p  ' E P  L  p L E , we may cover X by open sets on which E and Conversely, given are trivial, and get a unique map over each of these using the universal property of ordinary pro- jective space. By uniqueness, these maps glue together to give a map over all of . X E from To prepare for the next step we need at the least to know how to reconstruct E . For future use, we will treat an easy generalization. Write a line bundle on .m/ P O P E m -th tensor power of O is the .m/ .1/ . Thus (for any integer m ) the sheaf O for the P E P E    . / associated to the sheaf of Sym E sheaf on Proj -modules Sym Sym E E /.m/ on X , .  Sym E on . For any quasi-coherent sheaf F obtained by shifting the grading of P E we write F to denote F ̋ O .m/ . .m/ E P   E E The surjection ! O ,  .1/ , restricted to the fiber over a point .x;/ 2 P E P E  to its restriction to the subspace  sends a linear form on E . Thus any global sec- x x   of E tion gives rise to a global section Q  of O . The following result strengthens .1/ E P and extends this observation:

342 328 Chapter 9 Projective bundles and their Chow rings If Proposition 9.3. P E ! X is a projectivized vector bundle on X then for m  0  W m  O D Sym  E .m/ ; E P  i .m/ O  and R D 0 for i > 0 . E P  D 1 , we see that the map  W P E Taking X , together with the tautological m ! O line bundle .1/ , determines E . P E where has rank r C 1 Proof: U  X Suppose that E j E Š . Over an affine open set U r C 1 m 0 0  H , the natural maps Sym . E j are isomorphisms, / ! H O . O / j .m/ E P U U U i . O while H .m/ j , so the proposition follows immediately from the / D 0 for i > 0 E P U definition of the direct image functors. Remark. Proposition 9.3 is a direct generalization of the standard computation of 0 r H O . .m// — the case when X is a point. Though we will not make use of these facts, P the rest of the computation of the cohomology of line bundles on a projective space, and Serre duality, also generalize, and one can show that 8 m  ˆ i for E Sym 0; D < i .m/ R  O D for 0 0 < i < r 1; P E  ˆ : r 1 m Sym E for i D r: k (Here we adopt the convention that D 0 for k < 0 .) As a part of our computation Sym E E P E of the Chow ring of in the next section, we will see that every line bundle on P  L L O has the form  .m/ for a unique line bundle ̋ on X and integer m ; that P E Pic . P E / Š Pic X ̊ Z . From the push-pull formula of Proposition B.7, we get a is, computation of the direct images of any line bundle:  i i   L ̋ O .m//: O .m// D L ̋ R R . .  P E P E  ́ See Dieudonn e [1969, p. 308] for equivalent material, with references to EGA. relative duality . For example, setting Serre duality also generalizes to a V r ! D 1/ r E . =B E P r D R .! we have , and more generally O /   B =B E P r 1 . M R / H om . O .! ̋ M  / /; D   B for any line bundle M on P E . See Altman and Kleiman [1970], in particular Theorem 3.8, for most of this.  Supposing that has a global section  ¤ 0 , the proof of Proposition 9.3 shows E Q  of O such that the corresponding section .1/ vanishes on the locus of pairs .x;/ P E that  in a vanishes on  ; thus the divisor . Q / meets a general fiber of  W P E ! X x hyperplane. It will not in general meet every fiber of P E ! X in a hyperplane, however;

343 Maps to a projective bundle Section 9.2 329   x 2 X , and the divisor . Q / may have zeros P E will contain the  E of the section 1 . D  corresponding fibers E .x/ . / P x X that are projective Using these ideas, we can characterize the schemes over bundles:  W Y ! X be a smooth morphism of projective schemes whose Proposition 9.4. Let r P (scheme-theoretic) fibers are all isomorphic to . The following are equivalent: P E is the projectivization of a vector bundle (a) on X . D Y E W Y ! X is a projective bundle; that is, it is locally isomorphic to a product in (b)  . X the Zariski topology on r (c) on Y whose restriction to each fiber Y Š There exists a line bundle P L of  is x r isomorphic to .1/ . O P r (d) There exists a Cartier divisor intersecting a general fiber Y  Š P  of Y in D x a hyperplane. Proof: Condition (a) clearly implies (b) and (c): The projectivization of a vector bundle is locally trivial in the Zariski topology, since a vector bundle is, and comes with the line O bundle . .1/ P E r 1  Š U  P .U/ for any It is clear that (b) implies (d): Just take an isomorphism r 1 r and take Š P U X , choose a hyperplane  P H  D the closure in Zariski open of U  H . Y Also, it is easy to see that (c) and (d) are equivalent: If D is a divisor as in (d), the r L D O . By the constancy of .D/ , restricted to a general fiber, is O .1/ line bundle Y P L the Euler characteristic of a sheaf in a flat family (Corollary B.12), the restriction of to r .1/ . any fiber is O P Conversely, if L is a line bundle as in (c), tensoring with the pullback of an ample we can assume the existence of a nonzero global section of line bundle from , whose X L zero locus will be the divisor of part (d). To complete the argument we take L as in part (c), and we must prove that Y is as in 1 1 r X H 2 . L we have / D p part (a). For any . O , so Theorem B.5 shows H .1// D 0 p P 0 r  . L is a vector bundle whose fiber at p is H E . that .1// WD O  P ̨ W We claim that there is an isomorphism ! P E commuting with the projections Y to X . By Proposition 9.2 we can define the morphism ̨ by giving a line bundle that  is a subbundle of E , or equivalently a line bundle that is a homomorphic image of     D   L .  E     . L ! L coming from the definitions of  There is a natural map and     Restricted to the fiber over a point p this map becomes the surjection E ! ̋ O p P E / . p  .1/ O ! , so  L  P be the corresponding L is surjective. Let ̨ W Y ! E  P / . E p morphism.

344 330 Projective bundles and their Chow rings Chapter 9 is an isomorphism on each fiber of ̨  The map because it restricts to the map r r 1 r . This shows that P j O  P .1/ j given by the complete linear series ̨ is Š ! .p/ P a set-theoretic isomorphism. ̨ To prove that ̨ is a scheme-theoretic isomorphism, we need to show that if  O q 2 P E then the map of local rings ̨ Y W to a point y 2 O ! carries ;q E P Y;y is an isomorphism. Of course it is enough to prove this after completing both rings. D Set . By smoothness, the completions of both local rings are isomorphic p .y/ y O to ;:::;z ŒŒz çç . Since ̨ commutes with the projections, it induces the identity 0 r X;p modulo the maximal ideal of O , and thus induces an isomorphism. X;p We can also use Proposition 9.2 to see when two vector bundles give the same projective bundle: 0 Corollary 9.5.  Let P E ! X and  X W be a scheme. Two projective bundles W 0 X on are isomorphic as X -schemes if and only if there is a line bundle L X ! P E 0 ̋ E . In this case the line bundle O such that E . D 1/ corresponds under the L E P 0 0 L / ̋ O isomorphism to  . . 1/ . E P 0 . Tensoring the tautological sub- E Proof: D L ̋ E Let L be a line bundle, and set 0 0 0 0 0 1 0  O E  D , we get a subbundle bundle L ̋  . E with  . L 1/ ! / E P 0 1 0 0 / ̋ O L . By Proposition 9.2 this determines a unique morphism E . 1/ !  .  E P 0 -schemes ' W P E ! of P E such that X  0 1 0 . 1/ D  O . L ' O / ̋ . 1/: E P E P The inverse map is defined similarly. The proof that they are inverse to each other is / P . L ̋ E ! ! P E corresponds to the original subbundle P E that the composite  . 1/   O E . E P 0 0 0 P  is a vector bundle on W E E X ! X be the Conversely, suppose that , and let 0 W P E projection. If ! P E is an isomorphism commuting with the projections to X , ' n n it follows that P O .1/ to itself preserves the bundle then since any isomorphism from P n  0 n .1/ restricts on each fiber P . E O .1/ / Š P ' to the bundle O . By Corollary B.6, E P P x 0  0 ̋ .1/ D  O . L / . Thus ' X O on L .1/ for some line bundle E P P E  0 0 0 0 0 . .1// . .1// D   O D L ̋ ' E O P E P E   0   .1/ D ' L O ̋ E P  0 1 L D ' ̋  O .1/ P E   D L ̋ O  .1/ E P   E D ; ̋ L   0 and also O 1/ D  . . L / ̋ O ' , as claimed. . 1/ E P E P

345 Chow ring of a projective bundle 331 Section 9.2 9.3 Chow ring of a projective bundle We now turn to the central problem of this chapter: to describe the Chow ring of a P E ! X . We will see that the Chow groups of Y depend only Y projective bundle D E , but the ring structure reflects the Chern classes of on the rank of E . ̈ As we mentioned in Section 2.1.4, the K unneth theorem holds for the Chow ring of the product of any smooth variety with a projective space. Thus, if r C 1 r P D Y O D X  P . / r P then r ̋ A.Y/ A. P A.X/ / Š Z r C 1 Œç=. ̋ Š Z / A.X/ Z 1 r C /; A.X/Œç=. Š r is the pullback of the hyperplane class on . In particular, P where  r M i D A.Y/  A.X/ 0 i D as groups. (Given that the pullback map A.X/ ! A.Y/ is injective, here and in what  as a subalgebra of A.Y/ , suppressing the “  ;” for example, follows we think of A.X/ 2 ̨ˇ 2 A.X/ and ˇ ̨ A.Y/ , we mean with when we write products of the form  ̨/ˇ 2 . .) A.Y/ The general case is not much more complicated: 1 E be a vector bundle of rank r C Let on a smooth projective scheme X , Theorem 9.6. 1  D c be the projection. The map . O X ! .1// 2 A and let . P E / . Let  W P E E P 1  W ! A. P E / A.X/  is an injection of rings, and via this map C 1 r r A.X/Œç=. c A. P C c //: . E / E CC / Š E . 1 r 1 C r C 1 ̊ given by ! A. P In particular, the group homomorphism / A.X/ . ̨ 7! ;:::; ̨ / E r 0 P i    . ̨ is an isomorphism, so that / i r M i A. / P E A.X/  Š 0 i D as groups. It is worth remarking that much of the statement of Theorem 9.6 remains true r is smooth: If E is a vector bundle of rank X C 1 over an without the assumption that  X and P E D Proj . Sym E arbitrary scheme / its associated projective bundle, then we  have a well-defined line bundle .1// O .1/ on P E such that O D c . restricts to the 1 E P E P

346 332 Chapter 9 Projective bundles and their Chow rings hyperplane class on each fiber, and we can show that r M i P E A. /  Š A.X/ 0 i D as groups, just as in the smooth case (see Fulton [1984, Chapter 3]). (Note that in this A. P setting we do not have a ring structure on / , but multiplication by the or A.X/ E class  is still well-defined since it is the Chern class of a line bundle.) It was one of the insights of Grothendieck [1958] that Theorem 9.6 could be inverted the Chern classes of E as the coefficients in the unique expression of and used to define C r 1 r 1;;:::; as a linear combination of the classes  (or rather to prove the existence of classes satisfying the axioms of Theorem 5.3). The original definitions of Chern and Stiefel–Whitney classes in the 1930s came from topology. They did not mention degeneracy loci, but could be directly related to that characterization of the classes; as we have seen in Chapters 6 and 7, this is closer to the way Chern classes are thought of and used in practice. As a definition, however, it has the drawback of depending on the existence of global sections. (This is a problem only in the algebro-geometric context; in 1 the continuous or settings, thanks to partitions of unity there is never a shortage of C sections.) While it is possible to define Chern classes for bundles with enough sections via degeneracy loci, and even (as we illustrate in Section 5.9.1) to prove basic properties such as the Whitney formula in that setting, in order to have a full toolkit of techniques for calculating Chern classes it is necessary to extend the definition to arbitrary bundles, and for this the Grothendieck–Serre definition is better. We isolate part of the proof of Theorem 9.6 that will be useful elsewhere: ̨ 2 A.X/ , then Lemma 9.7. Let the hypotheses be as in Theorem 9.6. If  r D i if , ̨ i D .  ̨/  0 if i < r: i i . By the push-pull formula (Proposition B.7), ̨/ D  .  / ̨ . If i < r , then Proof:   i r . , we see similarly that / is zero for dimension reasons. If i D r   / . must be   0 A a multiple .X/ mŒXç X . Let  be the class of a point 2 of the fundamental class of r  ç D  X ./ D Œ P E x f the class of the fiber P E . Intersecting both sides Š P 2 and x x r / . of the equality  D mŒXç with  and taking degrees, we have  r r r D . deg /  / D deg . .  Œ P m ç/ D 1;  since the restriction of  to a fiber is the hyperplane class. In fact, we have encountered this construction before, in the proof of Lemma 5.12. L r i be the map A.X/ P / ! A. W Let Proof of Theorem 9.6: E i D 0 X i r i . 7! ˇ  ˇ/ ;  i

347 Chow ring of a projective bundle Section 9.3 333 L r ' and let W : A.X/ ! A. P E / be the sum of the multiplications by powers of  0 D i X i ;:::; ̨ / 7! W . ̨ ' ̨ :  r 0 i i ' By Lemma 9.7, the composite is upper-triangular with ones on the diagonal; in is a monomorphism. particular, ' To prove the additive part of Theorem 9.6, it now suffices to show that the subgroups i generate A. P E  additively. This is a relative version of the fact that the linear / A.X/ subspaces of a projective space generate its Chow ring, and the proof runs along the same lines. In the case of a single projective space, we used the technique of dynamic n projection to degenerate a given subvariety Z  P to a multiple of a linear space; we do the same thing here, but in a family of projective spaces. Z  P E is a k -dimensional subvariety, we say that Z We start with a definition. If has footprint if the image W D .Z/ has dimension l , or equivalently if the general l  W ! W has dimension k l . fiber of the map Z and footprint Z P E is a subvariety of dimension k If l , then Lemma 9.8.  X 0 Z C  n Z B i i B  P E such that: for some subvarieties i C 0 k r l for a class D ŒZ (a) ̨ ç ̨ 2 A.X/ .  (b) Each B . has footprint strictly less than l i Applying the lemma repeatedly, we can express the class of an arbitrary sub- i  , establishing the group isomorphism variety as a sum of classes of the form ̨ L i . E P Š A.X/ / A.  By Corollary 9.5, replacing E with its tensor product with a line Proof of Lemma 9.8:  does not change c E , but has the effect of replacing the class  by   L bundle P . L / . 1 In particular, it does not affect the truth of our assertion, so we can assume from the  is generated by global sections. E outset that x  .Z/  X and a general collection This done, we choose a point of ;:::; 2 0 r  global sections of E , making sure that the  satisfy two conditions: i (a)  . .x/;:::; E .x/ are independent, that is, they span the fiber r x 0 (b) .x/ is disjoint from the fiber .x/ D  D  E The zero locus P . D 0/  x 0 l k Z . D Z \ P E x of Z over x x U  X These are both open conditions; let x 2 X where they hold. Note be the locus of in particular that, by the first condition, the bundle P E is trivial over U , the sections r . ;:::;  giving an isomorphism P E P Š U  0 r U

348 334 Chapter 9 Projective bundles and their Chow rings r A P E of Š U  P Now consider the one-parameter group of automorphisms t U given, in terms of this trivialization, by the matrix   I 0 k l 1 C : I  0 t r C l k z D Z \ P E Let be the preimage of U in Z (note that .Z/ \ U ¤ Z , since ¿ U z ); let Z x be the closure of the image A 2 . .Z/ Z/ and let Z be the limiting cycle, 0 t t 1 ı 0 , of the subvarieties t . In other words, let ˆ as  A Z  P E be the incidence ! t correspondence ı 1 z .t;p/ Df ˆ  P E j t ¤ 0 and p 2 A 2 . A Z/ gI t ı . let and let Z ˆ be the fiber of ˆ over t D 0 be the closure of ˆ 0 What does look like? Over the open subset U  Z the original cycle Z has been X 0  . There is thus a unique flattened to a multiple of the zero locus 0 D  DD r 1 C l k 0 D Z Z dominating W of .Z/ , and it is the closure of the intersection of component 0 1 U/ DD \ .W the common zero locus   D 0 with the preimage  . r 1 C k l  Now, we have arranged for E to be generated by global sections, so that the linear series j O , by Bertini .1/ .1/ j has no base locus. Since the  O are general sections of i P E P E l  of them intersects the C k D  D  D 0 of r the common zero locus r C l 1 k 1  .W / k -dimensional subvariety of P E with subvariety generically transversely, in a k l r k C l ; moreover, since this intersection is fibered over  with fibers P  ŒW ç class , W it is irreducible. In sum, l r k C 0 ŒZ ç D mŒW ç   m for some multiplicity . To complete the proof we note that we do not need to know what happens over U \ W in W , because any component of Z W the complement of not dominating 0 l . necessarily has footprint smaller than r C 1 From this description of the Chow groups we see that we can write as a linear  combination of products of (pullbacks of) classes in A.X/ with lower powers of  — that is,  f of degree r C 1 over A.X/ . Thus the ring satisfies a monic polynomial A.X/Œç A.X/Œç=.f / A. P E / factors through the quotient homomorphism . Since ! L i A.X/Œç=.f / Š A .X/ as groups, it follows that the map A.X/Œç=.f / ! A. P E /  is an isomorphism of rings. It remains to identify the polynomial f . Let S D O be the Q . 1/ , and let E P  , a bundle of rank  S cokernel of the natural inclusion ! r . We have an exact E sequence  ! S !  Q E ! 0 ! 0:  Identifying with  A.X/ as before, we have A.X/ c. S /  c. Q / D c. E /

349 Chow ring of a projective bundle Section 9.3 335 by the Whitney formula (Theorem 5.3).  to be the first Chern class of the line bundle We defined the class .1/ , which O P E  S is the dual of D 1 ; thus , and we can write this as S c. / 2 1 E /  c. S / C c. D c. E /.1 C  C  Q / /: D c. is a vector bundle of rank Since , we conclude that Q r r C 1 r r 1 . / C c . E / D D c . E /  c Q CC c 0 . E / C c . E /; C 1 r r C 1 r 1 C 2 f so the polynomial is given by the formula in the theorem. L is a line bundle on P E Š P . E ̋ then Corollary 9.5 shows that / , but the X If L is different in the two representations; the two classes differ by multiplication  class . The relation between the two resulting descriptions of the Chow with the pullback of L ring is addressed in Exercises 9.30 and 9.31. Using Theorem 9.6, we can immediately compute the degrees of rational normal E X , scrolls, or, more generally, of any projectivized vector bundle over a curve P embedded by j .1/ j : O P E If ;:::;a are positive integers, then the degree of the rational normal Corollary 9.9. a r 0 P a is a vector bundle on a smooth curve / is S.a E scroll . More generally, if ;:::;a i r 0 and the line bundle O E .1/ X P E is very ample, then the degree of the image of P on E P under the embedding given by j O .1/ j is deg c . . E / 1 P E satisfy the equation S Note that degree and codimension of a scroll D codim C S S: deg 1 This is the minimal degree for any subvariety of projective space not contained in a 5 P , and any cone over it, also satisfy this equation, hyperplane. The Veronese surface in but these are the only “varieties of minimal degree.” See Harris [1995, Theorem 19.19]. If the rank of E is r C 1 then the dimension of P E is r C 1 , so the degree Proof: r C 1  O of the image of is .1/ j is deg E P under the embedding given by j . Since X E P 1 r C 1 E / D 0 for i > 1 , so  one-dimensional, we have c and D c . . E / . If X D P 1 i O D /; a . a / E O . ̊ ̊ 1 1 0 r P P P c . . E / D and a .1/ then deg S.a O ;:::;a by / is the embedding of P E r 0 E i 1 P k -plane over G .k;n/ 9.3.1 The universal In this section and the next, we will use Theorem 9.6 to give a description of the Chow ring of some varieties that arise often in algebraic geometry: the universal k -plane n along a linear space. over the Grassmannian G .k;n/ and the blow-up of P

350 336 Chapter 9 Projective bundles and their Chow rings G D .k; P V / be the Grassmannian parametrizing k - For the first of these, let G  planes in the projectivization of an .n C 1/ -dimensional vector space V , and P ƒ V be the universal plane ˆ let j G  P V Df p 2 ƒ g ; ˆ 2 .ƒ;p/ , via the projection  W ˆ ! initially introduced in Section 3.2.3. We can recognize ˆ G P of the universal subbundle on G , and use on the first factor, as the projectivization S . We will use the notation introduced above: We will Theorem 9.6 to describe A.ˆ/  via the pullback map  A.G/ , and denote the first A.ˆ/ with its image in identify 1 O .1/ by  2 A Chern class of the tautological bundle .ˆ/ . P S   V by restriction in turn on V gives rise to a section of S Note that a linear form l 2   V to each subspace of S  , and ultimately to a section of O , hence to a section of .1/ S P    O .1/ dual to the tautological inclusion O  S via the surjection 1/ , !  ! S . . S P P S z z P .k as the variety of pairs . D ƒ;/ with ˆ ƒ  V a Simply put, if we think of C 1/ - S z  ƒ a one-dimensional subspace, then we can define a  dimensional subspace and  of section O by setting .1/ S P l z .  ƒ;/ D l j : l  z In particular, we see that the zero locus of the section  is just the locus of . ƒ;/ such l that  is contained in the hyperplane Ker .l/  V , and hence the tautological class 1  O c via the .1// 2 A . .ˆ/ is just the pullback of the hyperplane class on P V D S 1 P W ˆ ! P V on the second factor . projection map  .k;n/ G Recalling the calculation of the Chern classes of the universal bundles on from Section 5.6.2 and applying Theorem 9.6, we conclude: n G D G .k;n/ be the Grassmannian of k -planes in P Let and Proposition 9.10. n n ˆ P ˆ the universal k -plane as above, with  W ˆ ! G and  W  ! P G the  projection maps. We have then 1 k k C 1 k k C 1 A.G/Œç=. C    . A.ˆ/ D CC 1/  /;  1;1 1;1;:::;1 1 1 of the  .ˆ/ is the tautological class, or equivalently the pullback via  A where 2 n P . hyperplane class in k The two special cases occurring most commonly are the cases n 1 of the D universal hyperplane and the case k D 1 of the universal line. In the first case, n  n 2 P p ˆ  P Df j .H;p/ 2 H g ; n  and if we let be pullback to ˆ of the hyperplane class in P , we have ! n n 1 n C 1 n n C 1 n . ; ! ; Œ!;ç=.! Z CC 1/ D ! A.ˆ/ /:

351 Chow ring of a projective bundle Section 9.3 337 ƒ q Ä 3 P from the line ƒ . Figure 9.2 The fiber over a point under the projection of We have written the ideal of relations in this way to emphasize the symmetry, but it is n C 1 1 C n .ˆ/  . Note that when a C b D dim or D redundant: we could drop either ! 1 , we have 2n  .n 1/ 1;n/; .n;n 1 if .a;b/ D or b a  D .! deg / otherwise, 0 n  n   P P ˆ which we could also see from the fact that is a hypersurface of bide- . .1;1/ gree The universal line will also come up a lot in the following chapters; in this case we have 2 A. G A.ˆ/ D  /:  C  .1;n//Œç=. 1;1 1 a b c  We will leave it to the reader to calculate the degrees of monomials  of top  1 1;1 degree a C 2b C c D dim .ˆ/ D 2n 1 in Exercise 9.33. n 9.3.2 The blow-up of P along a linear space In Section 2.1.9 we saw how to describe the Chow ring of the blow-up of projective space at a point. We can now analyze much more generally and systematically the n n Chow ring of the blow-up P D Bl P Z D P V along any linear of projective space ƒ r 1 subspace ƒ Š . The key is to realize Z as the total space of a projective bundle. P To understand the picture, first recall that the blow-up is the graph of the rational map n n r n n r - P given by projection from ƒ . Thus Z  P   P P . We will show W ƒ n r to the second factor makes ! P Z Z into a projective bundle. Cer- that the projection r -dimensional projective space (see Figure 9.2). tainly, each fiber of the projection is an n .n r/ -plane Ä  P Concretely, if we choose an disjoint from ƒ , we can write n Ä .p;q/ 2 P Z  Df j p 2 ƒ;q g :

352 338 Chapter 9 Projective bundles and their Chow rings n r C 1 n  Ä Š ƒ;q P P . If we 2 The fiber over a point is thus the linear subspace q n 0 V V ƒ corresponds to an r -dimensional linear subspace P as  V and Ä write P , then r 1/ -dimensional subspace W . The fiber of Z corresponds to a complementary .n C 0 Ä corresponds to the subspace spanned by V and the one-dimensional subspace q over 2 0 q in W . Here V Q is fixed, while the one-dimensional subspace varies q corresponding to . This suggests that W over all such subspaces of Z is the projectivization of the bundle 0 r n r n . .V O ̊ 1/ ̋ / , which we will now prove: O P P 0 Proposition 9.11.  V be an r -dimensional subspace of an .n C 1/ -dimensional Let V , and let V vector space 0 n r n r 1/ ̊ .V E ̋ O /; D O . P P n r 0 is a vector bundle of rank so that 1 on r E .V / P C . The blow-up Z of P .V=V P D / n r 0 .V .r / , together with its projection to P 1/ -dimensional subspace , along the P n r ! P W E P . Under this isomorphism, the is isomorphic to the projective bundle  n ! P blow-up map corresponds to the complete linear series j O Z . .1/ j E P 0 0 0 Proof: W  V to V Choose a complement , so that V D W ̊ V V=V . With E as Š ̋ O in the proposition, the natural inclusion . 1/  .W induces an inclusion O / W P P W 0  .W ̋ O : O / ̊ .V E ̋ O ̋ V / D W P W P W P  0  . E V / H The dual map, which is a surjection, induces an isomorphism ! D  0  . Thus E is generated by its global sections and the complete linear series W V ̊ ! P .1/ j corresponds to a map O E j P V . E P 0 over a point q 2 P W is, as a subspace of V , equal to V The fiber of ̊ Q q , whose E 0 in of P V q Z P V . Thus, together with of the blow-up projectivization is the fiber over  W P E ! P W , we get a closed immersion ' W P E ! P V  P W the projection map P isomorphically to Z . E that maps the fiber of n r n n 1/  P Corollary 9.12. Let Z be the blow-up of an .r  -plane ƒ in P P . n r n 1 P Writing A ̨; and P for the pullbacks of the hyperplane classes on 2 .Z/ respectively, we have r 1 C n r C 1 r ; Z ̨ D /: Œ ̨;ç=. ̨ A.Z/ E  Z , the preimage of ƒ in Z , is With this notation the class of the exceptional divisor D  ̨: ŒEç r 0 n r , so the formula Proof: The Chern class of E . 1/ ̊ .V D ̋ O O ̨ 1 is / r n P P for A.Z/ follows at once from Theorem 9.6. Since  is the class of the preimage of n H P a hyperplane (which could contain ƒ ), and ̨ is represented by the proper  as claimed. ƒ , we have ŒEç D  ̨ transform of a hyperplane containing

353 Chow ring of a projective bundle Section 9.3 339 For example, in the case of the blow-up of the plane at a point we have 2 2 2 2 2 D  . 2 ̨ C ̨ D  ̨/ ; ŒEç D n ŒEç deg that is, minus the class of a point, as we already knew. But we can now compute in general (Exercise 9.38). 1 9.3.3 Nested pairs of divisors on P revisited We start by introducing two vector bundles that arise often in studying the geometry of rational curves; in particular, they will be a central object of study in Section 10.4.2. d 0 be the projective space of polynomials of H To begin with, let D O P P . .d// 1 P 1 P degree d in two variables modulo scalars — that is, divisors of degree . For any d on d d , then, we can define a vector bundle F on P informally by associating to each e  d D P divisor the vector space 2 0 F . I .e// D H D D 1 e P vanishing on D . Similarly, we can define a bundle E of polynomials of degree on d d informally by associating to each divisor 2 P P the quotient vector space on D 0 0 0 .e// . O . H .e//=H O . I D .e// E H D 1 D D D P of polynomials of degree e modulo those vanishing on D . To define these bundles 1 d  P  P precisely, let be the universal divisor of degree d , that is D d 1 D Df  P P j p 2 2 g ; .D;p/ D 1 d d d 1 1 and let  P W and  P P !  P  ! P P be the projection maps. We can W then take  D / . O F  .e/ ̋ I 1 D  P and  D ̨ . E O I / .e/ ̋ O 1  D P an application of the theorem on cohomology and base change shows that these have the d 1 fibers indicated, and that the exact sequence of sheaves on  P P 0 ! I 0; ! O ! O ! 1 d D D P  P d   tensored with the line bundle .e/ and pushed forward to P , gives the expected O 1 P exact sequence 0 ! F ! H 0 . O (9.1) 0 .e// ̋ O ! E ! 1 d P P d of bundles on P .

