1 3264 & All That Intersection Theory in Algebraic Geometry c David Eisenbud and Joe Harris
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 ) 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 , 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 ) and the library in S INGULAR (Decker et al. ). 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 .) 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  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. ) contains almost everything required. Other books geometry that cover this material include Undergraduate algebraic geometry (Reid ), Introduction to algebraic geometry (Hassett ), Elementary algebraic geom- etry (Hulek ) and, at a somewhat more advanced level, geometry, Algebraic (Mumford ), Basic algebraic geometry, I I: Complex projective varieties Algebraic geometry: a first course (Harris ). The (Shafarevich ) 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 ). (c) Faisceax An acquaintance with coherent sheaves and their cohomology. For this, alg (Serre ) 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 ) 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 . 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/ i h 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  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 .
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 ), 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 . 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  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/ Dh1i h 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/ i h 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 ) 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 ; 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 ), 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 . 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  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 ). 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  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 . 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  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 ), 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.  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 ), and its degree plane curves of degree has also been determined (Caporaso and Harris ). 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 . 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 ; and much more can be found, for example, in Beauville’s beautiful book on algebraic surfaces .
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 .
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 ). 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 ŒMçŒƒ 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: ŒNçŒƒ 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 ŒMçŒƒ 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 . 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, ŒMçŒƒ 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 ŒMçŒƒ , 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 ŒMçŒƒ 0 ç D ŒNçŒƒ 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 .
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  for basic geometry of the Grassmannian, Griffiths and Harris  for the basics of the Schubert calculus and Fulton  for combinatorial formulas, as well as the classic treatment in the second volume of Hodge and Pedoe . -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 <