354 340 Chapter 9 Projective bundles and their Chow rings D ˆ of the bundle E . This is a variety we P Consider now the projectivization E have encountered before, in Section 2.1.8: We can realize it as the subvariety d e .D;E/  P P j E  Df g D ˆ 2 1 . Moreover, under the inclusion of e on P d and of nested pairs of divisors of degrees e d e P   P ˆ , the pullback D P P E restricts to the in of the hyperplane class from  c tautological class . O D . .1// on P E E 1 P ˆ , and correspondingly the Chern We can use this to describe the Chow ring of e d d classes of P P Š . The key, as it was in Section 2.1.8, is to observe that E  ˆ abstractly, via the map d e d d e 0 0 ! P ̨  P P ; .D;D W / 7! .D;D C D P /:  e d e d P  , P and and P   be the pullbacks of the hyperplane classes on , Let , As we saw in Section 2.1.8, the pullback of the class  to ˆ is the sum respectively. 1 C d e P  C D 0 in A.  E / as  . We can then rewrite the relation   X e d C 1 d 1 i C e i e d C 1 i ; . 1/   0 D . D / i and we conclude that   e d C 1 i i 1/ c D . /  . : E i i To express this more compactly, we can write the total Chern class as e d C 1 : E / D .1 / c. In this form, it follows from the exact sequence (9.1) that   X 1 e C d i i  ; D F / c. D 1 d C e i / .1 so we have the Chern classes of F as well. 9.4 Projectivization of a subbundle  E X and F If E a subbundle then P F is a vector bundle on a smooth variety is naturally a subvariety of P E , and we can ask for its class in the Chow ring A. P E / . This will be a crucial element in understanding the Chow ring of a blow-up in general (Section 13.6); for now, it will allow us to answer Keynote Question (b).   P E ! X be the projection and let O be the universal E . 1/ Let W   E P p 2 P subbundle. A point lying over a point x 2 X corresponds to the one-dimensional E space that is the fiber of O if and only if this space is contained F . 1/ at p . Thus p 2 P E P F . In other words, p 2 in the fiber of F if and only if the composite map P   O F = / 1/ !  ' E !  W . E . E P

355 Projectivization of a subbundle Section 9.4 341 . We can view p ' vanishes at as a global section of the bundle   om 1/; O . E = F . Š O . H .1/ ̋  . E = F /: // E P P E P is scheme-theoretically If we write everything in local coordinates then we see that F ' P F is the same as the rank of defined by the vanishing of . Since the codimension of = F , it follows that Œ P F ç 2 A. P E / is given by a Chern class, which we can compute E using the formula for the Chern class of the tensor product of a bundle with a line bundle (Proposition 5.17): X is a smooth projective variety and F  E are vector bundles on If Proposition 9.13. s and r respectively, then X of ranks  F ç D c . // F Œ O = E .1/ ̋  P . E s r P s s r s 1 r r C /; CC E P . 2 A  D  s 1 r .  D . O where E .1// and P D c in c E = F /: Moreover, the normal bundle of P F E P 1 k k  . is .1/ ̋  O . E = F / P E This formula will be useful to us in many settings; for an immediate application, see Exercises 9.43 and 9.44. An important reason to consider projectivized subbundles is suggested by the ̨ W X ! P following characterization of sections. Giving a section — that is, a map E such that  ı ̨ is the identity — is the same as giving the image of the section; and we will therefore refer to the image as a section as well. Proposition 9.14. If L  E is a line subbundle of a vector bundle E on a variety X , ! then  P E is the image of a section X ! P E of the projection P E L X , and every P section has this form. P E over each point Informally: giving a section is the same as specifying point of X , that is, giving a one-dimensional subspace of each fiber of E . of that By the universal property of  W P E ! X , giving a map ̨ W X ! P E Proof: X ! X is the same as giving a line subbundle of E . “commutes with” the identity map 9.4.1 Ruled surfaces Recall that a ruled surface is by definition the projectivization of a vector bundle of rank 2 over a smooth curve. We can now answer Keynote Question (b): Proposition 9.15. A ruled surface can contain at most one irreducible and reduced curve of negative self-intersection.

356 342 Projective bundles and their Chow rings Chapter 9 X be a smooth curve, let W P E ! X be a ruled surface, and suppose that Proof: Let  P ;C E are two irreducible curves of strictly negative self-intersection. A fiber C  1 2 1 2  2 1 .x/ D  satisfies .Œxç Œ / D 0 , so the induced maps  W C X !  .x/ç are i 0 0 ! C X  ! C , and let ̨ W C C be the normalization of C finite. Let be the 1 1 1 1 1 corresponding map. Consider the pullback diagram ˇ  0 - P C D E ̨ P E P E  X 1  ? ? 0 - C X 1 ̨ 0 1 C  m† C / ˇ .C † represents a cycle D The preimage C D , where is a 1 1 1 1 1 X 1 m > 0 has no component in common with † section, and D . Hence 1 1 2 2  m ç deg D deg Œ† ç çŒˇ Œ† C çŒD ç deg Œ† 1 1 1 1 1  Œ† deg C ç  çŒˇ 1 1 deg Œˇ D † ç çŒC 1 1  2 ; D deg ŒC ç 1 2 Œ† . ç so deg < 0 1 Since a section pulls back to a section with the same self-intersection, we can 1 C repeat the process with a component of to obtain two sections † ˇ and † of 2 2 1 negative self-intersection. We can analyze this case using Proposition 9.14. Suppose that D P L †  P E . i i By Theorem 9.6, we have 2 //; E . C c A. A.X/Œç=. D / E P 1  D c because . O where / .1// . Now deg .c E . E // D deg  . .c c . E // D deg 1 P  1 E 1 1 2 c in degree 1. It then follows that deg  meets each fiber of D deg  . By . E /  1 Proposition 9.13, Œ† /; ç D  C c L . E / c . 1 1 i i so 2 2 deg ç 0 > D deg  Œ† : 2 deg c L . E / 2 deg C 1 i i Thus 2 deg L . (Exercise 9.50 strengthens this conclusion slightly.) > deg c / . E 1 i Supposing now that † ¤ † , we get an exact sequence 1 2 0 ! L 0; ̊ L ! ! E ! G 2 1 G is a sheaf with finite support; it follows that deg E  deg where , C deg L > deg E L 1 2 a contradiction.

357 Projectivization of a subbundle Section 9.4 343 By contrast, it is possible for a (nonruled) smooth projective surface to contain infinitely many irreducible curves of negative self-intersection; Exercises 9.45–9.47 show how to construct an example. It is an open problem (in characteristic 0) whether are bounded below, that S the self-intersections of irreducible curves on a surface ;C is, whether a surface can contain a sequence ;::: of irreducible curves with C 1 2 ́ ́ , J deg !1 . (In characteristic p > 0  C anos Koll .C ar has shown us an example, / n n described in Exercise 9.49.) 9.4.2 Self-intersection of the zero section on a scheme X may itself be considered as The total space of a vector bundle E  E Spec . Sym E A / over X . For various purposes it is useful to have a a scheme WD E X A that includes A E as an open compactification of , that is, a variety proper over subset, and we will describe the simplest such construction here. It is natural to try to compactify each fiber by putting it inside a projective space of the same dimension, and we can do this globally by taking the projectivization of the direct sum E ̊ O ; that is, we set X WD E . E ̊ O P /: X D E . Since c. E ̊ O r / be the rank of c. E / , we have Let X C 1 r r C A. E / c //: . D / A.X/Œç=. CC c E . E r 1 A  E is “the locus where the last coordinate is nonzero.” In terms of coordinates, E P E  Its complement is the divisor . E ̊ O , which we therefore call the “hyperplane / P X at infinity.” Since this is the locus where the section of O  .1/ corresponding to 1 2 O X E  E . / ̊ vanishes, we get O X D WD  O ç: .1// c Œ P E . 1 E (One can also see this from Proposition 9.13.)  P O  E is the locus where all the coordinates in E The section vanish; it is X . By Proposition 9.13, we have E , which we will call † thus the zero section of A 0 r r 1  E C c  . E / Œ† is a global ç CC c . More generally, if / D . E / C c . r 1 r 1 0 E .;1/ is a nowhere-vanishing section of E ̊ O , and the line subbundle section of , then X E , which we will call † it generates corresponds to a section of . Using Proposition 9.13  or the family , we see that , which gives a rational equivalence between † † and † 0 t  Œ† , then ç D Œ† r ç . If  vanishes in codimension 0  2 /: .Œ† E ç 猆 / D  . .ŒŒ† c  D çç/ D Œ./ ç 0  0 r  0  We claim that this formula holds in general:

358 344 Chapter 9 Projective bundles and their Chow rings Let be a vector bundle of rank on a smooth variety X , and let Proposition 9.16. E r W D . E ̊ O  / E X be the projection. Let  P X ! A . E /  E be the zero section, W ! X / D P . O with image † . We have 0 X 2 ç  / D c /; . .Œ† E r  0 and, for any class ̨ 2 A.X/ ,   ̨ D ̨c  . E /: r  Proof: By Proposition 9.13, r r 1 c C Œ† /: . E / ç D  CC c E . c . E / C 1 1 r 0 r , † P E  P . E ̊ O Since / , which has class  is disjoint from the hyperplane at infinity 0 X we get ç D 0 Œ† A. E / .) Thus . (This also follows from the computation of 0 1 2 r r E c C c // . E / D Œ† CC ç. . c C . E / ç Œ† 1 r r 1 0 0 D Œ† A. çc 2 . E / E /: 0 r 2 D From the push-pull formula we get ç E / D . , proving the Œ† / ç/c E .  / .Œ† c . r  0  0 r first assertion. For the second assertion, we use the fact that induces an isomorphism from † to  0  for any cycle  D  , and thus .ˇ \ Œ† . Thus ç/ ˇ ˇ on E X 0    2 D  /; . ̨Œ† ç/ D  . ̨Œ† ç  / D ̨c E . ̨   0  r 0   as required. See Theorem 13.7 for a generalization. 9.5 Brauer–Severi varieties  Y ! X that is isomorphic We defined a projective bundle to be a morphism W . to a product with projective space over Zariski open subsets covering the target X  was a product locally in Interestingly, if we had weakened the condition to saying that ́ etale, or analytic, topology on the , we would get in general a larger class of morphisms! X In this section, we will illustrate the difference with an example of a morphism that satisfies the weaker condition but not the stronger. Brauer–Severi variety over a variety X is a variety Y We start with a definition: A  W Y ! X such that all the (scheme-theoretic) together with a proper, smooth map r  are isomorphic to P fibers of , for some fixed r . Thus any projective bundle  W Y ! X is a Brauer–Severi variety. But, as we will see, the converse is false. ́  will be trivial locally in the etale (or, in It is in fact the case that such a morphism C , the analytic) topology, in the sense that every point x 2 X case the ground field is r 1 ́ Š such that U etale or analytic neighborhood .U/  U  P will have an . This is a

359 Brauer–Severi varieties Section 9.5 345 q C 0 C q L C C 0 M Figure 9.3 Local analytic triviality of the universal family of conics in the plane: 1 Š C 2 U  P Y via projection from q j 2 C 2 U . U C r consequence of the fact that has no nontrivial deformations. But it may not be trivial P locally in the Zariski topology. Here is an example: 5 2 be the space of conics in P Example 9.17. D Let V , and consider the universal P P conic  2 2 2 5 - P ˆ j p 2 Df .C;p/ 2 P C  P g  1 ? 5 P 4  ˆ as the total space of a P with its projections - to the two factors. We can realize i 2 ˆ  P : Indeed, bundle over is the projectivization of the rank-5 subbundle E  via 2 2  whose fiber E at a point p is the subspace of quadratic polynomials vanishing Sym V p . . (In particular, ˆ is smooth.) In these terms, the tautological class at D c 2 p O .1//  E P 1  1 .ˆ/ is the pullback of the hyperplane class  A . By Theorem 9.6, the divisor . O .1// 5 1 P 2 1 Š Z is generated by the pullbacks of the hyperplane classes class group .ˆ/ A 2 5 P . Note that these classes restrict to classes of degrees 2 and 0 on P from and any fiber of . Thus the intersection of the fiber of  has with any divisor on ˆ  1 1 even degree .  We now consider the projection . To obtain a map whose fibers are all isomor- 1 1 5 phic to , we let X  P be the open subset corresponding to smooth conics and P  W Y D ˆ let ! X be the restriction of  . By defini- to the preimage of X in ˆ 1 X 1  are smooth conics, and in particular isomorphic to P tion, the fibers of , so ˆ is a X Brauer–Severi variety over X . W Y  X is not a projective bundle. Indeed, if there were a We claim that ! 5  X  P nonempty Zariski open such that  W Y U ! U were isomorphic to U 1 U of the product U  P the projection to , then we could take a section of Y and U 5 ˆ ˆ meeting the general fiber of ˆ ! P take its closure in , obtaining a divisor in in a reduced point. This contradicts the computation above. Thus  W Y ! X is not a projective bundle.

360 346 Chapter 9 Projective bundles and their Chow rings  If we work over the complex numbers, we can see directly that is locally trivial in the analytic topology (and the same argument would work more generally for the 2 ́ P be a smooth conic. Choose lines L;M C 2 X such that L is etale topology). Let  0 \ and M \ L transverse to C C D ¿ . Over a sufficiently small analytic neighborhood 0 0 2 U 2 X we can solve analytically for a point q C of C \ L . The restriction of Y 0 C 1 is isomorphic to U  P to as U -schemes by the maps projecting a fiber C from q U C M (see Figure 9.3). to The conclusion of this example may be interpreted as a theorem in polynomial algebra: It says that there does not a exist a rational solution to the general quadratic polynomial . In other words, there do not exist rational functions X.a;b;c;d;e;f / , Z.a;b;c;d;e;f / such that Y.a;b;c;d;e;f / and 2 2 2 0: C cZ C C dXY C eXZ C f YZ  bY aX This is a generalization of the statement that the roots of a quadratic polynomial in one variable are not expressible as rational functions of its coefficients, though much stronger: Polynomials in several variables have many more solutions than polynomials in one variable! The same is true of polynomials of any degree d > 1 in any number of variables (Exercise 9.51). The set of Brauer–Severi varieties over a given variety X , modulo an equivalence relation that makes the projective bundles trivial, can be given the structure of a group, Brauer group X . There is another avatar of this group, as the group of called the of O Azumaya algebras modulo those that are the endomorphism algebras of vector over X bundles. Understanding the Brauer groups of varieties is an important goal of arithmetic geometry. See for example Artin [1982] for more about Brauer–Severi varieties, and Grothendieck [1966a] or Serre [1979] for more on the Brauer group. 9.6 Chow ring of a Grassmannian bundle X is any smooth variety and E is a vector bundle of rank n on X . Suppose that E Grassmannian , we can form the Generalizing the projective bundle associated to G.k; E / of k -planes in the fibers of E ; that is, bundle  E / Df .x;V / j x 2 X;V  E X: g G.k; ! x (As with a single Grassmannian, we can realize E / as a subvariety of the projec- G.k; V k P tivization E / . G.k; E / that extends both .) There is a description of the Chow ring of the description of the Chow ring of a projective bundle above and the description of the Chow ring of G.k;n/ given in Theorem 5.26; we will explain it here without proof.  As in the projective bundle case, there is a S   E defined tautological subbundle is the vector space G.k; E / ; this is a rank- k bundle whose fiber over a point .x;V / on

361 3 Conics in Section 9.6 347 P meeting eight lines  tautological quotient bundle Q D  E . E /= S be the . Let . As in the case of  V x E // is generated as an A.X/ -algebra by the A.G.k; projective bundles, the Chow ring , and also by the classes . S / Chern classes c . To understand the relations they c Q / . i i satisfy, consider the exact sequence  Q ! E ! ! ! 0:  0 S By the Whitney formula c. E / / Q D c. : / S c. Q has rank n k , the Chern classes c Since . Q / vanish for l > n k , and as in the l projective bundle case (above) or the case of G.k;n/ (Theorem 5.26) this gives all the relations: n be a smooth variety, and let Theorem 9.18. be a vector bundle of rank X on X . Let E G D G.k; E / ! X is the bundle of k -planes in the fibers of E then If    l ı c. / E n ç A.X/Œ ;:::; k C ; G dim ; l > A.G/ D 1 k   C  1 C ̇ 1 2 k l  f denotes the component of  of codimension l g  where has degree k . and k X is smooth, as long as In fact, the same formula holds without the assumption that one has developed the theory of Chern classes on singular varieties, as in Fulton [1984, Chapter 3] <  < r 0 < r < rank E , One can go further and, fixing a sequence of ranks 1 m flag bundle F .r is the set of all ;:::;r ; E / whose fiber over a point of X consider the 1 m flags of subspaces of the given ranks in E . There is again an analogous description of the Chow ring of this space. See Grayson et al. [2012] for this result and an interesting proof that is in some ways more explicit than the one we have given, even in the case of A.G.k;n// . 3 P meeting eight lines 9.7 Conics in 3 The family of plane conics in is naturally a projective bundle, and we will P now use this fact, together with Theorem 9.6, to compute the number of such conics 3 intersecting each of eight general lines L ;:::;L  P . 1 8 We start by checking that we should expect a finite number. There is a three- 3 P , and a five-parameter family of conics in each. Since parameter family of planes in two distinct planes intersect only in a line, the space of conics, whatever it is, should . 3 C 5 D 8 have dimension

362 348 Chapter 9 Projective bundles and their Chow rings 3 D L  P has codimension 1 in Next, the locus of conics meeting a given line L 3  is the image of the map given by .F C ;F the space of conics: If ;F P ;F , the / 3 1 0 2 meet the line Z 0 D Z have a common zero. D condition that is that F F C and 0 1 0 1 More geometrically: A one-parameter family of conics sweeps out a surface that meets L in a finite set, so a curve in the space of conics will intersect the locus of conics meeting a finite number of times. It is reasonable, then, to ask whether there is only a finite L number of conics that meet each of eight general lines and, if so, how many there are. 3 We will proceed as follows. First, as a parameter space for conics in P , we will 3  Q ! , whose points correspond to pairs .H;C/ with H a use a projective bundle P 3 H and C a conic in plane in P ; we will use the theory developed earlier in this chapter to calculate in its Chow ring. In particular, we will identify the class 2 A. Q / of the ı 8 deg of conics meeting a given line L , and compute the number Q ı D , our cycle  L L candidate for the number of conics meeting eight given general lines . i D meet To prove that this number is correct, we must show that the cycles L i transversely, and this requires a tangent space calculation. To do this, we will show 3 is in fact isomorphic to the Hilbert scheme H D H of / that our bundle Q P . 1 2m C 3 P having Hilbert polynomial p.m/ D 2m C 1 . This will allow us to subschemes of D prove the necessary transversality by describing the tangent spaces to in terms of the L general description of the tangent spaces to Hilbert schemes from Theorem 6.21; this is a special case of an important general principle explained in Exercise 9.60. 9.7.1 The parameter space as projective bundle 5 P Since the conics in a given plane naturally form a , and each conic is contained in 3 5 3  P a unique plane, it is plausible that the set of all conics in -bundle over P P , the is a 3 projective space of planes in . P 3  P , To make this structure explicit, consider the tautological exact sequence on which we may write as ;x / ;x .x ;x 0 1 2 3 4 ! .1/ 0: O ! O 0 ! S !  3  3 P P 3 3  3 4 O P P : S / D The projective bundle  P  P . is the family of 2-planes in P  3 P 3 3  over a point a D .a defined by ;:::;a is the plane / 2 P the fiber of P S H P  a 3 0 P  a D 0 . The dual S x is thus the space of linear forms on this plane, and, setting i i a 2  E . S WD / , the fiber of P E over the point a may be identified with the set of conics Sym 3 H . Note . We will therefore take Q D P E as our parameter space for conics in P in a 3 tautological family of conics in P that there is a 3  P E   P  P S X P E  3 P whose points are pairs consisting of a conic in a 2-plane and a point on that conic, with 3  3 and to P . P projections both to

363 3 Conics in meeting eight lines Section 9.7 349 P From the dual of the exact sequence above, we derive an exact sequence 4 4 2 0 ! O O ̋ . 1/ ! Sym . O 0: ! E / !  3 3   3 P P P 3  4 by , then, taking into account that P ! D 0 , If we denote the tautological class on ! the Whitney formula (Theorem 5.3) yields 4 2 3 4! c. E D 1 C / C 10! D C 20! 1=.1 : !/ 1 . Letting  2 A Q . We can now apply Theorem 9.6 to describe the Chow ring of / be the Q  O , we get .1/ of the pullback of E first Chern class of the tautological quotient to Q E P 3  6 5 2 4 3 3 C 4! A. C 10! P Q  C 20! /  /Œç=. / D A. 6 4 5 2 4 3 3 D C 10! ;  Z C 20! Œ!;ç=.!  4! /: C 9.7.2 The class of the cycle of conics meeting a line ı 1 of the divisor A ı . We next compute the class / 2 D D D using the technique of Q L undetermined coefficients. We know that ı D a! C b for some pair of integers a and b , and restricting to curves in Q a and b . Let Ä  Q be the gives us linear relations on 3 f  H g of conics in a general plane H  P C curve corresponding to a general pencil  ˆ Q be the curve consisting of a general pencil of plane sections f H  \ Q g and let  Q . We denote their classes in A . Q / by of a fixed quadric and ' respectively. 1 We claim that the following table gives the intersection numbers between our divisor !;;ı , and the curves : classes Ä;ˆ !  ı 1 1 0 ' 0 2 1 ' The calculation of the five intersection numbers other than is easy, and we leave to the reader the pleasure of working them out (Exercise 9.54). We can compute ' as the degree of the restriction of the bundle .1/ to the curve O P E ˆ ; equivalently, to show that ' D 0 we must show that T D O . ˆ . 1/ is trivial on E P 3 is a pair , with H a plane in Q .H;/ and  a one- To see this, recall that a point of P 0 dimensional subspace of . O H .2// ; the fiber of T over the point .H;/ is the vector H 0  . Now, if F 2 H space . O is the homogeneous quadratic polynomial defining .2// 3 P Q , we see that the restrictions of F to the planes H give an everywhere-nonzero section  of T over ˆ , proving that T j is the trivial bundle, as required. ˆ Given the intersection numbers in the table above, we conclude that ı D 2! C : There is also a direct way to arrive at this class, which we will describe in Exercise 9.55.

364 350 Projective bundles and their Chow rings Chapter 9 8 9.7.3 The degree of ı 8 i j  , we need to know the degrees of the monomials of degree 8. ! To compute ı 3  4 3 P is the class of a fiber of Q ! 0 ! D and  To start with, we have ! , and, since restricts to the hyperplane class on this fiber, we have 3 5  .! D 1: deg / 2 6 , we use the relation To evaluate the next monomial !  3 3 5 4 6 2 10! 20!   D  4! of Theorem 9.6, which gives 3 2 3 6 4 2 5 2  ! deg 10! D  / 20!  4! .! / D 4: . deg The same idea yields 7 8 D 6 and deg  .! D 4: deg / Putting these together we obtain         8 8 8 8 5 2 6 8 3 7 C 4 92: C  2 ! /  D C 8 D deg !  ! / C ..2! deg 3 2 1 3 i j Q ! P Writing for the projection, the numbers deg .!   W / computed above k may be interpreted (via the push-pull formula) as the degrees of the classes , which    of the bundle . See Definition 10.1 and, for an alternative Segre classes are called E computation, Proposition 10.3. 9.7.4 The parameter space as Hilbert scheme  ƒ is a smooth plane conic then the Hilbert polynomial of C is p.m/ D If C 3 1 . Let H WD H with this Hilbert 2m C be the Hilbert scheme of subschemes of P 1 2m C 3 ! H  polynomial, and let be the universal family. We have already described the C P 3 ! Q  P , and by the universal property of the tautological family of plane conics X  . Q ! H such that X D W  1/ C . Hilbert scheme there is a unique map 3 Q with its universal family X ! Q  P Theorem 9.19. is isomorphic to H with its 3 universal family H  P C via the map . ! C with We postpone the proof to develop a few necessary facts about subschemes p.m/ D 2m Hilbert polynomial 1 . To show that C is really a conic, we first want to C show that C is contained in a plane ƒ — that is, there is a linear form vanishing on C . 3 Since the number of independent linear forms on is 4 D p.1/ C 1 , it suffices to show P that the value h of the .1/ of the Hilbert function of C — that is, the dimension .S / 1 C C degree-1 part of the homogeneous coordinate ring of C — is equal to p.1/ .

365 3 Conics in meeting eight lines Section 9.7 351 P ƒ vanishes Once this is established we must show that a nonzero quadratic form on , and it suffices, for similar reasons as above, to show that C .2/ D dim .S h / on D 2 C C 3 is any subscheme with Hilbert . In fact, we will prove that if C  P 5 p.2/ D of 2m 1 , then the Hilbert function h polynomial .m/ D C is equal to p.m/ p.m/ C C for all m . This is contained in the following result: n  P Let be a subscheme, and let I Proposition 9.20. be its ideal sheaf and C C ç=I k Œx ;:::;x S its homogeneous coordinate ring. D 0 n C D S is p (a) .m/ If the Hilbert polynomial of 2m C 1 , then the Hilbert function of C C C is also equal to 2m S 1 . C (b) is the complete intersection of a unique 2-plane and a (non-unique) quadric C hypersurface. 1 (c) I H .m// D 0 for all m  0 . . C The form of the Hilbert polynomial implies that has dimension 1 and degree 2. C Proof: is a subscheme of degree 2 in the plane, Df D Ä g\ C x Thus a general plane section 0 Ä is either two distinct points or one double point. In either case, the Hilbert function of , we .m/ D 2 h m  1 . Writing S C for the homogeneous coordinate ring of for all C Ä S ! S whose kernel contains xS , whence have a surjective map Ä C C 2 .m/ h h D .m 1/  h .m/ Ä C C for m  1 . Since h , and .0/ D 1 , it follows that h 0 .m/  2m C 1 for all m  C C implies the same for all larger values. Since that a strict inequality for any value of m .m/ , the inequality above must be an equality for p h .m/ D 2m C 1 for large m D C C all 1 , proving the first statement.  m h D 3 , we see that C is be contained in The second statement follows. From .1/ C ƒ . From h , we see that .2/ D 5 a unique plane C lies on five linearly independent C quadrics; since at most four of these can contain , we see that C lies on a quadric ƒ 3 0 also has Hilbert function not containing ƒ . The subscheme C Q WD ƒ \ Q  P 0 C , and since C  C they are equal. 1 2m To prove the last statement, we use the long exact sequence in cohomology 0 0 0 1 1 n n H H . O . . .m// ! H I .m//: O O .m// ! H .m// . I . .m// ! H ! C C C P P 0 0 . I Since the last term is zero and the cokernel of the map .m// ! H H . O is .m// 3 C P 0 in S . But , it suffices to show that h the component of degree m O 1 .m// D 2m C . C C as C is defined in the plane by a quadratic hypersurface, we have also a sequence 1 0 0 0 .m// O . .m 2// ! H 2//; . O H ! .m// ! H .m . O O 0 H . ! 2 2 2 C P P P

366 352 Chapter 9 Projective bundles and their Chow rings O and, since the twists of have no intermediate cohomology, we get 2 P     m 2 C m 0 0 0 .m// O h . . O D h .m// O . h 2m 2// 1; D D .m C 2 2 C P P 2 2 as required. 3  H  P Proof of Theorem 9.19: over closed By Proposition 9.20, the fibers of C 3 3 P points of . Since this is also true for X  Q  P are precisely the distinct conics in , H the map W Q ! H is bijective on closed points. Q H is smooth. From the bijectivity Since is smooth, it now suffices to prove that , we see that dim H D dim Q D 8 , so it suffices, in fact, to prove that the tangent of H ŒCç space to has dimension 8. By Theorem 6.21, there is an isomorphism at each point 0 N H . T Š is a complete . Using Proposition 9.20 again, we know that / C 3 H ŒCç= P C= intersection of a linear form and a quadric. Thus N , and j .2// D . O O ̊ .1/ 3 3 3 C P C= P P 0 0 O the dimension of the tangent space is h C h .1// . O . .2// . C C 1 m . I By Proposition 9.20, , so the desired value is the H .m// D 0 for all 3 C= P sum of the values of the Hilbert function of C at 1 and at 2. Putting this together, we get T .2 8 D .2  1 C 1/ C dim  2 C 1/ D H ŒCç= as required. 9.7.5 Tangent spaces to incidence cycles To prove that the D intersect transversely we need to compute their tangent spaces L i L at the points of intersection. This task is made easier by the fact that, for general , i the intersection of the D of smooth conics, as we shall takes place in the locus U L i now prove: 3 L ;:::;L  P For a general choice of lines , no singular conic meets Lemma 9.21. 8 1 all eight. The family of singular conics has dimension 7, and the family of lines meeting a Proof: line, or a singular conic, has dimension 3. Thus the family consisting of 8-tuples of lines meeting a singular conic has dimension 7 C 3  8 D 31 , while the family of 8-tuples of lines has dimension 8  4 D 32 . Next we describe the tangent spaces to the cycles at points in U . Again, we use D L Q Š H at a point ŒCç corresponding to a conic the computation of the tangent space to 0 . T / N D H C . as 3 H ŒCç= C= P

367 3 Conics in Section 9.7 353 P meeting eight lines L C .p/ p L p , then a deformation of C Figure 9.4 If at a point is a conic meeting a line C D corresponding to a normal section if and only if .p/ is tangent to L .  remains in L 3 H  be a line and D Let  L the locus of conics meeting L . P Proposition 9.22. L 3 is a single reduced point,  is a smooth plane conic such that C \ L D f p g P If C D is smooth at ŒCç , and its tangent space at ŒCç is the space of sections of the then L p lies in the normal direction spanned by L ; that is, normal bundle whose value at   ˇ C T C T L p p 0 ˇ .p/ /  2 H 2 D . N D : T 3 L ŒCç P C= T C p See Figure 9.4 for an illustration. Proof: We prove Proposition 9.22 by introducing an incidence correspondence: For 3  P a line, we let L p Df .p;C/ 2 L  H j ˆ 2 C g : L The image of ˆ of under the projection  to the second factor is the cycle D H  2 L L L at the point .p;C/ is . By Lemma 6.23, the tangent space to conics meeting ˆ L 0 mod   .p/ j / : g C T . N Df  T ˆ T 2 .;/ L H 3 p p L .p;C/ C= P In particular, will carry its tan- will be smooth at .p;C/ ˆ  , and the projection 2 L 0  2 H 2 . N gent space injectively to the space of sections .p/ such that / 3 P C=  is one-to-one over T C/=T C . Since the map C .T L p , it follows that D is p p p 2 L smooth at ŒCç with this tangent space. This argument also applies to Hilbert schemes in a more general context; see Exercise 9.60. 3 Let C Corollary 9.23. . If L P ;:::;L be a smooth conic in are general lines meeting 8 1 at general points, then the cycles . ŒCç ;:::;D meet transversely at C  Q Š H D L L 8 1 Proof: By Proposition 9.22, it suffices to show that the eight linear conditions speci- fying that a global section of the normal bundle of C lie in specified one-dimensional subspaces at eight points of C are independent, for a general choice of the points and the subspaces. Since the rank of the normal bundle is 2, this is a special case of Lemma 9.24, proved below.

368 354 Chapter 9 Projective bundles and their Chow rings 0 Let X , and let V  H E . E / Lemma 9.24. be a vector bundle on a projective variety  ;:::;p 2 X are general points and V be a vector space of global sections. If p E p i 1 k i in the fiber E a general linear subspace of codimension , then the subspace of E 1 p at i p i / 2 V j .p Df  2 V W g has dimension i i W D max dim 0; dim .V / k g : f The obvious analog of this result fails if we allow V codim > 1 ; see Exercise 9.53. i Proof: D 1 , and note that if the Proceeding inductively, it suffices to show the case k had value in every hyperplane V  E at a dense set of points general section in V i p 2 X then V D p . 0 9.7.6 Proof of transversality 3 D Q ;:::;L   P If Proposition 9.25. L are eight general lines, then the cycles 8 1 L i intersect transversely. Proof: To start, we introduce the incidence correspondence 8 ¤ Df : I C/ 2 G .1;3/ †  Q j C \ L g ;:::;L ¿ for all i .L i 1 8 3  P L meeting a given smooth conic C is an irreducible hyper- Since the locus of lines G .1;3/ , we see via projection to Q that † is irreducible of surface in the Grassmannian dimension 32. Now, let  † be the locus of .L fail to ;:::;L † I C/ such that the cycles D 0 1 8 L i intersect transversely at ; this is a closed subset of † . By Corollary 9.23, † ŒCç ¤ † , 0 8 dim † , so for a general < 32 . It follows that † so does not dominate G .1;3/ 0 0 8 D 2 G .1;3/ / the cycles .L ;:::;L point are transverse at every point of their 1 8 L i intersection. In sum, we have proved: 3 There are exactly 92 distinct plane conics in P meeting eight general Theorem 9.26. lines, and each of them is smooth. As with any enumerative formula that applies to the general form of a problem, the computation still tells us something in the case of eight arbitrary lines. For one thing, it 3 eight lines, there will be at least one conic meeting ;:::;L says that if P L are any  1 8 all eight (here we have to include degenerate conics as well as smooth), and, if we assume that the number of conics meeting all eight (again including degenerate ones) is finite, then, assigning to each such conic C a multiplicity (equal to the scheme-theoretic degree T  D D , since the cycles of the component of the intersection H supported at ŒCç L L i are Cohen–Macaulay), the total number of conics will be 92. In particular, as long as the number is finite, there cannot be more than 92 distinct conics meeting all eight lines.

369 Exercises 355 Section 9.7 3 P , In Exercises 9.56–9.68 we will look at some other problems involving conics in including some problems involving calculations in / A. , some other applications of H the techniques we have developed here and some problems that require other parameter spaces for conics. 9.8 Exercises a Exercise 9.27. ;x ;:::;x Choosing coordinates on P corresponding to the mono- x 1 0 a 1 a a a ;s mials s 2  2 minors of the matrix ;:::;t , show that the   x  x x a 1 1 0 x x x  1 2 a . By working in local coordinates, vanish identically on the rational normal curve S.a/ show that the ideal I generated by the minors defines the curve scheme-theoretically. k Œx ;x Find a set of monomials forming a basis for the ring , and show that ;:::;x ç=I a 1 0 1 ad 1 . By comparing this with the Hilbert function of P , d C in degree it has dimension prove that I is the saturated ideal of the rational normal curve. Exercise 9.28. In order to do the same as we did in the previous exercise for surface a C b C 1 scrolls, prove that the Hilbert polynomial f P .d/ of the surface scroll S.a;b/  S satisfies   d C 1 f .d/  .a C b/ C d C 1: S 2 a C b C 1 be coordinates in P Let x  ;:::;x Exercise 9.29. . Prove that the 2 2 0 C b C 1 a minors of the matrix   x x x x   x x C 1 a C 2 1 1 a 0 a b C a x x  x x x  x 2 a a C 3 a 2 1 C b 1 C a C S.a;b/ . As in Exercise 9.27, show that the ideal vanish on a surface scroll generated I by the minors defines the surface scheme-theoretically. Then, using Exercise 9.28, prove that I is the saturated ideal of the surface scroll. Exercise 9.30. Let X be a smooth projective variety, E a vector bundle on X and P X its projectivization. Let L be any line bundle on X ; as we have seen, there is a ! E P E Š P . E ̋ L / , such that natural isomorphism   .1/ O O L .1/ ̋  Š : E P P . E ̋ L / Using the results of Section 5.5.1, show that the two descriptions of the Chow ring of agree. E D P P E ̋ L / .

370 356 Chapter 9 Projective bundles and their Chow rings Let  Y ! X be a projective bundle. Exercise 9.31. W A.X/ Show that the direct sum decomposition of the group given in Theorem 9.6 (a) Y Š depends on the choice of vector bundle E . with E P C i ̊ 1 ! A.X/ W (b) Show that if we define group homomorphisms by A.Y/ i i 7! . . ̨/; ̨ W . . ̨/;:::; ̨//;  i   A.Y/ given by then the filtration of  Ker . 0 /  Ker . D /  Ker . / A.Y/ . /  Ker 1 r 1 0 r E is independent of the choice of . A.Y/ Hint: . Give a geometric characterization of the cycles in each subspace of Exercise 9.32. In Example 9.17, we used intersection theory to show that there does not exist a rational solution to the general quadratic polynomial; that is, there do not exist rational functions Y.a;:::;f / and Z.a;:::;f / such that X.a;:::;f / , 2 2 2 bY C cZ C C dXY C eXZ C f YZ  0: aX To gain some appreciation of the usefulness of intersection theory, give an elementary proof of this assertion. Exercise 9.33. Let n .L;p/ 2 G .1;n/ ˆ P Df j p 2 L g  n P and , and let  be the pullbacks of the Schubert ,   be the universal line in 1;1 1 n 1 2 1 .1;n// G .1;n// ,  classes 2 A A . G 2 and the hyperplane class  2 A  . P . / 1;1 1 a b c   D  respectively. Find the degree of all monomials of top degree a C 2b C c 1;1 1 .ˆ/ 2n 1 . dim D 3 Exercise 9.34. of pairs consisting of a point p 2 P Consider the flag variety and a F 3 L  P ; that is, containing p line 3 3 : P F  G .1;3/ j p 2 L  P 2 g .p;L/ Df 1 2 3 -bundle over G .1;3/ may be viewed as a P P -bundle over P F . Calculate the , or as a Chow ring A. F / via each map, and show that the two descriptions agree. 3 .1;3/ Exercise 9.35.  G P is just the By Theorem 9.6, the Chow ring of the product tensor product of their Chow rings; that is 4 3 /:  G .1;3// D A. G .1;3//Œç=. P A. 3 F  of Exercise 9.34. In these terms, find the class of the flag variety  G .1;3/ P

371 Exercises Section 9.8 357 Generalizing the preceding problem, let Exercise 9.36. r 2 P F  G .1;r/ Df p 2 ƒ g : .0;k;r/ .p;ƒ/ j r  P G  Find the class of F . .0;1;r/ .1;r/ Generalizing Exercise 9.35 in a different direction, let Exercise 9.37. L Df 2 G .1;r/  G .1;r/ ˆ .L;M/ \ M ¤ ¿ g : j r Given that by Theorem 9.18 we have G .1;r/  G .1;r// Š A. G .1;r// ̋ A. G .1;r//; A. G ˆ G .1;r/  A. .1;r// for: in find the class of r D 3 . (a) r D 4 . r (b) (c) General r . n Exercise 9.38. Z be the blow-up of Z along an .r 1/ -plane, and let E  P be Let n ŒEç 2 A.Z/ . the exceptional divisor. Find the degree of the top power n n Z D Bl . P be the blow-up of Exercise 9.39. P Again let along an .r 1/ -plane ƒ ƒ In terms of the description of the Chow ring of given in Corollary 9.12, find the classes Z of the following: s (a) The proper transform of a linear space ƒ , for each s > r . P containing s P in general position with respect to ƒ The proper transform of a linear space (b) ƒ if s  n r , and transverse to ƒ if (that is, disjoint from r ). s > n s (c) In general, the proper transform of a linear space intersecting ƒ in an l -plane. P 3 3 D Bl along a line. In terms of the P Exercise 9.40. be the blow-up of P Let Z L description of the Chow ring of Z given in Corollary 9.12, find the classes of the proper 3 transform of a smooth surface P S of degree d containing L .  4 4 4 P Now let P D be the blow-up of Z Exercise 9.41. along a line, and let S  P Bl L be a smooth surface of degree d containing L . Show by example that the class of the proper transform of in Z is not determined by this data. For example, try taking S 4 D S.1;2/  P a cubic scroll, with S L either (a) a line of the ruling of S , or (b) the directrix of S , that is, the unique curve of negative self-intersection, and observe that you get different answers.

372 358 Chapter 9 Projective bundles and their Chow rings n n Let Exercise 9.42. P Z be the blow-up of P Bl along an .r 1/ -plane ƒ ; that D ƒ n r G .r;n/ of P -planes containing ƒ , we have  is, if we consider the subspace r r n n  Df P 2 .p;Ä/ j p 2 Ä g : Z P Z given in Corollary 9.12, find the class of Using the description of the Chow ring of n n r P  Z .  P Let C be a smooth curve, E a vector bundle of rank r on C and F ; Exercise 9.43.  E G two subbundles of complementary ranks and r s such that for general p 2 C the s F G are complementary in E . In terms of the Chern classes of the three fibers and p p p G 2 where F bundles, describe the locus of \ p : ¤ 0 C p p By using the result of Proposition 9.13 to calculate the class of the intersection (a) F in P G P P E . \ F G ! E . (b) By considering the bundle map ̊ To generalize the preceding problem: Let X be a smooth projective Exercise 9.44. F E r on X and variety of any dimension, ; G  E subbundles a vector bundle of rank of ranks a and b with a C b  r . Describe the locus of p 2 C where F , \ G 0 ¤ p p r C a b . assuming this locus has the expected codimension 1 We will see how to generalize this calculation substantially using the Porteous formula of Chapter 12; see Exercise 12.11. The following three exercises show one way to construct a surface with infinitely many reduced and irreducible curves of negative self-intersection. 2 and G be two general polynomials of degree 3 in P Let , and let F Exercise 9.45. t D V.t ;:::;p F C be the ;p G/ g p be the associated pencil of curves; let f C 1 1 1 2 0 9 t 2 P t 1 t P (that is, for all but countably base points of this pencil. Show that for very general 2 1 ) the line bundle O . p .p / many .C / is not torsion in Pic .C t / D A t 2 t 1 C t Exercise 9.46. Now let S be the blow-up of the plane at the points p — that is, ;:::;p 9 1 2 1 - P given by .F;G/ — and let P the graph of the rational map ;:::;E be E 9 1 ' W S ! the exceptional divisors. Show that there is a biregular automorphism that S 1 ! P commutes with the projection and carries E . S E to 2 1 Exercise 9.47. Using the result of Exercise 9.45, show that the automorphism ' of Exercise 9.46 has infinite order, and deduce that the surface S contains infinitely many irreducible curves of negative self-intersection. An amusing enumerative problem: In the circumstances of the preceding Exercise 9.48. 1 t 2 P be torsion of order 2 — that is, for will O exercises, for how many / .p p 1 2 C t how many t will O / .2p ? O / Š .2p 1 2 C C t t

373 Exercises Section 9.8 359 Let C g  2 over a field of characteristic Exercise 9.49. be a smooth curve of genus n  ! C be the Frobenius morphism. If Ä W  C C C is the graph of ' ' p > 0 ; let n 1 2 ŒÄ goes to ç 2 A and  C/ its class, show that the self-intersection .C . D / deg n n n n 1 !1 as . E e on a smooth Show that if Exercise 9.50. is a vector bundle of rank 2 and degree projective curve L and M sub-line bundles of degrees a and b corresponding to X , and P E with classes  and  sections of , then 2 2 a b and   C  e D 2e 2a 2b: D 2 2 deg deg  L C M  are distinct then  0 , with equality holding if In particular, if and E D L ̊ M . and only if N n Exercise 9.51. Let d in P be the space of hypersurfaces of degree . Using the P d > 1 analysis of Example 9.17 as a template, show that for the universal hypersurface N N n X 2 P  g! P j p P 2 Df ˆ .X;p/ d;n admits no rational section. 4 Exercise 9.52. Consider the flag variety 2 P F and a of pairs consisting of a point p 4  containing p ; that is, ƒ P 2-plane 4 4 p 2 ƒ F P Df g P .p;L/  G .2;4/: j  2 4 -bundle over -bundle over G .2;4/ , or as a G .1;3/ F P may be viewed as a . Calculate P A. / via each map, and show that the two descriptions agree. the Chow ring F Exercise 9.53. Show that the analog of Lemma 9.24 is false if we allow the V to i >1 : in other words, if V is a general linear subspace of  E have codimension i p i 0 E W  H m . , then the corresponding subspace / need not have dimension codimension i ̊ P 0 . E / 0;h m max . i Consider a bundle whose sections all lie in a proper subbundle. Hint: Calculate the remaining five intersection numbers in the table of inter- Exercise 9.54. section numbers on page 349 of Section 9.7.2. 1 ı D ŒD / ç 2 A Exercise 9.55. . H To find the class of the cycle of conics meeting a L H U H of pairs .H;/ 2 H line directly, restrict to the open subset  does such that not contain L (since the complement of this open subset of H has codimension 2, any relation among divisor classes that holds in U will hold in H ). Show that we have a map D ̨ U ! L sending a pair .H;/ to the point p D H \ L , and that in U the divisor W L is the zero locus of the map of line bundles  ! ̨ .2/ O T L sending a quadric Q 2  to Q.p/ .

374 360 Chapter 9 Projective bundles and their Chow rings Let Å H be the locus of singular conics. Exercise 9.56.  is an irreducible divisor in (a) Show that H Å . 1 A . H / as a linear combination of ! and 2 .  (b) Express the class ı Use this to calculate the number of singular conics meeting each of seven general (c) 3 P . lines in (d) Verify your answer to the last part by calculating this number directly. 3 Exercise 9.57. P Let be a point and F p  H the locus of conics containing the 2 p 2 F point is six-dimensional, and find its class in A . Show that p H / . . p Exercise 9.58. Use the result of the preceding exercise to find the number of conics 3 and meeting each of six general lines in P , the number of passing through a point p 3 p;q P conics passing through two points , and meeting each of four general lines in p;q;r and meeting each of two and the number of conics passing through three points 3 P . Verify your answers to the last two parts by direct examination. general lines in 3 A . Find the class in / of the locus of double lines (note that this is H Exercise 9.59. five-dimensional, not four!). n X  P a is a subscheme of pure dimension l and H Exercise 9.60. Suppose that n P of pure dimension component of the Hilbert scheme parametrizing subschemes of n n P P ; let ŒYç 2 H be a smooth point corresponding to a subscheme k < n  l in Y such that \ X Df Y g is a single reduced point, and suppose moreover that p is a smooth p point of both X and Y . Finally, let † .  H be the locus of subschemes meeting X X Use the technique of Proposition 9.22 to show that †  H is smooth at ŒYç , of X the expected codimension k l , with tangent space n   ˇ Y T T C X p p 0 ˇ n : / † D . T  N H 2 .p/ 2 X ŒYç P Y= T Y p The next few problems deal with an example of a phenomenon encountered in the preceding chapter: the possibility that the cycles in our parameter space corresponding to the conditions imposed in fact do not meet transversely, or even properly. 3 E H P Exercise 9.61. be a plane, and let  be the closure of the locus  H Let H 3 C  of smooth conics tangent to H . Show that this is a divisor, and find its class P 1 / ç 2 Œ E . H . A H 3 Exercise 9.62. Find the number of smooth conics in P meeting each of seven general 3 3 H ;:::;L  P L and tangent to a general plane . More generally, for  P lines 7 1 3 D 1;2 and 3 find the number of smooth conics in P k meeting each of 8 k general 4 3 and tangent to P  ;:::;H H general planes k . L lines  ;:::;L P 1 1 k 8 k

375 Exercises Section 9.8 361  4 For , why do the methods developed here not work to calculate the Exercise 9.63. k 3 3 8 number of smooth conics in general lines L P ;:::;L k meeting each of P  1 k 8 3 general planes H ? What can you do to find these ;:::;H and tangent to  P k 1 k numbers? (In fact, we have seen one way to deal with this in Chapter 8.) 4 P : Next, some problems involving conics in 4 Exercise 9.64. P Now let (again, defined to be complete K be the space of conics in 5 as a P -bundle intersections of two hyperplanes and a quadric). Use the description of K G to determine its Chow ring. over the Grassmannian .2;4/ Exercise 9.65. In terms of your answer to the preceding problem, find the class of the D of conics meeting a 2-plane locus ƒ , and of the locus E of conics meeting a line L ƒ 4  P L . 4 Exercise 9.66. meeting each of 11 general Find the expected number of conics in P 4 ;:::;ƒ 2-planes ƒ P  . 11 1 Exercise 9.67. Prove that your answer to the preceding problem is in fact the actual num- 4 ber of conics by showing that for general 2-planes ;:::;ƒ the corresponding  P ƒ 11 1 cycles D intersect transversely. ƒ i Finally, here is a challenge problem: 3 Let f S be a general pencil of quartic surfaces (that is, take  P Exercise 9.68. g 1 t t P 2 3 B general homogeneous quartic polynomials, and set S D ). A A C t B/  P and V.t 1 0 t How many of the surfaces S contain a conic? t

376 Chapter 10 Segre classes and varieties of linear spaces Keynote Questions n n (a) v be general tangent vector fields on Let ;:::;v . At how many points of P P 2n 1 is there a nonzero cotangent vector annihilated by all the v ? (Answer on page 366.) i If f is a general polynomial of degree d D 2m 1 in one variable over a field of (b) f m d -th powers as a sum of characteristic 0, then there is a unique way to write of linear forms (Proposition 10.15). If f and g are general polynomials of degree D 2m in one variable, how many linear combinations of f and g are expressible d m d -th powers of linear forms? (Answer on page 377.) as a sum of 4 If C  P (c) is a general rational curve of degree d , how many 3-secant lines does C have? (Answer on page 379.) 3 C  P (d) is a general rational curve of degree d , what is the degree of the surface If swept out by the 3-secant lines to C ? (Answer on page 380.) 10.1 Segre classes Our understanding of the Chow rings of projective bundles makes accessible the computation of the classes of another natural series of loci associated to a vector bundle. E is a vector bundle on a scheme X and We start with a naive question. Suppose that that E is generated by global sections. How many global sections does it actually take to generate ? More generally, what sort of locus is it where a given number of general E global sections fail to generate E locally? We can get a feeling for these questions as follows. First, consider the case where E is a line bundle. In this case, each regular section corresponds to a divisor of class c . . E / 1 If E is generated by its global sections, the linear series of these divisors is base point free, so a general collection of i of them will intersect in a codimension- i locus of class

377 Segre classes Section 10.1 363 i E . . That is, the locus where i general sections of / fail to generate E has “expected” c E 1 i and class . E / c . codimension i 1 has rank ; again, suppose that it is generated by global Now suppose that E r > 1 0 general sections, and let be the codimension-1 subset of E r sections. Choose X where the sections do not generate E . One can hope that at a general consisting of points p 0 the sections have only one degeneracy relation, so that on some open set U  point of X 0 0 0 they generate a corank-1 subbundle of E , and the quotient E = E  is a line bundle E X 0 . The sections of E yield sections of E = E , so if it is a line bundle they will vanish U on U ; that is, we should expect r C 1 general sections of in codimension 1 in to generate E E X . Continuing in this way (and assuming that away from a codimension-2 subset of i  r C i 1 sections of E might generate E away from a codimension- ), it seems that i r r C dim X sections might generate E locally everywhere. locus. In particular, r D O 1 C .1/ A case beloved by algebraists is that of . Here a collection of r E i V P general sections is a general map ' C i 1 r r ; ! O O .1/ V P V P r  .r C i that is, a general 1/ matrix of linear forms. The locus where the sections fail to generate is the support of the cokernel, which is defined by the r  r minors of the matrix. By the generalized principal ideal theorem (Theorem 0.2), the codimension of this locus is at most i , and in fact equality holds (as we shall soon see) whenever r C  i . In fact, the support of the cokernel is exactly the scheme defined by the ideal 1 of minors in this case (see Buchsbaum and Eisenbud [1977]). It turns out that the construction of projective bundles gives us an effective way of reducing this question (and many others) about vector bundles to the case of line bundles, P to the line bundle O .1/ on passing from E . To relate this line bundle to classes E P E , we push forward its self-intersections: on X Let X be a smooth projective variety, let E be a vector bundle of rank r Definition 10.1. X and  W P E ! X its projectivization, and let  on c -th Segre . O i . The .1// D E 1 P of is the class class E i r 1 C i s .X/; . E / D  / 2 A .  i and the (total) Segre class of E is the sum : s. D 1 C s E . E / C s / . E / C 2 1 (For a more general definition of the Segre classes, see Fulton [1984, Chapter 4].) The Segre classes give the answer to our question about generating vector bundles: Let E be a vector bundle of rank r on a smooth variety X that is Proposition 10.2.  ;:::; generated by global sections, and let be general sections. If X is the i 1 1 r i C and scheme where  ;:::; i has pure codimension X , then E fail to generate 1 i i C r 1 i ŒX / ç . . 1/ the class s E . is equal to i i

378 364 Chapter 10 Segre classes and varieties of linear spaces i . s . E / is represented by a 1/ We will prove here only the weaker statement that i X positive linear combination of the components of ; the stronger version is a special i (Theorem 12.4), which will be proved in full in Chapter 12. Porteous’ formula case of The proposition shows an interesting parallel between the Chern classes and the Segre classes of a bundle:  -th Chern class c The . E / is the locus of fibers where a suitably general bundle map i i ̊ i C 1 r ! E O X fails to be injective. i  -th Segre class s times the locus of fibers where a suitably general The E / is . 1/ i . i bundle map ̊ C i 1 r ! O E X fails to be surjective. The Segre classes may seem to give a way of defining new cycle class invariants of a vector bundle, but in fact they are essentially a different way of packaging the information contained in the Chern classes. Postponing the proof of Proposition 10.2 for a moment, we explain the remarkable relationship: The Segre and Chern classes of a bundle E Proposition 10.3. X are reciprocals of on one another in the Chow ring of : X 1 E E s. D /c. 2 A.X/: /  i , we deduce that . E . / D . 1/ Using the formula c / c E i i i  . / D . 1/ E s s . E /: i i 0 ! E ! F ! G ! Also, for any exact sequence of vector bundles, the 0 Whitney formula gives F / D c. E /c. G / , whence c. F / D s. E /s. G /: s. Proof of Proposition 10.3: If S and Q are the tautological sub- and quotient bundles  / and  D c P . S on / is the tautological class, then c. S E D 1  , so by the Whitney 1 formula  c. E / 2  / D c. Q  /: c. E E /.1 C  C D P C / 2 A. c. S /

379 Segre classes Section 10.1 365 X . Considering first the left-hand side, we see that We now push this equation forward to > 1 i < r . Q / is represented by a cycle of dimension c dim X , so it for the Chern class i maps to 0, while the top Chern class maps to a multiple of the fundamental class c . / Q 1 r 1  .c. Q of D X 2 A.X/ . — in fact, we saw in Lemma 9.7 that the multiple is 1. Thus //  On the other hand, the push-pull formula tells us that 2  2 .1 C  C  .c. C // D c. E /    E C  C  /.1 C /   D c. E /s. E /; completing the argument. n r n P and E D . O D X .1// For example, if , then P     1 1 2 r 1 C r C 3 2 / E D s. C  D : D 1  r C r 2 3 / C .1 / E c. 0 V V Let . E / ; suppose that dim H D n . Since E is Proof of Proposition 10.2: D ' W X ! G.n r;n/ generated by global sections, we have a natural map sending each point 2 X to the kernel of the evaluation map V ! E p , that is, the subspace p  is the pullback p . Via this map, E of sections of ' vanishing at Q of the universal E quotient bundle on G.n r;V / , and by Section 5.6.2 we have correspondingly   c D ' . .c. Q // D ' / . E /: i i In fact, we can see this directly: If ;:::;  , the E are general sections of V 2 1 i 1 r C locus where they fail to be independent will be the preimage of the Schubert cycle .W / † W  V is the span of , where V ;:::;  W , and, since the plane  C 1 i r 1 i 1 Œ' .† .W //ç of the preimage is the is general, by Kleiman transversality the class i Œ† D  . pullback of the class .W /ç i i  , the scheme ;:::; In the same way, if E are general sections of V 2 1 C r 1 i where they fail to span will be the preimage of the Schubert cycle .W / D X † i i 1 ; again, we can invoke ;:::; , where W  V is the span of  † .W / 1 r C i 1 1;1;:::;1 Kleiman to deduce that the X i and that have pure codimension i  D ' . ç /: ŒX i i 1 Finally, we saw in Corollary 4.10 that in the Chow ring of the Grassmannian we have .1 C  C /.1   C   / D 1; C 1 2 1;1 1 and combining these we have   X X 1 1 i  i  P 1/ . 1/ /;  E s. . ŒX ' ç D ' D D D i i 1 c. E /  i as desired.

380 366 Segre classes and varieties of linear spaces Chapter 10 n We can now answer Keynote Question (a). A tangent vector field on is a section P n  n P T of , so the question can be rephrased as: At how many points of D  do 2n n P P n n n T general sections of 1/ ? By Proposition 10.2, this is . fail to generate T times P P n n n / the degree of the Segre class / . s. T / T s D 1=c. T . And, as . By Proposition 10.3, n P P P n C 1 n / D .1 C / c. T , where we have seen (in Section 5.7.1),  is the hyperplane class P n on . Putting this together, P   1 n 2 C 2 n / T s. D D 1 .n C 1/ C  C ; P 1 n C 2 C .1 /  2n so the answer is . n 10.2 Varieties swept out by linear spaces We can use Segre classes to calculate the degrees of some interesting varieties “swept out” by linear spaces in the following sense. Let B be a smooth variety of n and ̨ W B ! G D G .k;n/ a map to the Grassmannian of k -planes in P dimension , m and let [ n X D P  ƒ ̨.b/ B 2 b n S B . Let P be corresponding to the points of the image of be the union of the planes in and G the universal subbundle on n P S Df .ƒ;p/ 2 G  P g j p 2 ƒ ˆ D k the universal -plane. Form the fiber product n D B  ; ˆ Df .b;p/ 2 B ˆ P  j p 2 ƒ g G B ̨.b/ with projection maps   n ˆ ! P B ; B so that we can write X .ˆ /: D B ˆ will be a Since m C k , we see from this that X is necessarily a variety of dimension B n P — of dimension at most m C k . In case it has dimension equal to m C k subvariety of that is, the map is generically finite of some degree d , or in other words a general point  of X lies on d of the planes ƒ by the — we will say that X is swept out d times ̨.b/ ƒ . planes ̨.b/ Assuming now that X has the “expected” dimension m C k , we can ask for its n . This can be conveniently expressed as the degree of a Segre class: degree in P

381 Secant varieties Section 10.2 367 Let Proposition 10.4. G .k;n/ be a smooth projective variety of dimension m , B   S D G .k;n/ any morphism and E D ̨ B G the pullback of the universal ̨ W ! . If G subbundle on [ n X P D  ƒ ̨.b/ 2 B b d times by the planes corresponding to points of B , then is swept out D deg .s .X/ . E //=d: deg m n 0 n Proof: L . O H 2 .1// is a homogeneous linear form on P If , then L defines a P  section of by restriction to each fiber of E D S E , and hence a section  . of O .1/ E P L B n 1 .H/  L H D V.L/  P The preimage given by of the hyperplane is the zero B n . Thus the pullback of the hyperplane class on P  under the map  is the locus of L B k C m deg c .1// on P E , and it follows that d  O .X/ D deg  D  tautological class D . E 1 P s E / , as required. . deg m is the number of points of its X Alternatively, we could argue that the degree of n m k/ -plane Ä  P intersection with a general ; since the class of the cycle .n m 1=d  G of † -planes meeting Ä is the Schubert class  .Ä/ 2 A k .G/ , this is m m  ̨  . Thus we have times the degree of the pullback m  d .X/ D deg ̨   deg m  Q / D deg c . ̨ m D s . E / deg m s. S / D 1=c. S / since c. Q / . D 10.3 Secant varieties n X  P The study of secant varieties to projective varieties is a rich one, with a substantial history and many fundamental open problems. In this section, we will discuss some of the basic questions. In the following sections we will use Segre classes to compute the degrees of secant varieties to rational curves. 10.3.1 Symmetric powers n m n ƒ X in P A is a linear space -secant Š P m  P -plane to a variety of k dimension m that meets X in k points, so it will be useful to introduce a classical -th construction of a variety whose points are (unordered) -tuples of points of X : the k k .k/ symmetric power X of X .

382 368 Chapter 10 Segre classes and varieties of linear spaces k .k/ X X Formally, we define by -fold product to be the quotient of the ordinary k k S , acting on X letters by permuting the factors. the action of the symmetric group on k k D A is any affine scheme, this means that X If Spec .k/ S k ̋ A ̋ ̋ A/ X /: WD Spec ..A .k/ is quasi-projective, is defined by patching together symmetric powers When X X of affine open subsets of X . The main theorem of Galois theory shows that when X .k/ k /= is a variety the extension of rational function fields .X / is Galois, and of k .X k . degree kŠ .k/ X One can show that such quotients are ! Y categorical : Any morphism k X -invariant morphism determines an S ! Y , and this is a one-to-one correspondence. k Further, the closed points of correspond naturally to the effective 0-cycles on X : they X p are usually denoted additively by p CC , where the p need not be distinct. 2 X i 1 k For these results, see Mumford [2008, Chapter 12]. k .k/ .k/ X ! X X is affine or projective if and only Since the natural map is finite, if X is. 1 D X A familiar example is the case : Here, X D Spec k Œtç , so A .k/ 1 S k ;:::;t Spec . k Œt /: / A ç . D 1 k This ring of invariants is a polynomial ring on the k elementary symmetric functions k 1 .k/ A . D A (see for example Eisenbud [1995, Section 1.3] for an algebraic proof), so . / Set-theoretically, this is the statement that a monic polynomial is determined by the set of its roots, counting multiplicity. 1 1 P . We could deduce it from the case of A , but instead A similar result holds for we give a geometric proof: k 0 1 .k/ Š / H P . O P . .k// D P Proposition 10.5. . 1 P 1 0 .1// as P H Proof: . O , the space of linear forms in 2 variables modulo We think of P 1 P scalars. The product of k linear forms is a form of degree k , which is independent of the order in which the product is taken. Thus multiplication defines a morphism 1 k 0 P that is invariant under the group / ' ! P H W . O is ' .k// . S . The morphism 1 k P finite and generically -to-one, so it has degree kŠ . kŠ 1 k .k/ P . is invariant, it factors through a morphism ' / ! W Since , and, since P 1 1 k .k/ / the degree of the quotient map ! . P . / P is kŠ , we see that is birational. Since k P is finite and birational, is an isomorphism. is normal and .k/ X is most useful when X is a smooth curve. One reason is The construction of given by the following result: .k/ Proposition 10.6. X is a variety and k > 1 , then X If is smooth if and only if X is smooth and dim X  1 .

383 Secant varieties Section 10.3 369 .k/ dim 0 , then X consists of a single reduced point, and X X is also a Proof: If D . X > 0 single reduced point. Thus we may assume that dim Away from the subsets where at least two factors are equal, the quotient map k .k/ is an unramified covering. Thus if ! is singular at a point p , and X X X ;:::;q the variety q CC 2 p;q are distinct points, then near X C q p 1 1 k 1 k 1 .k/ k X near .p;q ;:::;q looks like the product X ; in particular, it is singular. Thus / 1 1 k .k/ is singular then X is singular. if X .k/ is smooth and of dimension  2 . If X Now suppose that were smooth as well, X k .k/ k ́  the quotient map would be ! etale away from the diagonal in X X , a W X  W locus of codimension at least 2. But the differential , being a map !  T T d .k/ k X X between vector bundles of equal rank, would necessarily be singular in codimension 1, a contradiction. .k/ is a smooth curve then X will be smooth. This in fact It remains to see that if X 1 P described in Proposition 10.5: in the analytic follows from the special case X D p on any smooth curve X have neighborhoods iso- topology, any collection of points i 1 .k/ , and it follows that any point of P has a neighborhood X morphic to open subsets of k isomorphic to an open subset of P . C are central to the analysis of the The symmetric powers of a smooth curve C , as we will see illustrated in Appendix D. We can think of a point of geometry of .k/ 0 0 D  C , and use notation such as D [ D C and D \ D as a subscheme accordingly. .k/ In fact, is isomorphic to the Hilbert scheme of subschemes of C with constant C k — that is, zero-dimensional subschemes of degree (see Arbarello Hilbert polynomial k .k/ X > 1 X is singular, a point on X does dim or et al. [1985] for a proof). When X not H in general determine a subscheme of .X/ are often , and the Hilbert schemes k more useful. 10.3.2 Secant varieties in general In this subsection we will prove a basic result related to the dimension of secant varieties. Then we will state without proof some general results that may help to orient the reader. In the following two sections we will prove a number of results about the secant varieties of rational curves. r  P Let be a projective variety of dimension n not contained in a hyperplane. X  m r C Since general points of X are linearly independent, for any m  r we have a 1 rational map .m/ -  W X G .m 1;r/; p called the m -tuple p to the span CC , sending a general secant plane map m 1 m 1 r Š P ;:::;x p is given by  P ;:::;p . (In coordinates: If p , then D .x /  i;r i m 1 i;0 the maximal minors of the matrix .x -planes / .) We define the locus of secant .m 1/ i;j — that is, the closure X to be the image ‰  .X/ to G .m 1;r/ of the rational map  m

384 370 Chapter 10 Segre classes and varieties of linear spaces linearly independent points .m of the locus of .m 1/ -planes spanned by m 1;r/ in G . Finally, the variety X of [ r  D Sec .X/ P ƒ m 2 .X/ ‰ ƒ m -th secant variety of X . is called the m .ƒ ƒ : If Caution and ƒ \ X is finite, then deg 2 \ X/  m , but the converse is ‰ m false; Exercise 10.24 suggests an example of this. .m/ - m > 1 then the secant plane map  W X If n > 1 G .m 1;r/ is and p never regular: When a point on a variety of dimension 2 or more approaches 2 X 2 , the limiting position of the secant line p;q necessarily depends on q X another point is a curve and q the direction of approach. (When X , the limit is X a smooth point of T X .) This illustrates the point that — in this context, at least — always the tangent line q H the Hilbert scheme .X/ may be a better compactification of the space of unordered m m X than the symmetric power: When m D 2 , for example, the map -tuples of points on  W H Q .X/ ! G .1;r/ sending a subscheme of length 2 to its span is always regular. 2 r m and replace the embedding X  P Further, if we fix by a sufficiently high Veronese re-embedding, then every length- m X will span an m 1 plane, so the subscheme of H ! G .m 1;r/ will be regular. In this chapter, we will care only about the .X/ map m , so it does not matter which we use.  image of We begin with the dimension of Sec .X/ : m If m  r n , then the map is birational onto its image; in particular,  Proposition 10.7. .m/ mn dim X D has dimension . .X/ ‰ m This is slightly more subtle than it might at first appear. The first case would be 3  P the statement that if is a nondegenerate curve then the line joining two general C points of C does not meet C a third time. Though intuitively plausible, this is tricky to prove, and requires the hypothesis of characteristic 0. For the proof we will use the following general position result: r . If X  P Lemma 10.8 is a nondegenerate variety of (General position lemma) r n r P n dimension Ä  P Š a general linear subspace of complementary and dimension , then the points of Ä \ X are in linear general position; that is, any r n C 1 of them span . Ä We will not prove this here; a good reference is the discussion of the uniform position lemma in Arbarello et al. [1985, Section III.1]). Proof of Proposition 10.7: The proposition amounts to the claim that if p 2 ;:::;p m 1 X are general points, then the plane p . ;:::;p X they span contains no other points of m 1

385 Secant varieties Section 10.3 371 .m/  be the open subset of m -tuples of distinct, linearly X To prove this, let U independent points, and consider the incidence correspondence p CC p .p ;Ä/ 2 U  G Df n;r/ j ‰ : ;:::;p g 2 Ä .r m 1 1 m Via projection on the first factor, we see that ‰ is irreducible, and by Lemma 10.8 it .r dominates ; it follows that a general .r n/ -plane Ä containing m general G n;r/ r ;:::;p points 2 X is a general .r n/ -plane in P p , and applying Lemma 10.8 1 m 1/ -plane again we deduce that the .m ;:::;p contains no other points of X . p 1 m Let r G .m 1;r/  P ˆ 2 p 2 ƒ g .ƒ;p/ Df j r 1/ -plane in P be the universal , with projection maps .m  r - P ˆ  ? 1;r/: .m G Set 1 D  ˆ  .‰ ; .X// and  j D .X/ m m X .X/ ˆ m m -th secant variety .X/ so that the .X/ is the image of  the . We will call ˆ Sec m m X abstract secant variety . ˆ is irreducible of dimension mn C m 1 , Projection on the first factor shows .X/ m m dim Sec  mn C .X/ 1 , with equality holding when a general point on so that m . By way of language, .X/ lies on only finitely many m -secant .m 1/ -planes to Sec X m r expected dimension P X has dimension n we will call min .mn C m 1;r/ if  the of the secant variety .X/ < .X/ ; we will say that X is Sec -defective if dim Sec m m m min .mn C m 1;r/ , and defective if it is m -defective for some m . 5 Everyone’s favorite example of a defective variety is the Veronese surface in : P 2 5 P  is 2-defective. . P The Veronese surface Proposition 10.9.  X D / 2 In fact, the Veronese surface is the only 2-defective smooth projective surface! This much more difficult theorem was asserted, and partially proven, by Severi. The proof was completed by Moishezon in characteristic 0; see Dale [1985] for a modern treatment that works in all characteristics. Proof: The Veronese surface may be realized as the locus where a symmetric 3  3 matrix 1 0 z z z 0 1 2 A @ D M z z z 4 3 1 z z z 4 5 2

386 372 Chapter 10 Segre classes and varieties of linear spaces 5 M 2 P has rank 1 at two points , then M has rank at most 2 has rank 1. But if p;q C q M vanishes on the whole p at any point of the form . Thus the determinant of q det and M vanishes on the secant locus Sec , so the cubic form .X/ . p line spanned by 2 .X/  5 Thus dim Sec 1 D 4 , not  2 C 1 D 5 . 2 2 We can give a more geometric proof using a basic result introduced by Terracini [1911]: r Let X  P (Terracini’s lemma) be a variety and . Proposition 10.10 ;:::;p X 2 p m 1 X . If linearly independent smooth points of 2 Ä D p is any point in their ;:::;p p m 1 d span not in the span of any proper subset, then the image of the differential at the X 2 .X/ is the span .Ä;p/ ˆ point m T D Im d X;:::; T X p p X m 1 X at the points p X . In particular, if of the tangent planes to has dimension n and i r mn C m 1 , then X is m -defective if and only if its tangent spaces at  general m points are dependent. For a proof, see Landsberg [2012]. 5 We can use Terracini’s lemma to see that the Veronese surface X  P is 2-defective 5 H  P contains the tangent plane to as follows: A hyperplane X at a point p if and only if the curve H X is singular at p . Of course we can consider H \ X as a conic \ 2 P X , and, from the definition of the Veronese surface, we see that every conic in Š 5 P are dependent if and only if they are both appears in this way. Now, two planes in contained in a hyperplane. Putting this together with Terracini’s lemma, we see that to 2 show that X there is a conic is 2-defective we must show that given any two points in P 2 that is singular at both these points; of course, the double line passing through the in P points is such a conic. We can also use Terracini’s lemma to show that there are no defective curves: r C  P If is a nondegenerate reduced irreducible curve, then Proposition 10.11. D min .2m dim Sec 1;r/ for every m . .X/ m By Terracini’s lemma, it suffices to show that if p ;:::;p 2 C are general Proof: 1 m points then the tangent lines T and span r C are linearly independent when 2m 1  p i r when 1  r . 2m P 2 C p We have already seen in the proof of Theorem 7.13 that for a general point r C 1/p spans P the divisor (that is, a general point p 2 C is not inflectionary); it .r 2m 1 r spans a P follows that the divisor 2m  when 2m  r C 1 and spans P p when ;:::;p 2m r C 1 . By lower-semicontinuity of rank, it follows that for general p  1 m the divisor 2p CC 2p has span of the same dimension. 1 m

387 Secant varieties of rational normal curves 373 Section 10.3 The general question of which nondegenerate varieties are defective is a fascinating one, with a long history. Perhaps because of Terracini’s lemma, which relates the issue to the question of when multiples of general points impose independent conditions on ), the case of Veronese embeddings of projective interpolation problem polynomials (the spaces has attracted a great deal of attention. The following is a result of Alexander and Hirschowitz [1995]. The proof was later simplified by Karen Chandler, and an exposition of this version, with a further simplification, can be found in Brambilla and Ottaviani [2008]. The defective Veronese varieties are the following: Theorem 10.12. n  P . / is 2-defective for any n .  2 2  . P  / is 5-defective. 4 3  P  . is 9-defective. / 4 4  . P  / is 7-defective. 3 4  . P /  is 14-defective. 4 We will see in Exercises 10.26–10.29 that the Veronese varieties listed in the theorem are indeed defective (the hard part is the converse!). Note that by Terracini’s lemma Theorem 10.12 implies (and indeed is equivalent to) the following corollary: n Let p be any positive ;:::;p d be general points in A Corollary 10.13. , and let m 1  C n d n  C 1/ m.n . There exists a polynomial f of degree d on integer such that A n with specified values and derivatives at the points , except in the cases d D 2 and p i D .2;4;5/;.3;4;9/;.4;3;7/ and .4;4;14/ . .n;d;m/ 10.4 Secant varieties of rational normal curves We turn now from secant varieties in general to the special case of rational curves. Every rational curve is the projection of a rational normal curve, and its secant varieties are correspondingly projections of the secant varieties of the rational normal curve, so we will focus initially on that case. 10.4.1 Secants to rational normal curves We begin with the observation that finite sets of points on a rational normal curve are always “as independent as possible.” (This property actually characterizes rational normal curves, as we invite the reader to show in Exercise 10.31.) d Let C  P C be a rational normal curve. If D  Lemma 10.14. is a divisor of degree d is not contained in any linear subspace of d C 1 , then D  P m of dimension < m 1 .

388 374 Chapter 10 Segre classes and varieties of linear spaces C of length  d C 1 is linearly independent, Informally: Any finite subscheme  D 0 0 . in the sense that the map .1// ! H . O O H .1// is surjective. On an affine subset d D P 1 d 2 7! P ;:::;t the parametrization of the rational normal curve looks like / , .1;t;t of t is given by the 1 a so the independence of the images of any ;:::;a d C points 0 d nonvanishing of the Vandermonde determinant 0 1 d 1 a  a 0 0 Y B C : : : : : : : : det .a D /: a @ A : j i : : : d i d . H In Section 10.3.2 we described the secant map as a regular map on an open set, that is, as a rational map .m/ - G .m 1;d/: C W  One consequence of Lemma 10.14 is that the secant plane map has a natural extension to is actually a morphism, and in fact an injective map of sets. It is not hard to show that  to the image of  of the universal ˆ an embedding. We can thus regard the restriction C m 1 m 1 1 m .m/ / 1;d/ .m -bundle over . P G P as a D P -bundle over , and Sec P .C/ m is the image of this bundle. d  P Proposition 10.15. be a rational normal curve. When 2m 1  d , the Let C d is birational onto its image W .C/ !  P ˆ Sec ; more precisely, it is .C/ map m m C one-to-one over the complement of .C/ in Sec .C/ . Sec 1 m m d p 2 P Proof: is the image of two different points of ˆ Suppose a point .C/ , say m 0 0 0 < k < m D . ;p/ . Let k D dim . D \ D and / ; note that D;p/ 1 . Since the span . 0 D D has dimension 2m and 2 k , by Lemma 10.14 the union (as subschemes of 0 C ) of D and D of can have degree at most 2m k 1 . It follows that the intersection 0 0 D (again, as subschemes of C ) has degree at least k C 1 . Thus D \ D \ is a secant D -plane, and p 2 S . .C/  S .C/ k 1 m k Proposition 10.15 is not particularly remarkable in case 2m 1 < d : all irreducible, d C P nondegenerate curves have the property that when 2m 1 < d a general  point on the m -secant variety Sec C .C/ lies on a unique m -secant .m 1/ -plane to m , however, it is striking. For example, 1 D d 2m (see Exercises 10.33–10.35). In case 3 is the unique nondegenerate space curve whose secant the twisted cubic curve C  P 3 lines sweep out P only once (see Exercise 10.30).

389 Secant varieties of rational normal curves Section 10.4 375 10.4.2 Degrees of the secant varieties d C be a rational normal curve, and m any integer with 2m 1  d . Since  Let P .m/ C W G .m 1;d/ is regular, and ˆ ! ! Sec the secant plane map .C/ is  m C birational, it is reasonable to hope that we can answer enumerative questions about the geometry of the varieties .C/ . We will do this in the remainder of this section and Sec m .C/ . the next, starting with the calculation of the degree of Sec m There are a few cases that we can do without any machinery; for example: (a) Sec D C , so deg Sec . .C/ D d .C/ 1 1 (b) D 2 is not quite as trivial, but is readily done: The variety Sec The case .C/ has m 2 dimension 3, so its degree is the number of points in which it intersects a general 2 3/ -plane ƒ . As we saw in Exercise 3.34, the projection of C from ƒ to P .d will C birationally onto a plane curve map with nodes, and the points of ƒ \ S C .C/ 2 0  d 1 correspond to these nodes. Since has arithmetic genus C and geometric 0 2  d 1 . .C/ genus 0, we conclude that deg Sec D 2 2 d (c) is odd and m D .d C 1/=2 , then the secant locus is all of P Finally, if , so d D 1 . .C/ deg Sec m In order to go further, we use the Segre class technique of Proposition 10.4. To begin with, the map  1 .m/ .m/ 1 m P .C/ D .. P Š / .C/ / ˆ . P ! / ‰ Š m m m  ! , where H is the pullback H P S of the tautological subbundle on P has the form  m 1;d/ to P G . .m In fact, we have already seen this bundle before, in Section 9.3.3! To see this, let 0  , as the locus . O V D .d// . The rational normal curve lives naturally in P V H 1 P 1 given by evaluation at a point p 2 V . The span of a of linear functionals on C P of degree m on C is the space of linear functionals vanishing on those points, divisor D  0 of the subspace V V D H that is, the annihilator in . I . Thus the map .d// 1 D P D; m m   ! G .m 1;d/ sends D 2 P W to the subspace Ann .V , and it follows /  V P D   is the dual F that the pullback S F introduced in Section 9.3.3. We  of the bundle have from the results of that section that 1 / D c. F ; C d m 1 / .1 m 1 2 A where . P  / is the hyperplane class. The Segre class is the inverse, and taking the dual we have  d m C 1 F s. / D .1 C / : Finally, we can deduce:

390 376 Chapter 10 Segre classes and varieties of linear spaces d If P C is a rational normal curve, then, for 2m 1  d , Theorem 10.16.    d m C 1 .C/ deg Sec : D m m Note that in the case 2m 1 , the calculation reaffirms the conclusion of D d d -secant planes to m P sweep out exactly once. Proposition 10.15 that the C 10.4.3 Expression of a form as a sum of powers f and g are general polynomials We can now answer Keynote Question (b): If d D 2m in one variable, how many linear combinations of f and g of degree are expressible as a sum of -th powers of linear forms? m d This question is related to secants of rational normal curves, because if we realize 1 d -th powers P P then the curve of pure d as the projective space of forms of degree d on is a rational normal curve — it is the image of the morphism     d d d d d d 1 2 1 t; : s s 7! .s;t/ 3 t;:::;t P W ;ds P  2 2 (Note that we are relying here on the hypothesis of characteristic 0: If, for example, d is equal to the characteristic, then  is a purely inseparable map whose image is a line!) d p 2 P C lies on the plane spanned by distinct points q if and ;:::;q A point 2 m 1 p only if the homogeneous coordinates of can be expressed as a linear combination is a linear . Thus a form of degree d of the homogeneous coordinates of ;:::;q q m 1 -th powers of linear forms if and only if the corresponding point in m d combination of 1 d m -secant .m 1/ -planes to . P , and questions about the / lies in the union of the P expression of a polynomial as a sum of powers become questions about the secants. not Sec .C/ There is an important subtlety: It is the case that every point of m corresponds to a polynomial that is expressible as a sum of m d -th powers! For example, d are contained in Sec .C/ the tangent lines to . If d  3 , then no 2-plane in P C meets 2 the rational normal curve in four points, so a tangent line to C cannot meet any other secant line at a point off C . Thus the points on the tangent lines away from C are points of Sec that cannot be expressed as the sum of two pure d -th powers. .C/ 2 (The points on the tangent lines do have an interesting characterization, however: d At the point corresponding to the polynomial .t / , the tangent line is the set D f.t/ f and @f=@t , or equivalently the set of polynomials that have of linear combinations of d 1 roots equal to  .) By definition, Sec - .C/ contains an open set consisting of points on secant .m 1/ m distinct points of C . Further, by Proposition 10.15 a point in the m planes spanned by Sec .C/ lies on the span of a unique divisor of Sec open subset .C/ of Sec .C/ n m m 1 m degree m . Thus Theorem 10.16 yields the answer to Keynote Question (b), and even a generalization:

391 Special secant planes Section 10.4 377 If d 2m 1 , then the number of linear combinations of d 2m C 2 Corollary 10.17.  that can be expressed as the sum of m d -th powers is general forms of degree d pure  1 d m C deg Sec D .C/ . m m 10.5 Special secant planes r For a curve C  P other than a rational normal curve, it is interesting to consider in a dependent set of points; these are called special secant the subspaces that meet C . Examples of this that we will investigate below include trisecant and quadrisecant planes 3 4 lines to a curve , and trisecant lines to a curve C  P  . P C We start, as usual, with the question of dimension: When would we expect a curve r m 1 k  to contain m points lying in a C -plane? What would be the expected P P .m/ .m/  C dimension of the locus C m -tuples? of such k There are many ways to set this up. One would be to express the locus of such m p are the ;:::;p C 2 -tuples as a determinantal variety: If the coordinates of points m 1 rows of the matrix 0 1 ::: x x x 1;1 1;r 1;0 B C : : : : : : : : M D ; @ A : : : : ::: x x x m;1 m;0 m;r .m/ is just the locus where this matrix has rank then C m k or less. Now, in the space of k m .r C 1/ matrices, those of rank m k or less have codimension k.r C 1 m C  , so k/ .m/ m 1 k C we would expect the locus to have dimension of m -tuples spanning only a P k m k.r C 1 m C k/ D .k C 1/.m r k/ C r: An alternative in the case C C as the projection is a rational curve would be to express d d r 1 z z from a plane P C ! ƒ Š P C of a rational normal curve W C  . The  ƒ -secant .m 1 k/ m C then correspond to the m -secant .m 1/ -planes to -planes to z that intersect in a .k 1/ -plane, that is, the preimage under the secant plane map C ƒ .m/  W ! G .m 1;d/ of the Schubert cycle C † : g 1 k  ƒ/ \ .ƒ/ Df Ä 2 G .m 1;d/ j dim .Ä k k/ 1 C .r C m k.r 1 m C k/ , so again we would expect the This Schubert cycle has codimension C m k.r C 1 m C k/ . preimage to have dimension k > 0 ) are: The first three cases (with 3 trisecants to a curve C  P (a) (that is, r D 3 , m D 3 and k D 1 ), where we expect a one-parameter family; 3 (b) C  P 3 (that is, r D quadrisecants to a curve , m D 4 and k D 2 ), where we expect finitely many; and

392 378 Chapter 10 Segre classes and varieties of linear spaces 4 trisecants to a curve P C (that is, r D 4 , m D 3 and k D 1 ), where, again, we (c)  expect finitely many. That our expectations for the dimensions of these loci are indeed the case for general rational curves is shown in Exercise 10.36, though it is necessarily true of a general not point on any component of the Hilbert scheme of curves of higher genus, as shown in Exercise 10.37. In this section, we will show how to count the trisecants to a general rational curve 4 , answering Keynote Question (c), and we will determine the degree of the trisecant P in 3 surface of a general rational curve in , answering Keynote Question (d). We leave the P 3 to Exercise 10.38 for now; it will also be P case of quadrisecants to a rational curve in a direct application of Porteous’ formula in Section 12.4.4. 10.5.1 The class of the locus of secant planes is rational, the answers to all of the above questions come In case the curve C d  P directly from the answer to a question we have not yet addressed directly: If C m .m/ C ! Š P is a rational normal curve,  W .m 1;d/ the secant plane map and G ‰ .C/  G .m 1;d/ the image of  , what is the class Œ‰ ? .C/ç 2 A 1;d// . G .m m m m We have all the tools to answer this question at hand: We know that the pullback    of the dual of the universal subbundle E on G .m 1;d/ is the bundle S  whose S  D . S Chern classes we gave in Section 10.4.2. We know that / , so this says that  c i i 1   d m C i  m i i .  /; 2 A  D P  i 1 i m .m/  as usual is the hyperplane class in C Š P . Equivalently, since we also where S is s. S / D 1 C  know that the Segre class of C  , we have CC  2 1 d m   m C 1 d  i i m /:   (10.1) 2 A D . P  i i is an embedding), this G .m 1;d/ (and   generate the Chow ring of Since the classes i .C/ç explicitly Œ‰ determines the class of the image. We will use this idea to compute m m .m/ C in the cases below. It is an interesting fact that the map Š P  ! G .m 1;d/ W ̈ ucker embedding of the Grassmannian is the d -th Veronese map composed with the Pl m P on ; see Exercise 10.32 4 Trisecants to a rational curve in P 4 How many trisecant lines does a general rational curve of degree d in P possess? We already know the answer in at least two cases. First, there are no trisecant lines to a rational normal curve in the case d D 4 . If d D 5 , Proposition 10.15 says that a general 5 5 z P point lies on a unique 3-secant 2-plane to a rational normal curve p C  P 2 , and thus the projection of that curve from a general point has just one trisecant line.

393 Special secant planes 379 Section 10.5 5 d above: We use the fact The general case may be handled similarly to the case D 4 of degree that a rational curve is the projection of a rational normal curve P  C d d d z from a 5/ -plane ƒ  .d . The trisecant lines to C then correspond to P P C  z of degree d that meet ƒ . The trisecant lines to C 3-secant 2-planes to correspond to C  G .2;d/ of 2-planes meeting ƒ with the the intersection of the Schubert cycle † .ƒ/ 3 z z C . G .2;d/ of 3-secant 2-planes to cycle  . C/ ‰ 3 This gives the answer to Keynote Question (c): 4  P Proposition 10.18. is a general rational curve of degree d , then C has If C  2 d trisecant lines. 3 z is general, it is the projection of the rational normal curve from C Since C Proof: 5/ -plane ƒ . The number of trisecant lines is the number of points a general .d z .ƒ/ meets ‰ in which † . C/ . By Kleiman transversality, this is the degree of the 3 3  intersection class .C/ç , or equivalently the degree of the pullback   Œ‰ ; by the 3 3 3  d 2 above, this is . 3 3 Trisecants to a rational curve in P 3 We next turn to Keynote Question (d): If C  P is a general rational curve of 3 , what is the degree of the surface  P d swept out by the 3-secant lines to C ? S degree d D 3 Again we already know the answer in the simplest cases: 0 in the case d D 4 , since a smooth (a rational normal curve has no trisecants); and 2 in the case 3 .1;3/ on a quadric surface Q  P rational quartic is a curve of type , and the trisecants of C comprise one ruling of Q . z  . of a rational normal C/ be the projection To set up the general case, let C ƒ d 4 3 d z Š P C  P ; let L  P curve be a general line, and let from a general plane ƒ d 1 Ä D .L/   P be the corresponding .d 2/ -plane containing ƒ . The points of ƒ d z correspond to 3-secant 2-planes to intersection of C  with S that L P ƒ , and (a) meet Ä in a line. (b) intersect z † . ‰ C/ with the Schubert cycle These are the points of intersection of . .ƒ;Ä/ 2;1 3 Kleiman transversality shows that the cardinality of this intersection is the degree of the   pullback . . / 2;1 To evaluate this we express  as a polynomial in  and evaluate each ; ;::: 2 2;1 1 term using (10.1). Giambelli’s formula (Proposition 4.16) tells us that ˇ ˇ ˇ ˇ   2 3 ˇ ˇ  D    D 1 2 3 2;1 ˇ ˇ   1 0 (an equality we could readily derive by hand). By (10.1),      2 2 d d d 2   . deg   and deg  : D / / D . 1 3 2 2 3 1

394 380 Chapter 10 Segre classes and varieties of linear spaces Putting these things together, we have the answer to the question: 3 If  P is a general rational curve of degree d , then the degree C Proposition 10.19. is of the surface swept out by trisecant lines to C        2 2 d d d 1 d 2 : 2 D 3 3 1 2 10.5.2 Secants to curves of positive genus It is instructive to ask whether we could extend the computations of Sections 10.4 and 10.5 to curves other than rational ones. There is one key problem: in treating rational curves, we made essential use of the fact that the space of effective divisors of degree m 1 m is the variety P on , whose Chow ring we know. But when C has positive genus, the P .k/ A.C of symmetric powers of C are unknown. Chow rings / D 1 this is not an insurmountable problem; it is the content of In the case of g g  2 , however, it typically necessitates the Exercises 10.48 and 10.49. For genera use of a coarser equivalence relation on cycles, such as homology rather than rational equivalence. Given this framework, however, it is indeed possible to extend the results of this chapter to curves of arbitrary genus; see (Arbarello et al. [1985, Chapter 8]), where there are explicit formulas generalizing all the above formulas to arbitrary genus. 10.6 Dual varieties and conormal varieties We next turn to a remarkable property of projective varieties called reflexivity . A corollary of reflexivity is the deep fact that the dual of the dual of a variety is the variety itself. See Kleiman [1986] for a comprehensive account of the history of these matters (our account is based on that in Kleiman [1984]). We emphasize that the statements below are very much dependent on the hypothesis of characteristic 0; see the references above for the characteristic p case. n X  P be a subvariety of dimension Let k . If X is smooth, we define the conormal  n n variety P  P CX  to be the incidence correspondence  n n and .p;H/  P P Df j p 2 X 2 T CX X  H g : p  n n ı to be the closure in P CX  P If X of the locus is singular, we define CX of such pairs .p;H/ p is a smooth point of X . Whatever the dimension of X , the , where ı conormal variety will have dimension n 1 : it is the closure of the locus CX , CX which maps onto the smooth locus of X with fibers of dimension n k 1 . The dual  n  of X is the image of CX under projection on the second factor. P variety X  In these terms, we can state the main theorem of this section:

395 Dual varieties and conormal varieties Section 10.6 381 D T X L p r p q X  X X D T M q 2 2 P P  2  p at L D T Figure 10.1 The tangent line to X is the line dual to P . X  p n n   P (Reflexivity) is any variety and X .  P If X its dual, then  Theorem 10.20   n n n n   P  /  P P P CX is equal to C.X with the factors the conormal variety    .X reversed. It follows that D X — that is, the dual of the dual of X is X . /  is a plane curve, then the statement X For example, if D X says that if X  2 X X at the point L D T p X is the line is a smooth point then the tangent line to p 2    P of lines through p . More picturesquely put, the tangent lines to points near p 2 X “roll” on the point x . It is true more generally that the osculating k x -planes to a n  P smooth curve at points near p 2 X move, to first order, by rotating around the X .k .k 1/ osculating X at p while staying in the osculating C 1/ -plane to X -plane to at p (see Exercise 10.47). This picture, for plane curves, can be made precise as follows. Observe that if p 2 X 2 is a smooth point, then the tangent line  P T is the limit of the secant lines p;q as X p  2  X 2  P p approaches , this says that the tangent q . Applied to the dual curve X 2    P X T to the curve X line at a point L is the limit of the secant lines L;M  L 2  2   L . But the line L;M  P M 2 joining two points L;M 2 P X as approaches 2 2  2 L P P L;M dual to the point  \ M in P is the line in . corresponding to lines   D Now, the equality X means that the tangent line to X X at the point L D T X p p X is p 2 q 2 X corresponding to itself; this amounts to saying that the limit as p of the point of intersection r D T \ X approaches T itself, X is just the point p q p which is clear from Figure 10.1. Combining Theorem 10.20 with the argument at the beginning of Section 2.1.3, we see that the Gauss map of a smooth hypersurface is birational as well as finite: If X is a hypersurface whose dual is also a hypersurface, then the Corollary 10.21. n    X ! X G is birational, with inverse G P Gauss map W X W ! X . Thus if X  X X is a smooth hypersurface of degree d then the Gauss map is finite and birational, and  n 1 1/ . X d.d is a hypersurface of degree

396 382 Chapter 10 Segre classes and varieties of linear spaces This allows us, finally, to complete the proof of Proposition 2.9. Proof of Corollary 10.21: By Section 2.1.3, the dual of a smooth hypersurface is al- ways a hypersurface.   are both hypersurfaces then both G X If G and X are well-defined rational and X X  and G G maps. Since the graphs of are equal after exchanging factors, the two X X rational maps are inverse to each other, and are thus birational. As already noted in Section 2.1.3, the degree computation follows from the birationality of G . X One aspect of Theorem 10.20 may seem puzzling. The only way the dual of a n  P can fail to be a hypersurface is if the map variety X ! X  has positive- CX dimensional fibers — in other words, if every singular hyperplane section of has X positive-dimensional singular locus. This is a rare circumstance; as we will see in Exercise 10.42, it can never be the case for a smooth complete intersection, and, as we swept out by positive-dimensional X will see in Exercise 10.44, it can only happen for n linear spaces. But if we have a one-to-one correspondence between varieties X P  and their dual varieties, we seem to be suggesting that there are as many hypersurfaces n P as varieties of arbitrary dimension in ! The discrepancy is due to the fact that the duals of smooth varieties tend to be highly singular — see, for example, Exercise 10.45. There are many fascinating results about the geometry of dual varieties and conormal varieties. We recommend in particular Kleiman [1986], and the surprising and beautiful theorems of Ein and Landman (see Ein [1986]) and Zak [1991]. Ein and Landman proved, n X P for example, that for any smooth variety of dimension d in characteristic 0 the   .n 1/ dim X X is congruent to dim difference modulo 2!  As we mentioned earlier, it is relatively rare for the dual X of a smooth variety n P to not be a hypersurface. Exercises 10.43 and 10.46 give two circumstances  X where it does happen. 10.6.1 The universal hyperplane as projectivized cotangent bundle universal hyperplane The proof of the reflexivity theorem will make use of the n n  P P  ‰ Df .p;H/ j p 2 H g ; 2 k -plane introduced in Section 3.2.3 and analyzed further a special case of the universal V be an .n C 1/ -dimensional vector space in Section 9.3.1. To express it another way, let  W D V and its dual; we can then write W ‰ 2 P V  P .v;w/ j w.v/ D 0 g : Df Write the tautological sequence on P V as V 0 O ! Q . 1/ ! ! ̋ O ! 0: V P V P

397 Dual varieties and conormal varieties Section 10.6 383    W Thus . From the inclusion it follows that the line bundle O Q ̋ O . 1/ P Q V P n n    n  O . 1/ is the restriction from P  Q of the bundle  P on . In Section 6.1.1 P P 2  P V  P W may be regarded as the projectivization P Q we observed that inside ‰   ̋ O D P V P P W . W P V and To simplify notation, we write   for the projections from Z WD P V  P W W V   .b/ O O .a/ ̋  W  P O respectively, and we set . In P to .a;b/ WD and V W P V P Z W V    . Q / .0;  and O this language, . For our D . ‰ 1/ is the restriction of O 1/ Q P Z V present purpose, we want to give a more symmetric description. ! The map Proposition 10.22. ‰  P V may be described as the projectivization W V P  P . The tautological subbundle of of the cotangent bundle V V P  / . . 1/  O  P P V  V P V ‰  P V  P W D Z of O . is the restriction to 1; 1/ . Z Proof: The Euler sequence !  . 0 ! W ̋ O ! O 0 1/ ! V P V P V P that may be taken as the definition of  is the tautological sequence twisted by V P  Q 1/ , and in particular  D . O ̋ O . 1/ . By Corollary 9.5, we have V V P P P V  Q P  , with P Š   1/ D O O ̋  . 1/; O . P Q P  V P V  of O , as required. .0; 1/ ̋  and this is the restriction to 1/ ‰ 1; . 1/ D O . O Z P V Z V Proof of Theorem 10.20: X  P V is any subvariety then, over the open set where X If CX  ‰ is smooth, the conormal variety P  is the projectivized T D Proj Sym D V P V P  P D K conormal bundle , where K Proj Sym Ker  K . j /: ! WD X V P X itself is defined as the closure of this set. Over the open set The conormal variety of X  is smooth, K is the cokernel of the map of bundles where X T ! T ; j V X X P  j is equal to Sym T K so modulo the ideal generated by T , thought of as Sym X X P V T being contained in the degree-1 part j of the graded algebra Sym T j . Sheaf- X P V V X P is the image of the composite ifying, this means that the ideal sheaf of O CX in  P P V map u in the diagram   n T ̋ O . O ̋  1/  . 1/ T X   n n P P P V V u O P  n P

398 384 Chapter 10 Segre classes and varieties of linear spaces where the vertical map is the dual of P A dZ i i  n 1/  ! O . ;  P  n P V P n the tautological inclusion, tensored with . 1/ . O  P 1 .X/  \ of is a With these equations, we can tell whether a given subvariety ‰ C V . Let  W C ! ‰ be the inclusion. Set subset of CX   1/ . !  ;  W 1/ . O v D u ̋ O X P   P n n V P P and consider the diagram P dZ A d i i V  n 1/ j  j . O j   C C ‰ P  C n P P V d v j C d. / j C 1  j    X C C V From what we have said about the equations of the conormal variety, we see that  CX if and only if v j 0 D C ; since d is generically injective, we see that j C C V ı CX d  j C if and only if the composition v j is zero. C V C C  CX if and only if We will show that this condition is symmetric, so that   / (with the factors reversed). Since ‰ is defined by a hypersurface of C.X C .1;1/ , we have an exact sequence bidegree ' n n 0: . !  ! 0 1; 1/ ! !  j O ‰ ‰ P   W P W P P In coordinates, using the decomposition   n n   . D / / ̊  . P W W P  P P W V this becomes P P ; dA Z dZ A i i i i   n n / j  j  . ̊ . / ! . 1; 0 1/ j ! O ‰ ‰ W P ‰ P W P  P V W .d ;d / V W  ! ! 0: ‰ n 1/ D O . . 1; 1/ j C W  is the , we see that if ‰ Noting that ! O ‰ P  n  P W P P inclusion of any subvariety, then the composition P A dZ d i i V  n / !  1/ . ! O .  ‰ P  n P V P is the negative of the composition P Z dA d i i W  :  ! O . !  1/ . / W P  ‰ P n W P

399 Exercises 385 Section 10.6 It follows that the composition P dZ A d V i i  - - n 1/ j . j j  O   C ‰ P  C C n P V P d ?  C is zero if and only if the composition P Z dA d W i i  - -  . j  j 1/ j O  W  C C C P P ‰ P W W d ?  C is zero.   /  C C.X CX . If C If then the composite above is zero, and it follows that 0 , then C  CX if and only if C  CX  . Applying this argument to C D CX C ‰  D C.X and / , we obtain the desired equality. C 10.7 Exercises Exercise 10.23. Use the result of Exercise 9.36 (describing the class of the universal r G .k;r/  k ) to give an alternative proof of Proposition 10.4. -plane in P r X  P Exercise 10.24. be a variety, and ‰ (Improper secants) Let .X/  G .m 1;r/ m .m/ - W X the image of the secant plane map  G .m 1;r/ . Show by example that not every .m 1/ -plane ƒ such that deg .ƒ \ X/  m lies in ‰ .X/ . (For example, try m 5 3 with a trisecant line, with m D P .) X a curve in Prove Proposition 10.7 in the case of a nondegenerate space curve Exercise 10.25. 3 P — that is, that the line joining two general points of C C does not meet the curve  a third time — without using the general position lemma (Lemma 10.8). Exercises 10.26–10.29 verify that the Veronese varieties listed in Theorem 10.12 are indeed defective. n 2 0 0 2 n .2// P . I 2 .2// I Show that for the subspace H  H Exercise 10.26. . O p;q P q p of quadrics singular at p and q has codimension 2n C 1 (rather than the expected 2n C 2 ). n Deduce that any two tangent planes to the quadratic Veronese variety . P  / meet, and 2 n thus that . P  / is 2-defective for any n . 2 2 Show that for any five points ;:::;p 2 P Exercise 10.27. there exists a quartic p 5 1 T curve double at all five; deduce that the tangent planes to the quartic Veronese S p i 2 14 14  ), and hence . P surface /  P S are dependent (equivalently, fail to span P D 4 that S is 5-defective.

400 386 Chapter 10 Segre classes and varieties of linear spaces 3 Show that for any nine points ;:::;p p 2 P Exercise 10.28. there exists a quartic 9 1 surface double at all nine; deduce that the tangent planes X to the quartic Veronese T p i 34 34 3 . ), and hence threefold /  P  are dependent (equivalently, fail to span P D X P 4 X is 9-defective. that 4 p 2 P Finally, show that for any seven points there exists a ;:::;p Exercise 10.29. 1 7 X to the cubic T cubic threefold double at all seven; deduce that the tangent planes p i 34 34 4 D P  /  P P are dependent (equivalently, fail to span X Veronese fourfold ), . 3 X is 7-defective. and hence that Hint: This problem is harder than the preceding three; you have to use the fact that 4 through seven general points in there passes a rational normal quartic curve. P The following exercises can be solved using the following fact, the completeness 2 z of the adjoint series in P for plane curves: if , and C C is a nodal curve of degree d 0 .K its normalization, then we obtain the entire canonical series H by pulling back / z C 2 3 on P d vanishing on the nodes of C (see Arbarello et al. polynomials of degree [1985, Appendix A]). Exercise 10.30. Show that the twisted cubic curve is the unique nondegenerate curve 3 3 . (Note: This P p 2 P  C C such that a general point lies on a unique secant line to can be done without it, but it is easy if you apply the Castelnuovo bound on the genus of 3 a curve in P ; see Chapter 3 of Arbarello et al. [1985] for a statement and proof.) Exercise 10.31. Show that the rational normal curve and the elliptic normal curve of d degree 1 are the only nondegenerate curves C  P d with the property that every C d divisor of degree spans a hyperplane. on C d .m/ W C Let   C P Š Exercise 10.32. be a rational normal curve. Show that the map m C on m sending a divisor of degree to its span composed with the 1;d/ G ! P .m m ̈ d -th Veronese map on P . ucker embedding of the Grassmannian is the Pl m P is the associated to any monomial of degree d Hint: Show that the hypersurface in G preimage of a hyperplane section of 1;d/ .m d For the following three exercises,  P will be an irreducible, nondegenerate C curve and 2m 1 < d . The exercises will prove the assertion made in the text that a 1/ general point on the Sec -plane .C/ lies on a unique m -secant .m -secant variety m m C . to Exercise 10.33. Sec .C/ lies on Show by a dimension count that a general point of m only proper secants, that is, m 1 planes spanned by m distinct points of C . Exercise 10.34. -secant .2m 2/ -planes 2m Using Lemma 10.8, show that the variety of .2m/ contained in a C to C of divisors of degree 2m on C (equivalently, the locus 1 .2m 2/ -plane) has dimension at most 2m 2 .

401 Exercises Section 10.7 387 Sec Now suppose that a general point of lay on two or more Exercise 10.35. .C/ m 2m .2m 2/ -planes -secant planes. Show that the dimension of the variety of -secant m 2m 1 . to C would be at least r , and  is a general rational curve of degree d P k is a C Show that if Exercise 10.36.  number such that C k and m 1  k , then the locus of m -secant .m k 1/ -planes d r m k.r C 1 m k/ . has the expected dimension C By contrast with the last exercise, show that there exist components Exercise 10.37. 3 P H whose general point corresponds to a smooth, of the Hilbert scheme of curves in 3 C  with a positive-dimensional family of quadrisecant lines, nondegenerate curve P or with a quintisecant line. Compute the number of quadrisecant lines to a general rational curve Exercise 10.38. 3 . of degree d  P C  / In the notation of Section 10.5, the answer is the degree of the class . Hint:  2 deg 2;2 4 4 P A / . Express the class  in terms of the special Schubert classes .  (10.1) and use i 2;2 to evaluate it. n  P be a smooth surface of degree Let d , and let g be the genus S Exercise 10.39. 2 ; let e and f be the degrees of the classes c of a general hyperplane section of . T S / 1 S 2 c of lines tangent to . T .1;n/ / 2 A and .S/ . Find the class of the cycle T G .S/  1 2 S d e , f and g . (Note: From Exercise 4.21, we need only the intersection S , in terms of number deg .S/ç   .ŒT / ; do this using Segre classes.) 3 1 3 C  P g be a smooth curve of degree n and genus Exercise 10.40. , and S and Let 3 d two smooth surfaces containing C , of degrees  and e . At how many points of P T are S and T tangent? C Show the conclusion of Corollary 10.21 fails in characteristic p > 0 : Exercise 10.41. (a) Let k be a field of characteristic 2, and consider the plane curve 2 2 V.X  YZ/ C P D : 2    . P C Show that is a line, so that C  ¤ C is smooth, but the dual curve C e k has characteristic p > 0 , set Now suppose that the ground field D p (b) and q consider the plane curve 2 1 C q q q / Y X C V.YZ D  P C : Z  C is smooth, and that the double dual curve C Show that is equal to C , but that  G W C ! C is not birational! C

402 388 Segre classes and varieties of linear spaces Chapter 10 n X is a smooth hypersurface of  Exercise 10.42. We saw in Section 2.1.3 that if P  n   X must again be a hypersurface. Show P then the dual variety d > 1 degree n more generally that if is any smooth complete intersection of hypersurfaces of X  P  degrees then d > 1 will be a hypersurface. X i n  P Exercise 10.43. be a k -dimensional scroll , that is, a variety given as Let X the union [ D ƒ X b k 1 n f ƒ ; suppose that Š P of a one-parameter family of .k  P 1/ g -planes k  2 b (see Section 9.1.1). n H is a general hyperplane containing the tangent plane T P X to Show that if  (a) p p then the hyperplane section H \ X is reducible. X at a smooth point  3  n k C 2 when k  (b) Deduce that dim . X n This is a sort of partial converse to Exercise 10.43 above. Let P Exercise 10.44. X   X be any variety. Use Theorem 10.20 to deduce that if the dual is not a hypersurface, then X must be swept out by positive-dimensional linear spaces. n X  P Exercise 10.45. be a smooth hypersurface of degree d > 2 . Show that the Let  X is necessarily singular. dual variety 9 4 Exercise 10.46. D G .1;4/  P Let be the Grassmannian of lines in P X , embedded 9 ̈ in P ucker embedding. Show that the dual of X is projectively equivalent to by the Pl itself! X n  P Exercise 10.47. be a smooth curve, and for any k D 1;:::;n Let let X 1 W X ! G .k;n/  k p 2 X to its osculating be the map sending a point -plane. Show that the tangent k line to the curve  .X/  G .k;n/ at  is the (tangent line to the) Schubert cycle .p/ k k k -planes containing the osculating .k of -plane to X at p and contained in the 1/ osculating .k C 1/ -plane to X at p ; in other words, to first order the osculating k -planes move by rotating around the osculating 1/ -plane to X at p while staying in the .k .k C 1/ -plane to X at p . osculating If E is a smooth elliptic curve (over an algebraically closed field this Exercise 10.48. means a curve of genus 1 with a chosen point), the addition law on E expresses the k 1 . Verify this, and use it to give a as a P k -th symmetric power E -bundle over E k A.E / . description of k Exercise 10.49. Using the preceding exercise, find the degrees of the secant varieties of d an elliptic normal curve E  P .

403 Chapter 11 Contact problems Keynote Questions 3 3 P (a) , how many lines L  P S meet S in only  Given a general quintic surface one point? (Answer on page 391.) 2 G If D V.t is a general net of cubic plane curves, how F C t g f C t C H/  P (b) 2 1 0 t many of the curves will have cusps? (Answer on page 416.) C t 2 If is a general pencil of quartic plane curves, how D V.t g F C t C G/  P f (c) 0 t 1 many of the curves will have hyperflexes? (Answer on page 405.) C t If C g is again a general pencil of quartic plane curves, what are the degree and f (d) t genus of the curve traced out by flexes of members of the pencil? (Answer in Section 11.3.2.) Problems such as these, dealing with orders of contact of varieties with linear spaces, are known as contact problems . Their solution can often be reduced to the computation of the Chern classes of associated bundles. The most important of the bundles involved is a relative version of the bundle of principal parts introduced in Chapter 7 and described by Theorem 7.2. We will begin with an illustration showing how these arise. One point of terminology: We define the of a curve C on a smooth order of contact 2 X  X at p D C to be the length of the component of variety with a Cartier divisor z \ D the scheme of intersection p , or (equivalently) if  W C C ! C is the supported at normalization, the sum of the orders of vanishing of the defining equation of D at points z p C lying over p . If of is an isolated point of C \ D , this is the same as the intersection multiplicity m .C  D/ , and we will use this to denote the order of contact; however, we p will also adopt the convention that if  D then the order of contact is 1 , so that the C condition m . .C  D/  m is a closed condition on C , D and p p Finally, we reiterate our standing hypothesis that our ground field has characteristic 0. As with most questions involving derivatives, the content of this chapter is much more complicated in characteristic p , and many of the results derived here are false in that setting.

404 390 Chapter 11 Contact problems 11.1 Lines meeting a surface to high order 3 Consider a general quintic surface S . A general line meets S in five points; to P  require them all to coincide is four conditions, and there is a four-dimensional family of 3 S in P lines in . Thus we would “expect” there to be only finitely many lines meeting just one point. On this basis we would expect, more generally, that for a general surface 3 P  of any degree d  5 there will be a finite number of lines having a point of S . contact of order 5 with S As we shall show, this expectation is fulfilled, and we can compute the number. To verify the dimension statement, we introduce the flag variety 3 2 G .1;3/  P ˆ j p .L;p/ L g ; 2 Df G .1;3/ ; we can also realize ˆ as the which we think of as the universal line over P S of the universal subbundle S projectivization G .1;3/ . Next, we fix d  4 and on 3 2 ˆ and a surface S  P look at pairs consisting of a point of degree d such .L;p/ L S at p (or is contained in S ): that the line has contact of order at least 5 with N .L;p;S/ 2 ˆ  P Ä j m Df .S  L/  5 g ; p N 3 where P is the space of surfaces of degree d in P . Assuming d  4 , the fiber of Ä over any point .L;p/ 2 ˆ is a linear subspace N of codimension 5 in the space of surfaces of degree d . Since ˆ is irreducible of P Ä is irreducible of dimension , and hence that the fiber of dimension 5, it follows that N N ŒSç P over a general point is finite. Note that this includes the possibility that the Ä 2 D 4 : any line with fiber over a general point is empty, as in fact will be the case when d S must lie in S , but, as we saw in a point of contact of order 5 with a quartic surface Chapter 6, a general quartic surface contains no lines. In the case d 4 , correspondingly, D 34 projects with one-dimensional fibers to the hypersurface  P of quartics that do Ä † d contain a line. By contrast, we will see (as a consequence of Theorem 11.1) that if 5  N then the projection Ä ! P is generically finite and surjective. To linearize the problem, we consider for each pair .L;p/ 2 ˆ the five-dimensional vector space g p f germs of sections of O at .d/ L : E D .L;p/ germs vanishing to order  f at p g 5 To say that m is in the .S  L/  5 means exactly that the defining equation F of S p kernel of the map 0 ! . O E H .d// 3 .L;p/ P given by restriction of F to a neighborhood of p in L .

405 Lines meeting a surface to high order Section 11.1 391 To compute the number of lines with five-fold contact, we will define a vector bundle ˆ whose fiber at a point .L;p/ 2 ˆ is the vector space E E D on .L;p/ 3 0 5 on P will d of degree defined above, so that a polynomial F .d// . m .d/= O H L p  of E by restriction in turn to each pointed line .L;p/ . The zeros give a global section F  5 will then be the points .L;p/ 2 ˆ such that m , and — .S  L/  of the section p F assuming that there are no unforeseen multiplicities — the answer to our enumerative 5 . The necessary theory E / 2 A c .ˆ/ . problem will be the degree of the top Chern class 5 and computation will occupy the next two sections, and will prove: Theorem 11.1. If S is a general quintic surface, then there are exactly 575 lines meeting 3 S  P 4 is a general surface of degree d  S , in only one point. More generally, if 2 3 35d 200d then there are exactly C 240d lines having a point of contact of order 5 S with . d D 4 ! (In case Note that this does return the correct answer 0 in the case d 3 , the number is meaningless, since the locus of such pairs .L;p/ is positive-  dimensional.) 11.1.1 Bundles of relative principal parts The desired bundle E on ˆ is a bundle of relative principal parts associated to the map  ! G .1;3/; .L;p/ 7! ŒLç: W ˆ The construction is a relative version of that of Section 7.2; the reader may wish to review that section before proceeding. The facts we need are the analogs of some of the properties spelled out in Theorem 7.2. Suppose more generally that  W Y ! X is a proper smooth map of schemes, and L be a vector bundle on Y . Set Z D Y  let Y , the fiber product of Y with itself over X , and let  X ; be the diagonal, so that we W Z ! Y , with projection maps Å  Z 2 1 have a diagram  1 - - Y  D Y Z Å Y X   2 ? ?  - Y X m m -th order principal parts P The bundle of relative . L / is by definition Y=X 1 C m m  ̋ /  P . . L I L D O = /:  2 Z 1 Y=X Å

406 392 Contact problems Chapter 11 ! Y X and projections  With W Y   Y ! Y as above: W Theorem 11.2. i X m . L / The sheaf P Y , and its fiber at a point y 2 Y is the is a vector bundle on (a) Y=X vector space f g L germs of sections of y at j F y m . L / P D ; y Y=X  C 1 at y g germs vanishing to order f m 1 through  where ..y//  Y is the fiber of  D y . F y   . L  Š  L   (b) We have an isomorphism   2 1 1 C m   pushes forward to give a map ̋ L I O =  ! L  (c) The quotient map Z 1 1 Å  m  P L  Š  /; L L !   .   2 1 Y=X m 0 H  . L / is the section and the image of a global section G of P 2 whose . L / G Y=X y 2 Y is the restriction of G to a neighborhood of value at a point in F y . y 0 m (d) by order of vanishing L D L . For m > 1 , the filtration of the fiber P P / . L y Y=X Y=X m by subbundles that are the kernels of P . L / y corresponds to a filtration of at Y=X m k . L / ! P P surjections . L / for k < m . The graded pieces of this filtration Y=X Y=X are identified by the exact sequences m m m 1 . 0 ! ! P L (11.1) . L / ! P ̋ Sym / 0: . L / ! Y=X Y=X Y=X (11.1) The exact sequences in allow us to express the Chern classes of the bundles m . We will compute the . P L in terms of the Chern classes of L and those of  / Y=X Y=X latter in the case where Y is a projectivized vector bundle over X in the next section, and this will allow us to complete the answer to Keynote Question (a). Proof: Just as in the absolute case (Theorem 7.2), part (a) is an application of the theorem on cohomology and base change (Theorem B.5). Similarly, part (b) follows from statement (2) on page 525 in the appendix on cohomology and base change (Section B.2). Part (c) is also a direct analog of the absolute case. For part (d), consider the Å WD Å . As in the absolute case, we  Y  diagonal Y and its ideal sheaf I X Å Y=X 2 I  . I D have  (see Eisenbud [1995, Theorem 16.24] for the affine case, / =  1 Å Y=X Å to which the problem reduces). The sheaf   is a vector bundle on Y because Y=X Å is locally a complete intersection in Y  Y , it follows (see, for is smooth. Since X example, Eisenbud [1995, Exercise 17.14]) that 1 C m m 2 m . I /: D Sym = I I I = Å Å Å Å The desired exact sequences are derived from this exactly as in the absolute case.

407 Lines meeting a surface to high order Section 11.1 393 11.1.2 Relative tangent bundles of projective bundles m . L / To use the sequences P (11.1) , we need to to calculate the Chern class of Y=X T  W Y ! X is understand the relative tangent bundle . Recall first the definition: If Y=X  T !  W d a smooth map, then the differential is surjective. Its kernel is called T X Y of  , and denoted by the relative tangent bundle or, when there is no ambiguity, by T  ; its local sections are the vector fields on Y that are everywhere tangent to a fiber. T Y=X x X then the restriction T j Thus, for example, if 2 is the tangent bundle to 1 Y=X .x/  1 the smooth variety .x/  . One special case in which we can describe the relative tangent bundle explicitly is when W Y D P E ! X is a projective bundle (as was the case in the example of  Keynote Question (a), discussed in Section 11.1 above); in this section we will show how. Recall from Section 3.2.4 that if P V is a point in the projectivization P V of  2 , then we can identify the tangent space T a vector space P V with the vector space V  .;V=/ . As we showed, these identifications fit together to give an identification Hom of bundles /; Q D H T . S ; om V P are the universal sub- and quotient bundles. S O D Q . 1/ where and V P E This identification extends to families of projective spaces. Explicitly, suppose is a vector bundle on and P E its projectivization, with universal sub- and quotient X bundles S D O , we E . 1/ and Q . At every point .x;/ 2 P E , with x 2 X and   E x P Hom T E have an identification D Hom .; E , and these agree =/ D P . S / ; Q x x .x;/ .x;/  on overlaps of such open sets to give a global isomorphism: T /: Q ; Š H Proposition 11.3. . S om =X P E This is a special case of the statement that with notation as in the proposition the Proof: E / ! X is G.k; relative tangent bundle of the Grassmannian bundle T D H om . S ; Q /: /=X .k; Gr E Over an open subset where E is trivial this is an immediate consequence of the iso- morphism described in Section 3.2.4 between the tangent bundle of a Grassmannian and the bundle . S ; Q / , and as in that setting the fact that these isomorphisms om H are independent of choices says they fit together to give the desired isomorphism D H om . T ; Q / . S E /=X Gr .k; Using the exact sequence  ! S !  0 E ! Q ! 0; Proposition 11.3 yields an exact sequence  ! O ! T !  0; E ̋ 0 ! .1/ O E P E P P E =X

408 394 Chapter 11 Contact problems relative Euler sequence the . Applying the formula for the Chern classes of the tensor product of a vector bundle with a line bundle (Proposition 5.17), we arrive at the following theorem: is a vector bundle of rank r C 1 on the smooth variety X , then the E If Theorem 11.4. are T Chern classes of the relative tangent bundle E =X P k   X i r C 1 i k T / . c D . ; / E c i P k =X E i k i D 0 1 D P c . O via / .1// 2 A  . where E / and we identify A.X/ with its image in A. P E E 1 P the pullback map. 11.1.3 Chern classes of contact bundles Returning to Keynote Question (a), we again let 3 .L;p/ 2 G .1;3/  P g j p 2 L ˆ Df G , this is the . Via the projection  W ˆ ! G .1;3/ be the universal line over .1;3/ P E of the universal subbundle S on G .1;3/ . Let projectivization be the bundle on ˆ S given by 4  .ˇ .d//; O P E D 3 P .1;3/ G ˆ= 3 ˇ W ˆ ! P on the second factor. By Theorem 11.2, is the projection .L;p/ 7! p where 0 5 this has fiber H E . O . Thus, counting .d/= m D ˆ .d// at a point .L;p/ 2 L .L;p/ p multiplicities, the number of lines having a point of contact of order at least 5 with a d is the degree of the Chern class c . E / . general surface of degree 5 c , we recall first the description of the Chow ring of . E / To find the degree of ˆ 5 given in Section 9.3.1: Since D P S ! G .1;3/ ˆ G , and is the projectivization of the universal subbundle on .1;3/ ; . S / D   c c D . S / and 2 1 11 1 Theorem 9.6 yields 2 D A. .1;3//Œç=. A.ˆ/  G  C  /; 11 1 1 2 A where .ˆ/  O . Recall, moreover, .1/ is the first Chern class of the line bundle S P 3  ˇ D ˇ  ! , where  W ˆ ! P that the class is the can also be realized as the pullback 3 1 .L;p/ on the second factor and ! 2 A p . P projection / is the hyperplane class. 7! The relative tangent bundle T . By Theorem 11.4, its ˆ is a line bundle on .1;3/ ˆ= G first Chern class is c  . T : 2 / D 1 1 .1;3/ ˆ= G

409 Lines meeting a surface to high order Section 11.1 395 4  D E By Theorem 11.2, the bundle .ˇ P O has a filtration with .d// 3 P G .1;3/ ˆ= successive quotients    4 O ˇ O .d/; ˇ ̋  O .d/ ;:::; ˇ .d/ ̋ Sym  : 3 3 3 .1;3/ G ˆ= G .1;3/ ˆ= P P P m -th The bundle is dual to the relative tangent bundle T , so its  G .1;3/ .1;3/ G ˆ= ˆ= symmetric power has Chern class m Sym c. 2/: / D 1 C m.  1 .1;3 G ˆ=  With the formula c .ˇ O d .d// D , this gives 3 1 P  m c.ˇ O Sym ;  m .d/ ̋ D 1 C .d 2m/ C / 3 1 ˆ= G .1;3 P and altogether 4 Y .d D c. /: m .1 C / 2m/ C E 1 0 D m In particular, ..d . E / D d  ..d 2/ C  /: /  c 4/ C 2 4 /  ..d 6/ C 3 C /  ..d 8/ 1 1 1 1 5 We can evaluate the degrees of monomials of degree 5 in  and  by using the 1 Segre classes introduced in Section 10.1, and in particular Proposition 10.3: We have a b b a b . / D deg  //; .   deg / / D deg .s S . 1 a  1 1 1 S . / is the s -th Segre class of S . Combining Proposition 10.3 and the Whitney where k k formula, we have 1 C D s. S ; D c. Q / D 1 /   C 1 2 / c. S and so we have 4 4 deg / 2; D deg . . D / ˆ G .1;3/ 1 1 2 3 4 . / . D deg  deg 2; / D ˆ .1;3/ G 1 1 3 2 2 deg . / 1: D deg . D   / 2 ˆ .1;3/ G 1 1 5 and  are all zero: The remaining monomials of degree 5 in   since the D 0 1 1 4 5 .1;3/ is four-dimensional, while  Grassmannian  because the Segre D  G D 0 1  vanish above degree 2 (alternatively, since  D ˇ classes of ! is the pullback of a S 3 4 P  ). D 0 class on we see immediately that c E . Putting this together with the formula above for / , we get 5 deg . E / c 5  /..d d..d 2/ C  D /..d 4/ C 2 / deg 6/ C 3 4 /..d 8/ C 1 1 1 1  4 3 2 3 2 2 192/ d.35d  24d D C deg C 200d C 240/ d.50d  1 1 1 2 3 35d 200d D C 240d:

410 396 Chapter 11 Contact problems Assuming there are only finitely many and counting multiplicities, this is the number of lines having a point of contact of order at least 5 with S . To answer the keynote question, we need to know in addition that for a general 3 S S of degree d  surface all the lines having a point of contact of order 5 with  P 5 “count with multiplicity 1” — that is, all the zeros of the corresponding section of the bundle E on ˆ are simple zeros. To do this, we invoke the irreducibility of the incidence correspondence N .L;p;S/ P m j .S  L/  5 g ˆ  Df Ä ; p introduced in Section 11.1. By virtue of the irreducibility of , it is enough to show Ä .L;p;S/ 2 Ä the section of E corresponding to S has a simple that at just one point .L;p/ for which this is not the case, ˆ : Given this, the locus of .L;p;S/ zero at 2 Ä , will have strictly smaller dimension, and so being a proper closed subvariety of N P .L;p;S/ , Exercise 11.17 suggests cannot dominate . As for locating such a triple S one. We should also check that for general, no line has a point of contact of order at least 6 with S , or more than one point of contact of order at least 5; this is implied by Exercise 11.18. This completes the proof of Theorem 11.1. 11.2 The case of negative expected dimension In this section, we will describe a rather different application of the contact calculus developed so far: We will use it to bound the maximum number of occurrences of some phenomena that occur in negative “expected dimension.” 3 P We begin by explaining an example. We do not expect a surface of degree S  4 to contain any lines. But some smooth quartics do contain a line and some contain  d d contain? several. Thus we can ask: How many lines can a smooth surface of degree d We observe to begin with that the number of lines a smooth surface of degree can N P be the space of surfaces of degree contain is certainly bounded: If we let  4 , d and write N Df .S;L/ 2 P j  G .1;3/ † L  S g N for the incidence correspondence, then the set of points of over which the fiber P N N W † ! P for is finite of degree  m is a locally closed subset of P of the map  3 any . Since, as we saw in Section 2.4.2, a smooth surface in P of degree >2 cannot m contain a positive-dimensional family lines, by the Noetherian property the degrees of N the fibers over the open set U  P of smooth surfaces are bounded. We can thus ask in particular:

411 The case of negative expected dimension Section 11.2 397 3 What is the largest number  P Question 11.5. M.d/ of lines that a smooth surface S can have? d of degree Remarkably, we do not know the answer to this in general! The situation here is typical: there is a large range of quasi-enumerative problems where the actual number is indeterminate because the expected dimension of the solution set is negative. In general, almost every time we have an enumerative problem there are analogous “negative expected dimension” variants. For example, we can ask: n X  P Question 11.6. of (a) How many isolated singular points can a hypersurface have? d degree 2 (b)  P How many tritangents can a plane curve of degree d have? How many C hyperflexes? (c) How many cuspidal curves can a pencil of plane curves of degree have? How d many reducible ones? How many totally reducible ones (that is, unions of lines)? ́ We can even go all the way back to B ezout, and ask: How many isolated points of intersection can n Question 11.7. k linearly independent C n hypersurfaces of degree d in P have? Here there is at least a conjecture, described in Eisenbud et al. [1996] and proved in the case k 1 for reduced sets of points by Lazarsfeld [1994, Exercise 4.12]. For D a general discussion of these questions (and a more general conjecture), see Eisenbud et al. [1996]. All of these problems are attractive (especially Question 11.7). But we will not pursue them here; rather, we will focus on the original problem of bounding the number 3 of lines on a smooth surface in P , in order to illustrate how we can use enumerative methods to find such a bound. 3 P 11.2.1 Lines on smooth surfaces in S of degree d Since the number of lines on a smooth surface 4 is variable, it  cannot be the solution to an enumerative problem of the sort we have been considering. But we can still use enumerative geometry to bound the number. What we will do is to find a curve F on S whose degree is determined enumeratively and such that F contains all the lines on . In this we follow a line of argument proposed in Clebsch [1861, p. 106]. S A natural approach is to relax the condition that a line L be contained in S to the condition that L meets S with multiplicity  m at some point p 2 S . We can adjust m so that the set of points p for which some line satisfies this condition has expected dimension 1, defining a curve on the surface (as we will see, the right multiplicity is 4). Since this curve must contain all the lines lying on the surface, its degree — which we can compute by enumerative means — is a bound for the number of such lines.

412 398 Chapter 11 Contact problems Flecnode Ordinary node Figure 11.1 A flecnode is a node in which one branch has a flex at the node. 3 P if there exists a line L  flecnodal S p is First of all, we say that a point 2 p ; let F  S be the locus of such points. S having contact of order 4 or more with at (The reason for the name comes from another characterization of such points: for a , a general flecnodal point p 2 S general surface S will be one such that the intersection \ S S has a flecnode at p , that is, a node such that one branch has contact of order T p at least 3 with its tangent line; see Figure 11.1.) As we will show in Proposition 11.8, will always have the flecnodal locus of a smooth surface of degree d  3 F  S dimension 1. As we have observed, any line lying in S is contained in the flecnodal locus F . d D 3 any line meeting S with multiplicity  4 must lie in S , so Of course when S . To describe the locus of the flecnodal locus is exactly the union of the 27 lines in more generally, we again write for the incidence correspondence flecnodes on ˆ S 3 2 G .1;3/  ˆ Df j p 2 L g ; .L;p/ P 3 1 2 and we let .ˆ/ be the pullbacks of the corresponding classes on P ; and A 1 .1;3/ . Given a surface S , we wish to find the class of the locus G Df .L;p/ 2 ˆ j m Ä .L  S/  4 g : p 3 F  S is the image of Ä under the projection of ˆ to P Since the flecnodal locus , knowledge of this class will determine in particular the degree of . F  3 D P , consider the bundle Ä of O To compute the class of F . .d// 3 2 P .1;3/ G ˆ=  ˆ third-order relative principal parts of O whose  .d/ . It is a bundle of rank 4 on 3 2 P is the vector space of germs of sections of O p .d/ at fiber at a point , modulo .L;p/ L p : those vanishing to order at least 4 at 0 4 H F . O .d//: .d/= I D L .L;p/ p 0 A 2 H If . O , S .d// is a homogeneous polynomial of degree d defining a surface 3 P 3 A L  P then the restrictions of to each line  , of the bundle F yield a global section A whose zeros are the pairs .L;p/ such that L meets S with multiplicity  4 at p . has rank 4, the locus Since D 5 and F ˆ Ä (if not empty) is at least one- dim dimensional; if it has dimension exactly 1 then its class is the top Chern class 4 D c .ˆ/: . F / 2 A ŒÄç 4

413 The case of negative expected dimension Section 11.2 399 F by order of vanishing — We can calculate this class as before: We can filter the bundle that is, invoke the exact sequences (11.1) — and apply the Whitney formula to arrive at . F / D d  ..d 2/ C  c /  ..d 4/ C 2 /: /  ..d 6/ C 3 1 1 4 1 Of course, none of this will help us bound the number of lines on S if every point of S is a flecnode! The following result, which was assumed by Clebsch, is thus crucial for this approach. A proof can be found in McCrory and Shifrin [1984, Lemma 2.10]. For partial results in finite characteristic see Voloch [2003]. 3 S  P Proposition 11.8. is a smooth surface of degree d  3 over a field of If characteristic 0, the locus  Df ˆ j m Ä .L 2 S/  4 g .L;p/ p S has dimension 1. In particular, the general point of is not flecnodal. We will defer the proof of this proposition to the next section, and continue to derive F S of S our bound on the number of lines. By the proposition, the flecnodal locus  is a curve, whose degree is the degree of the intersection of with the class  . We can Ä evaluate this as before: 2 .F/ D d  ..d 2/ C  / /  ..d 4/ C 2 3 /  ..d deg 6/ C 1 1 1 d.11d D 24/: : S Putting this together, we have proven a bound on the number of lines in M.d/ of lines lying on a smooth surface The maximum number Proposition 11.9. 3 P is at most of degree d  3 S d.11d  24/ . In the case D 3 this gives the exact answer since d.11d 24/ d 27 . But D for d  4 the bound is not sharp: Segre [1943] proved the slightly better bound M.d/  d.11d 28/ C 12 . Even this is not sharp; for example with d D 4 we have d.11d C 12 D 76 , but Segre also showed that the maximum number of lines on a 28/ M.4/ D 64 . smooth quartic surface is exactly M.d/ simply by exhibiting a surface with Of course, we can give a lower bound for d d d d y a large number of lines. The Fermat surface C z V.x C w C / , for example, has 2 2 exactly lines (Exercise 11.25), whence M.d/  3d . This is still the record-holder 3d for general d . More is known for some particular values of d ; Exercises 11.26 and 11.27 exhibit surfaces with more lines in the cases d D 4;6;8;12 and 20 (respectively, 64, ` 180, 256, 864, and 1600 lines), and Boissi ere and Sarti [2007] find an octic with 352 (the current champion!).

414 400 Chapter 11 Contact problems 11.2.2 The flecnodal locus 3 It remains to prove that for any smooth surface S of degree d  3 the locus P  ˆ of pairs with m Ä .L  S/  4 has dimension 1. The following proof was  .L;p/ p shown us by Francesco Cavazzani: Proof of Proposition 11.8: Suppose on the contrary that the locus has a com-  ˆ Ä of dimension 2 or more, and let ;p / be a general point of Ä Ä . Since the ponent .L 0 0 0 0 over a point 2 Ä S consists of lines through p in T fiber of S , it has dimension at p p p most 1. By Theorem 7.11(a), there are only finitely many points over which the fiber , so must dominate S has positive dimension. Thus p Ä is a general point of S . By 0 0 \ T Theorem 7.11(b), the intersection . S has a node at p S 0 p 0 ˆ and writing down the defining We will proceed by introducing local coordinates on 3 3 Ä equations of the subset  P A and choose . To start with, we can find an affine open 3 3 .x;y;z/ so that the point p coordinates is the origin .0;0;0/ 2 A on and the line A 0 L g is the x .x;0;0/ T -axis; we can also take the tangent plane to be the plane Df S 0 p 0 D , and, given that the tangent plane section S \ T 0 , we can take p has a node at z S p 0 0 V.z;xy/ p \ T the tangent cone at - x to be the union S of the to the intersection S p 0 and y -axes. We can take coordinates in a neighborhood U of .L so that ;p ˆ / 2 .a;b;c;d;e/ 0 0 if .L;p/ is the pair corresponding to .a;b;c;d;e/ then p D .a;b;c/ and L Df .a C t;b C dt;c C et/ g : 3 Let in A f.x;y;z/ . If we write the restriction of f be the defining equation of S as to L X i ; .a;b;c;d;e/t t;b C dt;c C et/ D j D C f.a f ̨ i L  0 i ; ̨ ; ̨ and the four functions ̨ will be the defining equations of Ä in U . We want ̨ 1 0 2 3 ˆ ; we will actually prove the to show their common zero locus has codimension 4 in .L ;p / are independent. stronger fact that their differentials at 0 0 , L S/ T By the specifications above of p S and T C , we can write .S \ T , p 0 p p 0 0 0 0 3 4 z  u.x;y;z/ C xy  v.x;y/ C y f.x;y;z/  l.y/ C x D  m.x/: Note that since p we have u.0;0;0/ ¤ 0 , and since the tangent plane S is smooth at 0 S \ T 0 ; rescaling the coordinates, S has multiplicity 2 at p section we have v.0;0/ ¤ 0 p 0 u.0;0;0/ D v.0;0/ D 1 . Note by contrast that we may have m.0/ D 0 ; we can assume m this will be the case exactly when . .L  S/  5 p 0 0 in this expression to C t;b C dt;c C et/ in for .x;y;z/ .a Now, we can just plug f j write out , and hence the coefficients ̨ .a;b;c;d;e/ . This is potentially messy, but i L in fact it will be enough to evaluate the differentials of the ̨ at .L :p / — that is, at i 0 0

415 Flexes via defining equations Section 11.2 401 2 D .a;b;c;d;e/ .a;b;c;d;e/ . .0;0;0;0;0/ — and so we can work modulo the ideal That said, we have j f.a C t;b C dt;c D et/ D f C L 3 4 .a C t/.b C dt/v .c .b C dt/ C l.b C dt/ C .a C t/ et/u m.a C t/; C C and thus 2  c mod .a;b;c;d;e/ ̨ ; 0 2 e C b mod  C .a;b;c;d;e/ ̨ ; .c/ 1 2 mod .b;c;e/ C .a;b;c;d;e/ d ; ̨  2 and finally 2  4a ̨ m.0/ mod .b;c;d;e/ C .a;b;c;d;e/  : 3 the differentials of ;p are linearly ;:::; ̨ / at .L ̨ What we see from this is that 0 3 0 0 independent, unless m.0/ D 0 ; or in other words, if there is a two-dimensional locus †  ˆ of pairs .L;p/ such that m 5 .L  S/  4 , then in fact we must have m  .L  S/ p p .L;p/ for all 2 † . But we can carry out exactly the same argument again to show that if , then S there is a two-dimensional family of lines having contact of order 5 or more with S , and so on. We conclude all these lines in fact have contact of order 6 or more with has dimension 2 or more, then S must be ruled by lines; in other words, S must that if Ä be a plane or a quadric. 11.3 Flexes via defining equations 2 In our initial discussion of flexes in Section 7.5, we gave the curve C  P in 2 z  W question C ! P — that is, as the image of a map from a smooth parametrically 2 z z C , to P curve C p 2 , the normalization of C such that . We defined flexes as the points 2   P L/ the multiplicity m . . for some line L is at least 3 p This definition does not work well in families of curves. As we shall see, when a smooth plane curve degenerates to one with a node, a certain number of the flexes approach the node; but, according to the definition in Section 7.5, the nodal point will generally not be a flex, since in general neither branch of the node will have contact of order 3 or more with its tangent line. Thus to track the behavior of flexes in families we need a different way of describing them, related to the notion of Cartesian flexes described in Section 7.5.2. We will call the objects described below “flex lines” rather than flexes. 2 2  C of flex line to be a pair .L;p/ with L  P We define a a line and p 2 L P a point such that C and L intersect at p with multiplicity  3 ; that is, the set Ä of flex lines is the locus 2  2 m 2 .L;p/ Df  P Ä j P g .C  L/  3 : p

416 402 Contact problems Chapter 11 F Thus if is a flex line if and only is the vanishing locus of a polynomial , then C .L;p/ L p ; in other words, instead of to F if the restriction of vanishes to order at least 3 at C taking the defining equation of L and restricting to (or, more precisely, pulling back ), we are restricting the defining equation of to L . The two to the normalization of C C C is a general is smooth, but different in general: For example, if are the same when C , the tangent lines to the two branches are flex lines at the node. curve with a node at p , we To compute the number of flexes on a curve defined by a homogeneous form F ‰ to be the incidence correspondence define 2  2 2 ‰  P Df j p .L;p/ L g ; 2 P 2  , and consider P thought of as the universal line over 2  .d//; O . E D P 2  2 2 P ‰= P , whose fiber at a point is a rank-3 vector bundle on the three-dimensional variety ‰ .L;p/ 0 3 D . E .d//: .d/= I H O L .L;p/ p F gives rise to a section  The homogeneous polynomial of E , and the zeros of this sec- F tion correspond to the flex lines of the corresponding plane curve D V.F/ . Thus the C number of flex lines — when this number is finite, of course, and counting multiplicity — 3 / . E is the degree of 2 A c .‰/ . 3 Since the projection on the first factor expresses ‰ as the projectivization 2  P S ! P ‰ D 2  of the universal subbundle S , we can give a presentation of the Chow ring on P over .1;3/ in Section 11.1.3. Letting exactly as in the case of the universal line ˆ G 2  1 . A 2 / be the hyperplane class, we have P  2  2 2 A.‰/ /Œç=. D A.  C  P /; 1  where .ˆ/ is the first Chern class of the line bundle O A 2 .1/ . Recall from S P  can also be realized as the pullback  D ˇ Section 9.3.1, moreover, that the class ! ,  2 2 1 ˆ ! P where is the projection .L;p/ 7! p on the second factor and ! 2 A ˇ . P W / is the hyperplane class. and  as before by  We can also evaluate the degrees of monomials of degree 3 in using the Segre classes introduced in Section 10.1, and in particular Proposition 10.3: We have 2 3 2 3 D deg . deg / D 1 and deg . . / D deg . / / D 0: 2  2 is a hypersurface of (We could also see these directly by observing that   P P ‰ bidegree .1;1/ , and the classes  and  are the pullbacks of the hyperplane classes in the two factors.)

417 Flexes via defining equations Section 11.3 403 E by applying the exact sequences We can now calculate the Chern classes of (11.1) and using Whitney’s formula, and we get / / D d  ..d 2/ C E  ..d 4/ C 2/: . c 3 Hence deg . E // D 2d.d 2/ C d.d 4/ C 2d .c 3 3d.d 2/: D This shows that the number of flex lines, counted with multiplicity, is the same in the singular case as in the smooth case, whenever the number is finite. (Note that if D F defines a nonreduced curve, or a curve containing a straight line as a component, 0 the section defined by vanishes in the wrong codimension.) The present derivation F allows us to go further in two ways, both having to do with the behavior of flexes in families. In particular, it will permit us to solve Keynote Question (c). 11.3.1 Hyperflexes to a plane curve similarly: It is a pair .L;p/ such We define a C hyperflex line that C meet with multiplicity at least 4 at p . As with ordinary flex lines (and L and for the same reason), this definition is equivalent to the definition of a hyperflex given p is a smooth point of C in Section 7.5 when the point , but not in general: If a curve 2 P C has an ordinary flecnode at p (that is, two branches, one not a flex and the  C p other a flex that is not a hyperflex), then the tangent line to the flexed branch of at p is not a hyperflex in the sense of Section 7.5. Since will be a hyperflex line, though a general pencil of plane curves will not include any elements possessing a flecnode (Exercise 11.29), this will not affect our answer to Keynote Question (c). N To describe the locus of hyperflex lines in a family of curves, we denote by P the space of plane curves of degree d , and consider the incidence correspondence N .L ‰  P † j m .L;p;C/ 2  C/  4 g : Df p d  3 , the fibers of the projection † ! ‰ are linear spaces of dimension N 4 , When † N 1 ; in particular, it follows that a is irreducible of dimension from which we see that 2  P general curve of degree d > 1 has no hyperflexes. Furthermore, since for d  4 C N † ! P the general fiber of the projection is finite (see the proof of Theorem 7.13), the N N locus „ of curves that do admit a hyperflex is a hypersurface in P P in this case.  Keynote Question (c) is equivalent to asking for the degree of this hypersurface in the case d D 4 . We will actually compute it for all d . 2  2 P ‰  P To do this, we consider the three-dimensional variety as above, and  introduce the rank-4 bundle 2  3 D P E O . .d//  2 2 P ‰= P

418 404 Chapter 11 Contact problems is .L;p/ whose fiber at a point 4 0 O E .d/= I . H D .d//: L .L;p/ p F With this definition in hand, we consider a general pencil C t f G g t of 1 1 0 P 2 t 2 homogeneous polynomials of degree d . The polynomials F;G give rise to on P  ; sections of E , and the set of pairs .L;p/ that are hyperflexes of some element G F of our pencil is the locus where these sections fail to be linearly independent. Thus the 3 . . / 2 A number of hyperflex lines, counted with multiplicities, is the degree of .‰/ E c 3 E by order of vanishing, we arrive at We do this as before: Filtering the bundle the expression E / D .1 C d/.1 C .d 2/ C /.1 C .d 4/ C 2/.1 C .d 6/ C 3/: c. Thus 2 2 2 D .18d c 88d C 72/ E  . .22d 36/ / C 3 2 D 18d 66d C 36 6.d 3/.3d 2/: D d This gives zero when , as it should: a cubic with a hyperflex is necessarily D 3 reducible, and a general pencil of plane cubics will not include any reducible ones. We D 1 also remark that the number is meaningless in the cases d D 2 , since every d and point on a line is a hyperflex and a pencil of conics will contain reducible conics equal to the union of two lines. To show that the actual number of elements of a general pencil possessing hyper- flexes is equal to the number predicted, we have to verify that for general polynomials F G the degeneracy locus V. and ^  /  ‰ is reduced. We do this, as in the argument F G carried out in Section 7.3.1, in two steps: We first use an irreducibility argument to reduce the problem to exhibiting a single pair .L;p/ 2 ‰ such F;G of polynomials and a point is reduced at  / V. .L;p/ , then use a local calculation to show that there do that ^ F G and .L;p/ . indeed exist such F;G N For the first, a standard incidence correspondence suffices: We let P be the space d and G D G .1;N/ the Grassmannian of pencils of such of plane curves of degree curves, and consider the locus has a hyperflex line at . D ;L;p/ 2 G  ‰ j some C 2 D Df .L;p/ g : ‡ ‡ over .L;p/ is irreducible of dimension 2N 5 : It is the Schubert cycle The fiber of N is the codimension-4 subspace of .ƒ/  G , where ƒ Df C 2 P † j m g .L  C/  4 p 3 N .L;p/ . It follows that ‡ is irreducible P consisting of curves with a hyperflex line at 0 2 D dim of dimension . Now, if ‡ 2N  ‡ is the locus of . D ;L;p/ such that G V. is the pencil spanned by ^  D / is not reduced of dimension 0 at .L;p/ (where G F 0  ‡ is closed, G ), then, since ‡ F and 0 0 ¤ ‡ H) dim ‡ 2; < 2N ‡

419 Flexes via defining equations Section 11.3 405 0 0 ‡ then ‡ ‡ cannot dominate G . and it follows that if ¤ and Thus we need only exhibit a single V. .L;p/ ^  F;G / is reduced such that G F 2 2 with coordinates P at . We do this using local coordinates. Choose x;y so .L;p/  A 2 .0;0/ is the origin and L that A p is the line y D 0 . Set f.x;y/ D F.x;y;1/ D  g.x;y/ G.x;y;1/ . and D in a neighborhood of the point .L;p/ ‰ For local coordinates on , we can take the functions x;y and b , where D .x;y/ and L Df .x C t;y C p g : bt/ 2 t k E We can trivialize the bundle , so that the section in this neighborhood of .L;p/ of is given by the first four terms in the Taylor expansion of the polynomial E  F t;y C f.x around t D 0 . Thus, for example, the section associated to the C bt/ d 4 1 4 4 d D x F.x;y;z/ (that is, x C C y z D f.x;y/ polynomial ) is represented yz 4 y bt C .x C by the first four terms in the expansion of t/ : C 4 3 2 4x ;b C ;6x ;4x/; D x C .y  F P i j  a repre- x and the general polynomial y g.x;y/ gives rise to the section D i;j G sented by the vector a a .a x C a /: y C ; a C C a ; a b C C C y C 2a ; a x C 2;0 2;0 1;1 0;1 3;0 1;0 0;1 1;0 0;0 2 .) The section (Here we are omitting terms in the ideal .x;y;b/ is given by ^   G F 2 2 minors of the matrix  the   2 4 3 C 4x b 6x x C y 4x : C C C a a a a C 3;0 2;0 0;0 1;0 We have minors with linear terms a , y a x b , a 4a y 4a x and a b 3;0 0;0 2;0 0;0 1;0 3;0 a these are independent. This shows that the section and for general values of the i;j , as required. Thus: ^   vanishes simply at p G F In a general pencil of degree- d plane curves, exactly 6.d 3/.3d Proposition 11.10. 2/ will have hyperflexes; in particular, in a general pencil of quartic plane curves, exactly 60 members will have hyperflexes. 11.3.2 Flexes on families of curves We can also use the approach via defining equations to answer another question about flexes in pencils, one that sheds some more light on how flexes behave in families. Again, suppose that f C is a general pencil of plane curves of D V.t g F C t G/ 1 1 t 0 2 P t 2/ . The general member C degree of the pencil will have, as we have seen, 3d.d d t flex points, and as t varies these points will sweep out another curve B in the plane. We can ask: What are the degree and genus of this curve? What is the geometry of this curve

420 406 Contact problems Chapter 11 Figure 11.2 The singular elements of a pencil of conics are the pairs of lines joining the four base points. around singular points of curves in the pencil? We will answer these questions in this section and the next. To this end, we again write 2  2  ‰ Df .L;p/ 2 j p 2 L g ; P P and set 1 g 3  m  ‰ j : / .L  C Ä 2 .t;L;p/ Df P t p as the zero locus of a section of a rank-3 vector bundle on the Ä We will describe 1 . For  ‰ four-dimensional variety d  2 , we will show that Ä has the expected P Ä is smooth by completing dimension 1, and we will ask the reader to show that in fact the sketch given in Exercise 11.33. This will allow us to determine not only the class of Ä 1 2 under the projection P (which will give us the degree of its image  ‰ ! ‰ ! P B ) 1 but its genus as well. We will also describe the projection of P , which tells how to Ä the flexes may come together as the curve moves in the pencil. d D 2 , is easy to analyze directly, and already exhibits The case of a pencil of conics, some of the phenomena involved. As we saw in Proposition 7.4 and the discussion immediately following, a general pencil of conics will have three singular elements, each consisting of two of the straight lines through two of the four base points of the pencil. A smooth conic has no flexes, while the flex lines of a singular conic C are the pairs 2 with p 2 L  C . Thus the curve B , consisting of points p 2 P .L;p/ such that some .L;p/ is a flex line of some member of the pencil, is the union of the singular members of the pencil — that is, the union of the six lines joining two of the four base points. As such it has degree 6, four triple points, and three additional double points. However, the points of the curve Ä “remember” the flex line to which they belong, so Ä is the disjoint union of the six lines — a smooth curve, which is the normalization of B . The singularities of B are typical of the situation of pencils of curves of higher degree, as we shall see: In general, B will have triple points at the base points of the pencil and nodes

421 Flexes via defining equations Section 11.3 407 at the nodes of the singular elements of the pencil. In the case of conics, the projection 1 Ä has three nonempty fibers, each consisting of one of the singular members ! map P of the pencil. For general pencils of degree > 2 we shall see that the projection is a finite cover. E for the rank-3 vector bundle Returning to the general case, we again write 2  D P . E O .d//: 2  2 2 P ‰= P F and Writing , the sections  V for the two-dimensional vector space spanned by G F define a map of bundles and  G ̋ V ! E : O ‰ 1 1 1 via the projection  W P P  ‰ ! We now pull this map back to . If P  D P V ‰ ‰ is the projective line parametrizing our pencil, we also have a natural inclusion 1/ , ! . ̋ O O ; V P V V P 1 1 1 P  W P which we can pull back to the product  ‰ ! P ‰ . via the projection  Composing these, we arrive at a map   O  W . 1/ ! V ̋  I E O  ! 1 1 ‰  P P 1 2 P over the point  ‰ , this is the map that takes a scalar multiple of t .t;L;p/ F C t G 1 0 ). In particular, (modulo sections of to its restriction to .d/ vanishing to order 3 at L O p L the zero locus of this map is the incidence correspondence Ä .  Tensoring with the line bundle O  as a section of the .1/ , we can think of  1 P    O is thus given by the Chern class Ä .1/ ̋ bundle  E ; the class of 1 P  1 3  D c . ŒÄç O P  E ‰/: 2 A .1/ . ̋  / 1 3 P 1 1 1 the class of a point in A P  / , or its pullback to P We denote by  ‰ . Similarly, we .  and  , introduced as classes in A.‰/ above, to denote the pullbacks use the notation 1 of these classes to  ‰ . With this notation we have P   O 2/: C .1/ ̋  c. E / D .1 C  C d/.1 C  C .d 2/ C /.1 C  C .d 4/ 1 P Collecting the terms of degree 3, we get   c O / E .1/ ̋ .  1 3 P 2 2 2 2 2 2  C 2d .3d C ..3d /: 12d C 8/ C .6d 8/ C 2 8d/ D 2 To find the degree of the curve swept out by the flex points of members of P  B 2 L  P the family, we intersect with the (pullback of the) class ; we get  of a line    c deg . .B/ O 6: 6d .1/ ̋  D E / D  1 3 P Note that this yields the answer 6 in the case d D 2 , consistent with our previous analysis.

422 408 Chapter 11 Contact problems Ä . As We can use the same constructions to find the geometric genus of the curve 1 we observed in Proposition 6.15, the normal bundle to Ä ‰ is the in the product P  2  2   P ̋  O E . Since ‰  restriction to .1/   P of the bundle is a hypersurface Ä 1 P , its canonical class is of bidegree .1;1/ K I T 2 / D K D . 2 c D C  C   2 2 1 ‰ ‰  P P it follows that K 2: D 2 2 1 ‰  P By the calculation above, 3; . E / D 3 C c 6/ C .3d 1 and so we have 8/ D . C .3d K : C / j Ä Ä We have seen that the degree of  j ; similarly, we is 3d.d 2/ , and deg . j 6 / D 6d Ä Ä can calculate 2 2 C 3d 6d: 12d C 8/ D .6d 8/ deg .3d D / j . Ä Altogether, we have 2 2 D deg .K 2g.Ä/ / D 24d 78d C 48; Ä and so 2 D 12d 39d C 25: g.Ä/ d D 2 this yields g.Ä/ D 5 , as it should: As we saw, in this case Note that when 1 consists of the disjoint union of six copies of P Ä . 11.3.3 Geometry of the curve of flex lines We will leave the proofs of most of the assertions in this section to Exercises 11.34– 11.38; here, we simply outline the main points of the analysis. We begin with the geometry of the plane curve traced out by the flex points of the B 2 . We have already — that is, the image of the curve Ä under projection to P curves C t seen that the degree of is 6d 6 B . The singularities of B can be located as follows: At each base point p of the pencil, all members of the pencil are smooth. We will see in Exercise 11.34 that three members of the pencil have flexes at , so that B has a triple point at each base point of the pencil. p 2 B occur at points p 2 P C that are nodes of the curve The only other singularities of t containing them. As we have seen, at such a point the tangent lines to the two branches are each flex lines to C is two-to-one there; as we will verify in , so that map Ä ! B t Exercise 11.35, the curve B will have a node there.

423 Cusps of plane curves Section 11.3 409 Ä Since the projection is the normalization, these observations give another ! B 2 : There are in general base points of the d Ä way to derive the formula for the genus of 2 1/ nodes of elements C of our 3.d pencil, and as we saw in Chapter 7 there will be t pencil, so that the genus of Ä is 2 2 .B/ 3d D p 3.d 1/ g.Ä/ a 2 2 1 8/ 3d .6d 7/.6d 3.d 1/ D 2 2 D 39d C 25: 12d We can study the geometry of the curve Ä in another way as well: via the projection 1 ! on the first factor. Since a general member of our pencil has 3d.d 2/ flexes, Ä P 1 3d.d 2/ cover of the line P parametrizing our pencil. Where is this is a degree Ä ̈ ucker formula of Section 7.5.2 shows that if C cover branched? The Pl is smooth it t can fail to have exactly 2/ flexes only if it has a hyperflex, in which case the 3d.d hyperflex counts as two ordinary flexes. Such hyperflexes are thus ordinary ramification 1 ! points of the cover Ä . P That leaves only the singular elements of the pencil to consider, and this is where it gets interesting. By the formula of Section 7.5.2, a curve of degree d with a node has genus one lower, and hence six fewer flexes (in the sense of that section), than a smooth curve of the same degree. If C is a singular element of a general pencil of plane curves, t 0 t three of the flex lines of the curves ! approach each of the tangent lines t then as C 0 t to the branches of C at the node (Exercise 11.38). Thus each of the tangent lines to the t 0 1 branches of C at the node is a ramification point of index 2 of the cover Ä ! P . t 0 We can put this all together with the Riemann–Hurwitz formula to compute the Ä yet again: Since there are 6.d 3/.3d 2/ hyperflexes in the pencil, and genus of 2 1/ singular elements, 3.d 2 2 D 2  3d.d 2/ C 6.d 3/.3d 2/ C 4  3.d 1/ 2g.Ä/ ; and so 2 D 3d.d 2/ C 3.d 3/.3d 2/ C 2  3.d 1/ 1 C g.Ä/ 2 D 39d C 25: 12d 11.4 Cusps of plane curves As a final application we will answer the second keynote question of this chapter: 2 P How many curves in a general net of cubics in have cusps? This will finally complete 9 our calculation, begun in Section 2.2, of the degrees of loci in the space P of plane cubics corresponding to isomorphism classes of cubic curves. Solving this problem requires the introduction of a new class of vector bundles that further generalize the idea of the bundles of principal parts.

424 410 Chapter 11 Contact problems ordinary cusp of a plane curve We start by saying what we mean by a cusp. An C such that, in an analytic neighborhood of in the over the complex numbers is a point p p 2 3 in suitable (analytic) coordinates. y can be written as C D 0 plane, the equation of x k other than the complex numbers, If we were working over an algebraically closed field C at p we could say instead that the completion of the local ring of is isomorphic to 2 3 x k / , and this is equivalent when k D C . Similar generalizations can be ŒŒx;yçç=.y made for many of the remarks below. It is inconvenient to do enumerative geometry with ordinary cusps directly, because the locus of ordinary cusps in a family of curves is not in general closed: ordinary cusps 2 3 n can degenerate to various other sorts of singularities (as in the family y C x tx when t ! 0 ). For this reason we will define a cusp of a plane curve to be point where the Taylor expansion of the equation of the curve has no constant or linear terms, and where the quadratic term is a square (possibly zero). As will become clearer in the next section, this means that a cusp is a point at which the completion of the local ring of the curve, in some local analytic coordinate system, has the form 2 y D k ŒŒx;yçç=.ay O C terms of degree at least 3 /; C;p a where is a constant that may be equal to 0. From the point of view of a general net of curves of degree at least 3, the difference between an ordinary cusp and a general cusp, in our sense, is immaterial: Proposition 11.13 will show that no cusps other than ordinary ones appear. It is interesting to ask questions about curves on other smooth surfaces besides 2 P . Most of the results of this section can be carried over to general nets of curves in any sufficiently ample linear series on any smooth surface, but we will not pursue this generalization. 11.4.1 Plane curve singularities Before plunging into the enumerative geometry of cusps, we pause to explain a little of the general picture of curve singularities. Let p 2 C be a point on a reduced curve. In an analytic neighborhood of the point, C looks like the union of finitely many branches, each parametrized by a one-to-one map from a disc. Over the complex numbers these maps can be taken to be parametrizations by holomorphic functions of one variable; in general, this statement should be interpreted y C O is reduced of the local ring O p of to mean simply that the completion at C;p C;p and the normalization of each of its irreducible components (the branches) has the form k ŒŒtçç , where k is the ground field (if our ground field were not algebraically closed then the coefficient field might be a finite extension of the ground field). These statements are part of the theory of completions; see Eisenbud [1995, Chapter 7].

425 Cusps of plane curves Section 11.4 411 A ) Node ( ) Cusp ( A ) Tacnode ( A 3 2 1 Figure 11.3 The simplest double points. It is a consequence of the Weierstrass preparation theorem that over the complex numbers two reduced germs of analytic curves are isomorphic if and only if the comple- tions of their local rings are isomorphic, so we will use the analytic language although we will work with the completions. See Greuel et al. [2007] for more details. The theory of singularities in general is vast. But what will concern us here are dou- ble points of curves, and in that very limited setting we can actually give a classification, which we will do now. To begin with, we have already defined the notion of the of a variety multiplicity 2 X in Section 1.3.8. One consequence is that if X has multiplicity 2 X p at a point (“ p is a at of X ”), then the Zariski tangent space T p X has dimension double point p C 1 . In particular, if p is a double point of a curve C then X T dim C D 2 , so that dim p p an analytic neighborhood of C is embeddable in the plane, and hence the completion in y k O of C at p has the form of the local ring ŒŒx;yçç=.g.x;y// , where g has O C;p C;p leading term of degree exactly 2. . We say that p C be a double point of a reduced curve C 2 Let Definition 11.11. y Š O -singularity p of C (or that C has an A is -singularity at p ) if an A n n C;p 1 n 2 C k x / , that is, if in suitable (analytic) coordinates, C has equation ŒŒx;yçç=.y 1 2 C n x D 0 around p . y p For example, a double point is a node ( C has two smooth branches meet- 2 C ) if and only if C has, in suitable analytic coordinates, equation ing transversely at p 2 2 x y D 0 , and is thus an A ordinary cusp -singularity. Similarly, an is a point 1 2 3 ordinary y p x with local analytic equation D 0 ( A 2 -singularity), and an C 2 4 2 is a point with local analytic equation x tacnode y 0 ( A -singularity); this D 3 looks like two smooth curves simply tangent to one another at p . In general, if p 2 C is odd, then a neighborhood of is an n D 2m C 1 -singularity and p in C con- A n m C 1 at p (we can write sists of two smooth branches meeting with multiplicity 2m C 2 2 m C 1 m C 1 x y /.y C x D .y x / ), while if n is even then C is analytically irreducible at . p Proposition 11.12. Over an algebraically closed field any double point of a plane curve C is an . 1 -singularity for some n  A n

426 412 Chapter 11 Contact problems We work in the power series ring C , and we must show that if a power Proof: ŒŒx;yçç has nonzero leading term of degree 2 then, after multiplication by a unit series f.x;y/ n 1 2 C x C y . Since and a change of variables, it can be written in the form ŒŒx;yçç of 2 2 with C ax any nonzero quadratic form over C may be written (modulo scalars) as y 2 2 yg has the form f D y C C g.x/ C , we may assume that a .x/ C y f g for .x;y/ 2 2 1 g , g , we reduce and g .x;y/ with g g .0;0/ D 0 . Multiplying f by the unit 1 some 2 2 1 2 0 y . Making a change of variable of the form y g D 0 g .x/ (called a to the case D 2 1 g ), we can raise the order of vanishing of ; repeating this Tschirnhausen transformation 1 g 0 as well. But if g has order operation and taking the limit we may assume that D 1 C 1 , then, by Hensel’s lemma (Eisenbud [1995, Theorem 7.3]), n has an .n C 1/ -st g 2 C ax root of the form C . We may take this power series to be a new variable, and x 2 n C 1 , as required. after this change of variables we get y f.x;y/ D x N In the space P parametrizing all plane curves of given degree d , we can estimate the dimension of the locus of curves having certain types of singularities, at least when the degree of the curves is large compared with the complexity of the singularity (this is an open problem when the degree is small; see Greuel et al. [2007] for more information): N Let be the space of plane curves of degree d  k , and let P Proposition 11.13. N 2 C  Å  be the set of pairs .C;p/ such that P has an A . -singularity at p P k k N 2 k P 2 in is locally closed and has codimension C  P Å . Its closure is irreducible, k N 2 and contains in addition the locus .C;p/  P  of pairs P such that C has ˆ N 2 P  P multiplicity 3 or more at  and the locus „ of pairs .C;p/ such that p lies p on a multiple component of C ; in fact, [ Å ˆ [ „ [ D : Å l k k  l N ! Note that since the projection P Å on the first factor is generically finite, the k N P will have codimension k in Å image of . Thus, among all plane curves we will see k curves with a node in codimension 1, curves with a cusp in codimension 2 and curves with a tacnode in codimension 3; all other singularities should occur in codimension 4 and higher. Finally, note that the situation is much less clear when k is large relative to d ; for example, as we mentioned in Section 2.2, it is not known for all k whether and d d with an A there exists an irreducible plane curve of degree -singularity. In particular, k 2 d > 6 whether there exists an irreducible plane curve C  P it is not known for of degree with an A d -singularity (this is the largest value allowed by the 1/.d 2/ .d genus formula).

427 Cusps of plane curves Section 11.4 413 11.4.2 Characterizing cusps As in the case of the simpler problem of counting singular elements of a pencil of curves, the first thing we need to do to study the cusps in a net of plane curves is to linearize the problem. The difficulty arises from the fact that even after we specify a 2 p P it is not a linear condition on the curves in our linear system to have a 2 point 2 and a line  P p . It becomes linear, though, if we specify both the point L cusp at p 3 . through p with which we require our curve to have intersection multiplicity at least  2 P Thus we will work on the universal line over 2  2 P ‰  Df .L;p/ j p 2 L g ; 2 P which we used in Section 11.3 above. In the present circumstances, we also want to 2 parametrizing subschemes of . P ‰ / H as a subscheme of the Hilbert scheme think of 2 2 2 2 . P of degree 2. Specifically, it is the locus in / of subschemes of P P supported at H 2 2 2 ‰ a single point: We associate to Ä D Ä of degree 2  P .L;p/ the subscheme L;p 2 p with tangent line Ä D L  P T . supported at p For a given point .L;p/ 2 ‰ , we want to express the condition that the curve 2 V./ P  of the line bundle L D O associated to a section have a cusp .d/ on D C 2 P p with m at .C  L/  3 . This suggests that we introduce for each .L;p/ the ideal J p L;p of functions whose zero locus has such a cusp; that is, we set 3 2 C J D m I ; L;p p Ä 2 Ä D Ä .  P L is the subscheme of degree 2 supported at p with tangent line where L;p E ‰ whose fiber at a point .L;p/ is We want to construct a vector bundle on 0 2 = . E D ; L H L ̋ J /: P L;p .L;p/ 2 2 to P  To do this, consider the product and  . Let , with projection maps ‰ and P ‰  2 1 2 2  2 2 ‰ be the graph of the projection map P P   P  ! P  — in other words, ‰ Å 2 ..L;p/;q/ 2 ‰  P Å j p D q g : Df 2 2  B  P Likewise, let be the universal scheme of degree 2 over ‰  H . We . P Ä / 2 then take   2  3 D E =  I L ̋  I C L I 1  2 2 2 2  P  Å=‰ Ä=‰ P by the theorem on cohomology and base change (Theorem B.5), this is the bundle we want. A global section of the line bundle L gives rise to a section of E by restriction. Given 0 D corresponding to a three-dimensional vector space V  H a net . L / , we get three sections of , and the locus in B where they fail to be independent — that is, where some E linear combination is zero — is the locus of .p;/ such that some element of the net has a cusp at p in the direction  . In sum, observing that two elements of a general net

428 414 Chapter 11 Contact problems d > 2 cannot have cusps at the same point, and that a general cuspidal curve of degree has a unique cusp (we leave the verification of this fact to the reader), the (enumerative) c E / . In the remainder of this answer to our question is the degree of the Chern class . 3 section we will calculate this. One remark before we launch into the calculation. We are using here the fact that we can characterize the condition that a curve by saying that C contains have a cusp at p C Œx;yç=J ; the parameter space ‰ can be viewed as Spec a scheme isomorphic to k L;p 2 . We could use the same technique to count curves parametrizing such subschemes of P with tacnodes; this is sketched in Exercises 11.46–11.48. If we try to apply the same techniques to count curves with other singularities, have an -singularity however, we run into trouble. For example, the condition that A C 5 , in the classical terminology) is that C contain a scheme isomorphic to (an oscnode 3 2 Œx;yç=.y;x / . But the parameter space for such subschemes of the plane is not k Spec 3 Spec k complete (schemes isomorphic to / can specialize to the “fat point” Œx;yç=.y;x 2 Spec k ), and if we try to complete it in the most natural way, by scheme Œx;yç=.x;y/ does not extend as a E taking the closure in the Hilbert scheme, the relevant bundle bundle to the closure. This problem is addressed and largely solved in Russell [2003]. 11.4.3 Solution to the enumerative problem 2  2 We start by recalling the description of the Chow ring A.‰/ of  P ‰ from  P Section 11.3: We have 2  2 2 3 2 2 D  C  P / A.‰/ Z Œ;ç=. /Œç=. ; D  C  /; A.  2 2 ‰  where and  is the pullback of the is the pullback of the hyperplane class in P 2 ‰ as the projectivization of the universal hyperplane class in (equivalently, if we view P  2 P , the first Chern class of the line bundle O S .1/ ). The degrees of subbundle on P S and  are monomials of top degree 3 in  2 2 3 3 deg . . / D 1 and deg . / / D deg . deg / D 0: D E we want to relate it to more familiar Now, in order to find the Chern class of bundles. To this end, we observe that the inclusions 3 2 , m J , ! m ! L;p p p and the corresponding quotients L L L p p p ! ! 2 3 L L m m L J p p p L;p p p globalize to give us surjections of sheaves ̨   1 2  I . L / P ! E and E ! / L . P  2 2 2 2 P P

429 Cusps of plane curves 415 Section 11.4 the composition ˇ  2  1  L . !  P / P . L / 2 2 2 2 P P is the standard quotient map of Theorem 7.2. Consider the corresponding inclusion   2 D  Ker . ̨/ , ! Ker .ˇ/ T . Sym ̋ L /: 2 2 P 2   Sym T L  What is the image? It is the tensor product of with the sub-line bundle of 2 2 P whose fiber at each point .L;p/ is the subspace spanned by the square of the linear form 2 2 2 on vanishing on T at L  T T P ! . In other words, the inclusion T P L , P T p p p p p 2 gives rise to a sequence each point .L;p/ ‰    (11.2) T 0; ! T ! ! 0 ! ˇ N  2 2 P ‰= P   ˇ whose fiber at T where N is the space of linear forms is the sub-line bundle of .L;p/ 2 P 2 2 P vanishing on T on L  T as the “relative conormal bundle” T P (we can think of N p p p 2  2 2   P  ! P P of the family ). Taking symmetric squares, we have an inclusion ‰  2 2  ! ˇ Sym Sym T N , ; 2 P L we arrive at an inclusion and tensoring with the pullback of 2    2 , ! ˇ ˇ Sym Sym ̋ T N L /; ̋ L . 2 P 2  P ˇ whose image is exactly . . L /= E 2 P c. E / . To begin with, we We can put this all together to calculate the Chern class 2 P know the classes of the bundle . L / from Proposition 7.5: We have 2 P 2  2 2 6 C : D .1 C .d 2// . D 1 // 6.d 2/ C 15.d 2/ P  L c.ˇ 2 P Next, the Chern class of the line bundle can be found from the sequence (11.2) : N We have      . ˇ N T c / c D T c 1 1 1 2 2  P P ‰= D 2 C / 3 .  ; D  c . T comes from Theorem 11.4. Thus where the equality  C / D 2 1 2  ‰= P 2  / N ̋ ˇ c. L Sym D 1 C .d 2/ 2;

430 416 Contact problems Chapter 11 2 2   . L /= E Š Sym ̋ N and since ˇ P L the Whitney formula gives ˇ 2 P  2  ˇ c P L . / 2 P D / c. E 2  ˇ ̋ L / N Sym c. 2 2 1 C 15.d 2/ C  6.d 2/ D 1 .2 .d 2// 3 X 2 2 k .d 2/ C  2/ / .1 C 6.d D : .2 2// 15.d 0 k D is . E / In particular, the third Chern class c 3 3 2 2 2 D .2 .d 2// c C 6.d 2/.2 .d 2// . C 15.d 2/ E  / .2 .d 2//; 3 and taking degrees we have 2 2 2 24.d 2/ 2/ 6.d C 2/ C 24.d 12.d 2/ c / deg . 2/ 30.d D E 3 2 D 36d C 24: 12d We have thus proven the enumerative formula: Proposition 11.14. of curves of degree d The number of cuspidal elements of a net D 2 , assuming there are only finitely many and counting multiplicities, is on P 2 36d C 24 D 12.d 1/.d 12d 2/: Of course, to answer Keynote Question (b) we have to verify that for a general net there are indeed only finitely many cusps, and that they all count with multiplicity 1. The first of these statements follows easily from the dimension count of Proposition 11.13. The second can be verified by explicit calculation in local coordinates, analogous to what we did to verify, for example, that hyperflexes in a general pencil occur with multiplicity 1; alternatively, we can use the method described in Section 11.4.4 below. d D 1 and 2, as it should. And, in the Note that the formula yields 0 in the cases d D 3 , we see that a general net of plane cubics will have 24 cuspidal members, case answering Keynote Question (b). Equivalently, we see that the locus of cuspidal plane 9 cubics has degree 24 in the space P of all plane cubics, completing the analysis begun in Section 2.2. 2 Note that there was no need to restrict ourselves to nets of curves in P ; a similar analysis could be made for the number of cusps (possibly with multiplicities) in a sufficiently general net of divisors associated to a sufficiently ample line bundle L on any . (Here the role of ‰ would be played by the projectivized tangent bundle surface T .) S P S We leave this version of the calculation to the reader; the answer is that the number of D D . L ;V / on a surface S is cuspidal elements in a net of curves 2 2 C 12c deg .12 2c /; C 2c 1 2 1 . As always, this number is subject to the usual  D c / . L / and where T D c . c i i 1 S

431 Cusps of plane curves Section 11.4 417 caveats: it is meaningful only if the number of cuspidal curves in the net is in fact finite; in this case, it represents the number of cuspidal curves counted with multiplicity (with multiplicity defined as the degree of the component of the zero scheme of the E ). supported at corresponding section of .p;/ 11.4.4 Another approach to the cusp problem There is another approach to the problem of counting cuspidal curves in a linear system, one that gives a beautiful picture of the geometry of nets. It is not part of the overall logical structure of this book, so we will run through the sequence of steps involved without proof; the reader who is interested can view supplying the verifications as an extended exercise. L a very ample line bun- S To begin with, let be a smooth projective surface and  j L j be a general two-dimensional subseries, corresponding to a three- dle; let D 0 V  H . L / . We have a natural map dimensional vector subspace  2 ! V S D P W ' P  1  '  S of the lines ; the preimages  P V V are to the projectivization of the dual L C  S of the linear system the divisors . If we want, we can think of the complete linear D n 0  as giving an embedding of S in the larger projective space P system j P H L . L / j , D and the map ' as the projection of S corresponding to a general .n 3/ -plane. Now, the geometry of generic projections of smooth varieties is well understood in low dimensions. Mather [1971; 1973] showed that these are the same in the algebro- geometric setting as in the differentiable; in the latter context the singularities of general projections of surfaces are described in Golubitsky and Guillemin [1973, Section 6.2]. 2 n ! P The upshot is that if is the projection of a smooth surface S  ' W from a S P .n 3/ -plane, then: general  R The ramification divisor S of the map ' is a smooth curve.    is the birational image of P V The branch divisor B R , and has only nodes and  ordinary cusps as singularities. ́ 2 etale locally around any point p In fact, S , one of three things is true. Either: ́ (i) The map is p ... R ). etale (if 2 is a point of .x;y/ .x;y The map is simply ramified, that is, of the form / (if p 7! (ii) not lying over a cusp of B ). R  The surface S is given, in terms of local coordinates .x;y/ on P V (iii) around '.p/ , by the equation 3 z xz y D 0:

432 418 Chapter 11 Contact problems (This is the picture around a point where three sheets of the cover come together; in a neighborhood of '.p/ the branch curve is the zero locus of the discriminant 3 2 '.p/ .) 27y 4x , and in particular has a cusp at The interesting thing about this set-up is that we have two plane curves associated to it, lying in dual projective planes:  ' of the map P . (a) The branch curve V B  (b) V parametrizing divisors in the net , we have the discriminant In the dual space P D P V , that is, the locus of singular elements of the net. Å curve  What ties everything together is the observation that  P V the discriminant curve Å   V B . To see this, note that if L  P V P is a is the dual curve of the branch curve  B (in particular, not passing through any of the singular points of ), line transverse to B 1 '  S will be smooth: This is certainly true away from points then the preimage .L/ ́ , where the map ' is B etale, and at a point p 2 L \ of we can take local coordinates B  1 on with L given by y D 0 and B by x D 0 ; at a point of ' V .x;y/ .p/ the P 2  ́ S will either be V etale or given by z P D x . A similar calculation shows cover ! 1 B at a smooth point then  conversely that if L .L/ will be singular. is tangent to Pl ucker formulas for plane curves . These say At this point, we invoke the classical ̈ 2  P that if is a plane curve of degree d > 1 and geometric genus g having ı nodes C    ı has degree d C and cusps as its only singularities, and the dual curve  nodes and  and  cusps as singularities, then  d D 1/ 2ı 3; d.d     2ı .d 3 1/ ; D d d     1 1 2/ ı  D g D .d : 1/.d .d 2/ ı 1/.d  2 2 ff. See, for example, Griffiths and Harris [1994, p. 277 ]. Given these, all we have to do is write down everything we know about the curves R , B and Å . To begin with, we invoke f W X ! the Riemann–Hurwitz formula for finite covers : If  is a rational canonical Y  with divisor D , the divisor of the pullback Y form on  will be the preimage of D f plus the ramification divisor R  X ; thus  1 2 C R K A K .X/: D f X Y In our present circumstances, this says that   K D ' ̋ O .R/ I K S S P V   .1/ since the pullback , we can write this as ' O is equal to L V P 3 D L K .R/; S  D c . T R is / c  D c or, in terms of the notation . L / , the class of and 1 1 1 S 1 D c .S/: ŒRç 3 2 A C 1

433 Cusps of plane curves Section 11.4 419 g of the curve : Since R is smooth, by Among other things, this tells us the genus R adjunction we have 1 / R  .R C K C 1 D g S 2 1 D C C 3/.2c 1 C 3/ .c 1 1 2 2 2 1 .9 D C 2c C 9c 1: / C 1 1 2  P D '.R/ It also tells us the degree B V of the branch curve : This is the d  R with the preimage of a line, so that intersection of 2 D .c C 3/ D 3 d C c : 1 1 of the discriminant curve Å  P V : This is the Finally, we also know the degree e number of singular elements in a pencil, which we calculated back in Chapter 7; we have 2 e 3 C 2c C c : D 1 2 Å We now have enough information to determine the number of cusps of . Let and  denote the number of nodes and cusps of Å respectively. First off, the geometric ı Å genus of is given by 1 g .e 1/.e D 2/ ı ; 2 and the degree d of the dual curve is d D e.e 1/ 2ı 3: Subtracting twice the first equation from the second yields  2g d C 2.e 1/ D 2 2 2 2 C C 2c 9 C 2 .3 C C c c / C D 9c C 2c 1/ 2.3 1 2 1 1 1 2 2 C 12c ; C D C 2c 12 2c 2 1 1 agreeing with result stated at the end of Section 11.4.3. Note that this method also gives us a geometric sense of when a cusp “counts with multiplicity one;” in particular, if all the hypotheses above about the geometry of the map ' are satisfied, the count is exact. This also gives us a formula for the number of curves C in the net with two nodes. This is the number ı Å , which we get by subtracting three times of nodes of the curve g d : this yields the equation for above from the equation for 3 d 3g e.e ı 1/ C D .e 1/.e 2/; 2 where d , e and g are given in terms of the classes ;c by the equations above. and c 2 1 d D 1 and Note that the formula returns 0 in the cases D 2 , as it should, and in d 9 d D 3 it gives 21 — the degree of the locus of reducible cubics in the P the case of all cubics, as calculated in Section 2.2. Exercises 11.49–11.51 describe an alternative (and perhaps cleaner) way of deriving the formula for the number of binodal curves in a net, via linearization.

434 420 Chapter 11 Contact problems 11.5 Exercises 4 d Let be a general hypersurface of degree   6 . How many P X Exercise 11.15. 4  will have a point of contact of order 7 with X ? P L lines 3  S be a general surface of degree d  P . Using the dimension Exercise 11.16. Let 2 counts of Proposition 11.13 and incidence correspondences, show that: p in a dense open subset U  S , the intersection S \ (a) For S has an ordinary T p . double point (a node) at p Q the S such that for p 2 Q There is a one-dimensional locally closed locus  (b) intersection \ T S has a cusp at p . S p (c) Ä of points p 2 S , lying in the closure of Q , such that the There will be a finite set intersection \ T S S has a tacnode at p . p (d) U;Q and Ä ; that is, no singularities other than nodes, S is the disjoint union of . S cusps and tacnodes appear among the plane sections of ˆ be the universal line over G .1;3/ Exercise 11.17. E the bundle on ˆ intro- Let and 3 L  L be the line X be the point D X P D 0 , and let p 2 duced in Section 11.1. Let 3 2 ˆ . By trivializing the bundle in a neighborhood of .L;p/ Œ1;0;0;0ç E and writing 2 everything in local coordinates, show that the section of E coming from the polynomial 2 5 2 4 X C X . C X .L;p/ has a simple zero at X X X 2 3 1 0 1 0 3 d Let S be a general surface of degree   4 . Show that, for any Exercise 11.18. P 3  : L and any pair of distinct points p;q 2 L line P m . .S  L/  5 (a) p (b) . .S  L/ C m m .S  L/  6 p q 3 is called an A point  P Exercise 11.19. S Eckhart point of S p on a smooth surface S \ T . Recall that in Exercise 7.42 we saw S has a triple point at p if the intersection p 3 that a general surface  P of degree d has no Eckhart points. S Show that the locus of smooth surfaces that do have an Eckhart point is an open (a) d C 3 1 / . 3 P  subset of an irreducible hypersurface in the space of all surfaces. ‰ 3 S  (b) Show that a general surface that does have an Eckhart point has only one. P (c) Find the degree of the hypersurface ‰ . 4 Exercise 11.20. Consider a smooth surface  P . Show that we would expect there S to be a finite number of hyperplane sections H \ S of S with triple points, and count the 1 number in terms of the hyperplane class  2 A .S/ and the Chern classes of the tangent bundle to S .

435 Exercises Section 11.5 421 Applying your answer to the preceding exercise, find the number of Exercise 11.21. 4 P hyperplane sections of with triple points in each of the following cases:  S 4 is a complete intersection of two quadrics in . (a) S P (b) S is a cubic scroll (Section 9.1.1). 2 5 . (c) P S / is a general projection of the Veronese surface P  .  2 In each case, can you check your answer directly? 3 S  P a general surface of degree d , find the degree of the surface For Exercise 11.22. 3 S having a point of contact of order at least 4 with swept out by the lines in P . The following exercise describes in some more detail the geometry of the flecnodal 3 Ä of a smooth surface S  P  , introduced in Section 11.2.1; we will use the locus S notation of that section. 3 Exercise 11.23. S  P be a general surface of degree d . Let (a) Find the first Chern class of the bundle . F (b) Ä is smooth, and that the projection Ä ! C is generically Show that the curve one-to-one. Ä . (c) Using the preceding parts, find the genus of the curve Show, on the other hand, that the flecnodal curve of S is the intersection of S with a (d) surface of degree 11d 24 , and use this to calculate the arithmetic genus of C . Can you describe the singularities of the curve C ? Do these account for the discrep- (e) and of C Ä ancy between the genera of ? N 3 N be the space of surfaces of degree d  4 in P Exercise 11.24. and ‰  P Let P the M.d/ of locus of surfaces containing a line. Show that the maximum possible number 3  P by considering of degree d is at most the degree of ‰ lines on a smooth surface S S and a general second surface T . Is this bound better or worse the pencil spanned by than the one derived in Section 11.2.1? d d d d 3 the Fermat surface S  D V.x Exercise 11.25. C y Show that for C z d C w  / d 3 2 contains exactly 3d lines. P Exercise 11.26. For F.x;y/ any homogeneous polynomial of degree d , consider the 3 given by the equation  P surface S F.x;y/ F.z;w/ D 0:

436 422 Chapter 11 Contact problems 1 ̨ preserving the polynomial F (that If is the order of the group of automorphisms of P 2 S to itself), show that d C ̨d lines. F is, carrying the set of roots of contains at least L D and L Hint: are the lines z D w D 0 and x if y D 0 respectively, and 1 2 D ' ! L L any isomorphism carrying the zero locus F.x;y/ D 0 to F.z;w/ W 0 , 2 1 consider the intersection of S with the quadric [ Q D p;'.p/: ' L p 2 1 3 Using the preceding exercise, exhibit smooth surfaces P S of  Exercise 11.27. degrees 4, 6, 8, 12 and 20 having at least 64, 180, 256, 864 and 1600 lines, respectively. 2 4 4 4 z C D C Verify that the Fermat quartic curve V.x /  P Exercise 11.28. has C y 12 hyperflexes and no ordinary flexes. Recall that a node p if one of C of a plane curve is called a flecnode 2 Exercise 11.29. C p has contact of order 3 or more with its tangent line. Show that the branches of at N 4 of all plane curves of degree d  the closure, in the space P , of the locus of curves with a flecnode is irreducible of dimension 2 . N How many elements of a general net of plane curves of degree d will Exercise 11.30. have flecnodes? Exercise 11.31. Verify that for a general pencil f C of plane curves D V.t g F C t G/ 1 0 t d , if is a hyperflex of some element C of degree of the pencil, then: .L;p/ t m .C 4  L/ D (a) ; that is, no line has a point of contact of order 5 or more with any t p element of the pencil. is a smooth point of C (b) p . t (c) p is not a base point of the pencil.  and Using these facts, show that the degeneracy locus of the sections  of the G F E bundle introduced in Section 11.3.1 is reduced. Let f C be a general pencil of plane curves of D V.t g F C t Exercise 11.32. G/ 0 1 t 2 d . If degree 2 P is a general point, how many flex lines to members of the pencil p f C ? g pass through p t For Exercises 11.33–11.38, we let f D V.t be a general pencil of F C t g C G/ 1 0 t plane curves of degree d , 2  2 P ‰ Df  P .L;p/ j p 2 L g 2 be the universal line and 1 Df .t;L;p/ 2 P Ä  ˆ j m : .L  C g /  3 t p 2 2  P under the projection be the image of Ä Let Ä ! ˆ ! P B ; that is, the curve traced out by flex points of members of the pencil.

437 Exercises Section 11.5 423 First, show that Ä Exercise 11.33. is indeed smooth, by showing that the “universal flex” N .C;L;p/ j .C  L/  3 g P †  ˆ Df m p N d plane curves) is smooth and invoking P is the space parametrizing all degree- (where Bertini. Can you give explicit conditions on the pencil equivalent to the smoothness ? Ä of 2 2 Exercise 11.34. If is a base point of the pencil, show that exactly three members p P p B has an ordinary triple point at p . of the pencil have a flex point at , and that the curve 2 p P C is a node of the curve If a point Exercise 11.35. containing it, the tangent 2 t lines to the two branches are each flex lines to , so that the map Ä ! B is two-to-one C t B has correspondingly a node at p . there. Show that the curve Finally, show that the triple points and nodes of Exercise 11.36. described in the B B preceding two exercises are the only singularities of . C be an element of our pencil with a hyperflex .L;p/ . Show that Exercise 11.37. Let t 1 the map Ä is simply ramified at .t;L;p/ , and simply branched at t . ! P Let C L be an element of our pencil with a node p ; let L Exercise 11.38. and be the 1 t 2 / C p . Show that .t;p;L , and that these at 2 Ä tangent lines to the two branches of i t 1 Ä ! are ramification points of weight 2 of the map (that is, each of the lines L P i is a limit of three flex lines of nearby smooth curves in our pencil, and these three are t is a branch point cyclically permuted by the monodromy in the family). Conclude that 1 of multiplicity 4 for the cover ! P . Ä Exercise 11.39. Let f C including a g be a general pencil of plane curves of degree d t cuspidal curve . (That is, let C C D V.F/ be a general cuspidal curve, C V.G/ D 1 0 0 C f D V.F C a general curve and g the pencil they span.) As t ! 0 , how many tG/ t C flexes of approach the cusp of C has a tacnode? ? How about if C 0 0 t The following series of exercises (Exercises 11.40–11.44) sketches a proof of Proposition 11.13. Suppose that p 2 C is an A Exercise 11.40. -singularity for n  3 . Show that the n 0 0 C D Bl is C of C at p has a unique point q lying over p , and that q 2 blow-up C p z A p at -singularity. Conclude in particular that the normalization an C ! C of C 2 n has genus ̆  1 z D p : . p C/ 1/ .n C .C/ a a 2 - Let S be a smooth surface and C  S a curve with an A Exercise 11.41. 2n 1 singularity at p . (a) Show that there is a unique curvilinear subscheme Ä  S of degree n supported at 2 p C  S at such that a local defining equation of lies in the ideal I p . Ä

438 424 Chapter 11 Contact problems z z If Bl (b) S is the blow-up of S along Ä , show that the proper transform D C of C S Ä z is smooth over p and intersects the exceptional divisor E transversely twice at in S z smooth points of . S z D D S is any such curve then the image of Conversely, show that if in S has an (c)  -singularity at p . A 1 2n A -singularities. This is the Exercise 11.42. Prove the analog of Exercise 11.41 for 2n same statement, except that in the second and third parts the phrase “intersects the z transversely twice at smooth points of E exceptional divisor ” should be replaced with S z “is simply tangent to the exceptional divisor S and does not meet at a smooth point of E E otherwise.” L be a line bundle on a smooth surface S , and assume that for any Exercise 11.43. Let  S of degree n supported at a single point we have Ä curvilinear subscheme 1 2 H ̋ L . I D 0: / Ä 0 of curves in the linear series  P H Å . Show that the locus / - j L j with an A L k k 0 k in P H / . L singularity is locally closed and irreducible of codimension for all k  2 . 2n Deduce from the above exercises the statement of Proposition 11.13. Exercise 11.44. Show that if L is the n -th power of a very ample line bundle, then the Exercise 11.45. 2 1 . L ̋ I condition H / D 0 is satisfied for any curvilinear subscheme Ä  S of degree Ä n=2 D j L j is a general net in the complete linear or less. Conclude in particular that if j j associated to the fourth or higher power of a very ample bundle then no curve series L 2 D has singularities other than nodes and ordinary cusps. C The following three exercises sketch out a calculation of the number of curves C  S with a tacnode in a suitably general three-dimensional linear system. (Here, as in the case of cusps, when we use the term “tacnode” without the adjective “ordinary” we include as well singularities that are specializations of ordinary tacnodes, that is, A , triple points or points on multiple components.) -singularities for any n  3 n Exercise 11.46. S be a smooth surface and L a line bundle on S . Let B D P Let be T S the projectivization of the tangent bundle of S , which we may think of as a parameter space for subschemes Ä  S of degree 2 supported at a single point. Construct a vector bundle E on B whose fiber at a point Ä 2 B may be naturally identified with the vector space 0 2 H / . L E L ̋ I D = Ä Ä Exercise 11.47. D A.B/ of B In terms of the description of the Chow ring P T given in S Section 11.4.2, calculate the top Chern class of the bundle constructed in Exercise 11.46.

439 Exercises Section 11.5 425 Using the preceding two exercises, find an enumerative formula for the Exercise 11.48. L that have a tacnode. If number of curves in a three-dimensional linear series D j j 3 P d , apply this to find the expected number of plane  S is a smooth surface of degree sections with a tacnode. Check your answer by calculating the number directly in the D 2 and 3. cases d The following three exercises describe a way of deriving the formula for the number of binodal curves in a net via linearization. We begin by introducing a smooth, projective 2 compactification of the space of unordered pairs of points 2 P : We set p;q 2 2  2 z 2 L ˆ P Df .L;p;q/  P j  P p;q ; g z be the quotient of and let ˆ by the involution .L;p;q/ 7! .L;q;p/ . To put it ˆ 2 ˆ differently, with L  P consists of pairs a line and D  L a subscheme of .L;D/ 2 ˆ is the Hilbert scheme of subschemes of P with Hilbert degree 2; or, differently still, polynomial 2. (Compare this with the description in Section 9.7.4 of the Hilbert scheme 3 of conic curves in — this is the same thing, one dimension lower.) P 2  ˆ Observe that the projection P expresses ! as a projective Exercise 11.49. ˆ  2 bundle over P , and use this to calculate its Chow ring. 2 Exercise 11.50. as the Hilbert scheme of subschemes of P Viewing of dimension 0 ˆ E and degree 2, construct a vector bundle ˆ whose fiber at a point D is the space on 2 0 D H E . O I .d/= .d//: 2 .L;p;q/ D P ˆ (What would go wrong if instead of using the Hilbert scheme as our parameter space 2 we used the Chow variety — that is, the symmetric square of P ?) Express the condition 2 be singular at C  P that a curve V.F/ p and q in terms of the vanishing of an D associated section  .L;p;q/ of E on H at F Exercise 11.51. Calculate the Chern classes of this bundle, and derive accordingly the formula for the number of binodal curves in a net.

440 Chapter 12 Porteous’ formula Keynote Questions ef 1 M P M be the space of e  f matrices and Let Š  M the locus of matrices (a) k k or less. What is the degree of M ? (Answer on page 433.) of rank k 3 n  the projection of S be a smooth surface and f W S ! P Let P S from a (b) n 4 n general plane P  P p . At how many points Š 2 S will the map f fail to ƒ be an immersion? (Answer on page 438.) 3 3  P (c) be a smooth rational curve of degree d . How many lines Let  P C meet L C four times? (Answer on page 441.) 12.1 Degeneracy loci We saw in Chapter 5 that the Chern class c on . F / of a vector bundle F of rank f i X F is generated by global sections, as a smooth variety can be characterized, when general sections of D e C 1 f F become dependent: the class of the scheme where i Specifically, if the locus where a map e O ! ' W F X fails to have maximal rank has the expected codimension , then c . F / is the class of the i i e  e minors of a matrix representing ' . A similar scheme that is locally defined by the result holds for Segre classes. We can substantially extend the usefulness of this characterization in two ways: e k locus of a map O by considering the rank- , and ! F for arbitrary k  min .e;f / X e by replacing O with an arbitrary vector bundle E . In this chapter we will do both: X We henceforth consider the class of the scheme M .'/ where a map of vector bundles k ' W minors of a ! F has rank  k , locally defined by the ideal of .k C 1/  .k C 1/ E

441 Degeneracy loci Section 12.1 427 . Such loci are called ' e and f for degeneracy loci matrix representation of . We write F and the ranks of E , respectively. ef and ' In the “generic” case, where is the X is an affine space of dimension gen f matrix of variables, the codimension of the locus M  .' e / is map defined by an gen k k/.f k/ (Harris [1995, Proposition 12.2] or Eisenbud [1995, Exercise 10.10]). .e In general we say that k/ is the expected codimension of M .e .'/ . In k/.f k M .'/ under the assumption that it this chapter we will give a formula for the class of k has the expected dimension Such a formula was first found by Giambelli in 1904, in the special case where E ́ are both direct sums of line bundles. Ren e Thom observed more generally in F and the context of differential geometry that when .'/ M has the expected codimension k its class (suitably construed) depends only on the Chern classes of E and F . This was made explicit by Porteous (see Porteous [1971], which reproduces notes from 1962), giving the expression now called Porteous’ formula. (The formula might more properly be called the Giambelli–Thom–Porteous formula; we have chosen to call it the Porteous formula for brevity and because that is how it appears in much of the literature.) The result was proven (in a more general form, in which one specifies the ranks of the ' E ) in the context of algebraic geometry by restriction of to a flag of subbundles of Kempf and Laksov [1974]. The form of the expression is interesting in itself: Porteous’ formula expresses .'/ç as a polynomial in the components of the ratio c. F /=c. ŒM / E . k D 0 , and suppose that the To get an idea of what is to come, consider the case k M .'/ , where the map ' induces the zero map on the fibers, has the expected locus 0  . The map ' E ef ̋ F , may be regarded as a global section of the bundle codimension .'/ is the locus where this global section vanishes; thus its class is M and the locus 0  . E c ̋ F / . ef L  E E The splitting principle makes it easy to understand ̋ F / : If c D . L and i ef L   M . were sums of line bundles, then E F ̋ F D L would be the sum of the M ̋ j i i  L / D 1 C ̨ c. and c. M , / D 1 C ˇ ̨ , then If we write L ˇ ̋ M C / D 1 c. j j i j i i j i so, by Whitney’s formula, Y  ̋ F / D E c. .1 C ˇ /; ̨ i j i;j and in particular Y  D ̋ F / E . c .ˇ /: ̨ i j ef i;j ̨ This expression is symmetric in each of the two sets of variables and ˇ , so it can j i be written in terms of the elementary symmetric functions of these variables, which are and F . If we think of the ̨ E as the roots of the Chern polynomial the Chern classes of i 2  . E / WD 1 C c / . E /t C c F . E /t , then C , and similarly for c ̋ . F / c c is . E 2 1 t t ef the classical resultant of c // . E / and c F . (See for example F / , written Res . .c /;c . E . t t t t t

442 428 Chapter 12 Porteous’ formula Eisenbud [1995, Section 14.1] for more about resultants and their role in algebraic geometry.) By the splitting principle, the result we have obtained holds for all maps of vector bundles: X are vector bundles of ranks e and f on a smooth variety and , If F Proposition 12.1. E then  c ̋ F / D Res /;c .c //: . E . F E . t t t ef F / and c . E / c each have constant coefficient 1, and in this case . The polynomials t t we can express the resultant differently. WD . ; ;:::/ We first introduce some notation. For any sequence of elements 0 1 e e . / D det D . / Å , we set , where e;f in a commutative ring and any natural numbers f f 0 1   e f f 1 f C 1 C B C B C   B C f f 1 2 f C e B C : : : B C : e : : : : . / D WD : : B C : : : f B C B C : : : : : : : : B C : : : : @ A   e f f C 2 e f C 1 f e a.t/ D 1 C a t t CC a b t If and b.t/ D 1 C b CC t Proposition 12.2. 1 1 e f are polynomials with constant coefficient 1, then     b.t/ a.t/ e ef f Å 1/ Å D .a.t/;b.t// ; Res D . t e f b.t/ a.t/ Œb.t/=a.t/ç denotes the sequence of coefficients .1;c ;:::/ ;c of the formal power where 2 1 2 D 1 C c series t C c . t b.t/=a.t/ C , and similarly for Œa.t/=b.t/ç 2 1 We will give the proof in Section 12.2. ; A.X/ , we write Œ ç for the sequence For any element is 2 ;:::/ , where . 1 i 0 the component of of degree i . The next corollary gives the expression of the top Chern class of a tensor product that we will use: Corollary 12.3. E and F are vector bundles of ranks e and f If X , on a smooth variety then   / F c. e  : / Res D .c c . E /;c . . F // D Å F E ̋ t t t ef f / E c. ' W E ! F is a homomorphism that vanishes in expected codimension In particular, if ef , then   c. / F e .'/ç D Å ŒM : 0 f c. E /

443 Porteous’ formula for Section 12.1 429 M .'/ 0 Porteous’ formula for the class of an arbitrary degeneracy locus follows the same pattern: ' W E ! Let be a map of vector bundles of (Porteous’ formula) . F Theorem 12.4 f on a smooth variety X . If the scheme M .'/ ranks  X has codimension e and k k/.f k/ , then its class is given by .e   c. F / k e Å D .'/ç ŒM : k k f / E c. f k 1 < f ; in this case Å k D e The formula is easiest to interpret in the case . / k e  1 is the determinant of the 1 matrix     1 f e C F / c. c. / F 1 ; D D f 1 C e / c. / E c. E k g ̨ for the codimension- k part of a Chow class where we write 2 A.X/ . Specializing f ̨ e f e C 1 f c. F /=c. E / g D O then further, if E D c , so we recover the charac- / F . 1 f C e X f F O , then terization of Chern classes as degeneracy loci (Theorem 5.3). If instead D X e C 1 f e C 1  f /=c. D f 1=c. E / g E /ç g , so we recover f , the Segre class s c. / F E . 1 C f e f  ! E . the characterization of the Segre class as degeneracy locus of a map O X 1 More generally, represents an obstruction to the existence of /ç E Å Œc. F /=c. C 1 f e W E ' F : If ' were an inclusion of vector bundles, an inclusion of vector bundles ! F = E would be a vector bundle of rank equal to rank F rank E D f e , so then 1 e 1 C f D Œc. /=c. E /ç Df c. F /=c. E / g Å . 0 F f C 1 e The proof of Theorem 12.4 will be given in Section 12.3: After a reduction to a ŒM as the image of .'/ç “generic case,” we will express ŒM is a map from . /ç , where 0 k 0 S of rank e k to a bundle F a bundle of rank f on a Grassmannian bundle over X ; e k 0 i /ç /=c. S / g f in the matrix D c. F under the pushforward to , the entry Œc. F /=c. E X f k i /=c. f / g F will be replaced by c. , yielding Porteous’ formula. E M .'/ 12.2 Porteous’ formula for 0 In this section we will prove the resultant formula of Proposition 12.2, and thus complete the proof of Corollary 12.3. Q Since the polynomial .ˇ Proof of Proposition 12.2: has no repeated factors, ̨ / j i ̨ and ˇ it divides any polynomial in the that vanishes when one of the ̨ is equal to i i j e ˇ has this vanishing property. Indeed, . We first show that Å one of the .b.t/=a.t// j f if a.t/ and b.t/ have a common factor 1 C , then dividing a.t/ by this root gives a polynomial a.t/ such that b.t/ D g.t/ a.t/ a.t/

444 430 Chapter 12 Porteous’ formula < f is a polynomial of degree . If we write the ratio b=a as a power series b.t/ 2 t t C c 1 c C C ; D 2 1 a.t/ c and substitute the power series t 1 into this expression, we get a power series C C 1 2 1 C c t a.t/c.t/ c g.t/ t D C D C 1 2 c whose coefficients C a c CC a D c vanish for i  f . It fol- c 1 i i e C 1 1 i 1 e i e .b.t/=a.t// , and thus lows that the vector a ;:::; a D ; 1/ is annihilated by . 1 1 e f e e .b.t/=a.t// D 0 . D det D .b.t/=a.t// Å f f It follows that Y e / d Res ̨ .a.t/;b.t// D d D .b.t/=a.t// .ˇ Å t i j f i;j for some polynomial in the ̨ d and ˇ . j i Q Q a.t/ .1 C Writing , we see that the coefficient of t/ and b.t/ D D ̨ t/ .1 C ˇ i j j P k k t D b.t/=a.t/ c is homogeneous of degree k in the variables in the power series t k e ̨ , and thus every term in the determinant Å ;ˇ , as does .b.t/=a.t// has degree ef j i f Res .a.t/;b.t// . It follows that d is a constant. t e becomes D 0 , we see that If we take all the D b.t/ , and D a .b.t/=a.t// b.t/=a.t/ i f  Q e e / .b , ˇ lower-triangular; in this case its determinant is D Res D .a.t/;b.t// j t f j d D so . 1 12.3 Proof of Porteous’ formula in general 12.3.1 Reduction to a generic case We first explain how to reduce the proof to a case where a slightly stronger hypothesis holds: (a) M .'/ is of the expected dimension .e k/.f k/ . k (b) .'/ is reduced. M k The points x 2 X where the map ' ; has rank exactly k are dense in M .'/ (c) x k equivalently, M . k/ .'/ has codimension >.e k/.f 1 k To do this, consider the map  1 ' E W ! E ̊ F taking E onto the graph Ä  E ̊ F of ' . The original map ' is the composition of ' with the projection to F .

445 Proof of Porteous’ formula in general Section 12.3 431 0 W WD G.e; E ̊ F / ! X , and we write X We now form the Grassmannian bundle   for the tautological subbundle of rank E ̊ F /  e . Since is an inclusion of S ! . bundles, the universal property of the Grassmannian guarantees that there is a unique 0 u X W such that the pullback under X of the tautological inclusion map ! map u  ̊ . E ̊ F / on the Grassmannian is W E ! E S ! , and thus the pullback  F   0 !  ' . E ̊ F / !  W F is ' . It follows that M of the composite map .'/ D S k 1 0 .' // . u .M k 0 E ̊ F is the kernel of the projection to Since , the points of M E .'  / are the F k 0  X meets the fiber of such that the fiber of S points  x E in dimension at least e k . 2 † With notation parallel to that of Chapter 4, this is the Schubert cycle WD † : / E . k f .e k/ where the bundles in question are trivial, by the X It is defined over any open subset of same determinantal formula that defines the corresponding Schubert cycle in the case of M vector spaces. A look at this formula shows that .'/ as schemes. † D k where the bundles E and F Over an open set in X are trivial, the Grassmannian 0 C G.e; ̊ F / is the product of X with the ordinary Grassmannian D E f / X G.e;e † is the product of X with the corresponding Schubert cycle and the Schubert cycle G.e;e C f / . As was mentioned in the discussion of the equations of Schubert in varieties after Theorem 4.3, these varieties are reduced, irreducible and Cohen–Macaulay 0 .' (Hochster [1973] or De Concini et al. [1982]). In particular, / D † is reduced, M k 0 .e k/ . Moreover, M irreducible and Cohen–Macaulay of codimension D .' k/.f / 1 k 0 0 1/.f † X 2 x has codimension .e k C , so the points k/ > codim † / E . k f k C 1/ .e 0 0 where . has rank exactly k are dense in M / .' ' 0 k x 0 is a Cohen–Macaulay subvariety, we can apply the Cohen– / D † Because .' M k  0 Macaulay case of Theorem 1.23 and conclude that u ŒM ŒM D .' .'/ç /ç . Since k k  0 A.X S / ! A.X/ is a ring homomorphism, and since the Chern classes of u and W  F E and F respectively, we see that it suffices to pull back to the Chern classes of  0 . ' prove Porteous’ formula for the map k D 0 12.3.2 Relation to the case 0 ' ' Replacing as above, we may assume that ' satisfies hypotheses (a), (b) and by (c) of the previous section. We next linearize the problem by introducing more data. To say that x 2 M .'/ k means that there is some k -dimensional subspace of F , and by that contains ' / . E x x x ' . E / when x is a general point of assumptions (a) and (c) the subspace is equal to x x . .'/ . To make use of this idea, we introduce the Grassmannian M W G.e k; E / ! X  k     F E for the tautological rank- k subbundle. Let  W S !  We write E !  S     W / where  be the composite map. The locus in .'/  k; E !  F F factors G.e  S k through the tautological rank- . E /=  may also be described as M , the ./ quotient 0 is x where  vanishes. It follows that the map from M .'/ ./ to M locus of points 0 k

446 432 Chapter 12 Porteous’ formula .'/ is reduced, M surjective and generically one-to-one. Since we have assumed that k ./ç as ŒM we can compute the class .'/ç  . ŒM 0  k M dim ./ , we have that From the fact that  is generically one-to-one on ./ D M 0 0 dim dim k; E / . Since G.e X C k.e k/ , it D dim dim .e k/.f k/ M X .'/ D k k/f ./ has the expected codimension .e M , and thus by Corollary 12.3 follows that 0    / c. F e k  Å ŒM ./ç ŒM D : D .'/ç    0 k f / c. S 12.3.3 Pushforward from the Grassmannian bundle It remains to compute Completion of the Proof of Theorem 12.4:    / c. F k e : Å   f S / c.  . E Let /= S . By Whitney’s formula, Q D  E / c. ; / D c. S c. / Q so     F / c. c. / F c. / F  Q c. Q D D  c. / /:  / c. E / / c. E S c. The point is that we have isolated the factors in the entries of the matrix e k  /ç Œ D .c. F /=c. E //c. Q f that are pullbacks from . To take advantage of this, we expand the determinant X