3264 & All That - Intersection theory in algebraic geometry

633 Pages·2011·2.375 MB·English

by  Eisenbud D., Harris J.

This is page i
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3264 & All That
Intersection Theory in Algebraic Geometry
(cid:13)c David Eisenbud and Joe Harris
July 11, 2011
This is page v
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Contents
0.1 Why you should read this book . . . . . . . . . . . . . . . 1
0.2 Why we wrote this book . . . . . . . . . . . . . . . . . . . 3
0.3 What’s with the title? . . . . . . . . . . . . . . . . . . . . . 3
0.4 What’s in this book . . . . . . . . . . . . . . . . . . . . . . 4
0.4.1 Overture . . . . . . . . . . . . . . . . . . . . . . . . 4
0.4.2 Second beginning . . . . . . . . . . . . . . . . . . . 4
0.4.3 Using the tools . . . . . . . . . . . . . . . . . . . . . 5
0.4.4 Further developments . . . . . . . . . . . . . . . . . 6
0.4.5 Relation to “Intersection Theory” . . . . . . . . . . 7
0.4.6 Keynote Problems . . . . . . . . . . . . . . . . . . . 7
0.5 What you should know before you begin . . . . . . . . . . 7
0.6 What’s in a scheme? . . . . . . . . . . . . . . . . . . . . . . 8
1 Overture 11
1.1 The Chow Ring . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.1 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.2 Rational equivalence. . . . . . . . . . . . . . . . . . 13
1.1.3 First Chern class of a line bundle . . . . . . . . . . 15
1.1.4 First results on the Chow Group . . . . . . . . . . . 16
1.1.5 Intersection products . . . . . . . . . . . . . . . . . 16
Multiplicities . . . . . . . . . . . . . . . . . . . . . . 19
1.1.6 Functoriality . . . . . . . . . . . . . . . . . . . . . . 20
Proper Pushforward . . . . . . . . . . . . . . . . . . 20
Flat and Projective Pullback . . . . . . . . . . . . . 22
1.2 Chow Ring Examples . . . . . . . . . . . . . . . . . . . . . 23
vi Contents
1.2.1 Varieties built from Affine Spaces . . . . . . . . . . 23
Open Subsets of Affine Space. . . . . . . . . . . . . 23
Affine stratifications . . . . . . . . . . . . . . . . . . 24
1.2.2 The Chow ring of Pn . . . . . . . . . . . . . . . . . 25
Degrees of Veronese varieties . . . . . . . . . . . . . 27
Degree of the dual of a hypersurface . . . . . . . . . 28
1.2.3 Products of projective spaces . . . . . . . . . . . . . 28
Degrees of Segre varieties . . . . . . . . . . . . . . . 30
The class of the diagonal . . . . . . . . . . . . . . . 31
The graph of a morphism . . . . . . . . . . . . . . . 32
1.2.4 The blowup of Pn at a point . . . . . . . . . . . . . 33
1.2.5 Loci of singular plane cubics . . . . . . . . . . . . . 39
The locus of reducible cubics . . . . . . . . . . . . . 41
Triangles . . . . . . . . . . . . . . . . . . . . . . . . 42
Asterisks . . . . . . . . . . . . . . . . . . . . . . . . 43
1.3 Chern Classes . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.3.1 Chern Classes as Dependency Loci. . . . . . . . . . 44
1.3.2 Whitney’s Formula and the Splitting Principle . . . 46
1.3.3 The Chern Character . . . . . . . . . . . . . . . . . 47
1.4 Chern Class Examples . . . . . . . . . . . . . . . . . . . . . 47
1.4.1 The 27 Lines on a Cubic Surface . . . . . . . . . . . 47
1.4.2 First Computations . . . . . . . . . . . . . . . . . . 49
Sums of Line Bundles . . . . . . . . . . . . . . . . . 49
Duals . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.4.3 The Cotangent Bundle . . . . . . . . . . . . . . . . 49
The Genus of a Curve on a Surface . . . . . . . . . 52
The Self-Intersection of a Curve on a Surface . . . . 53
Linked curves in P3 . . . . . . . . . . . . . . . . . . 53
The topological Euler Characteristic . . . . . . . . . 54
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2 Introduction to Grassmannians and Lines in P3 65
2.1 Introduction to Grassmann Varieties . . . . . . . . . . . . . 65
2.1.1 The Plu¨cker embedding . . . . . . . . . . . . . . . . 66
2.1.2 Covering by affine spaces; local coordinates . . . . . 69
2.1.3 Universal sub and quotient bundles . . . . . . . . . 71
2.1.4 The Tangent Bundle of the Grassmann Variety. . . 73
2.2 The Chow Ring of G(1,3) . . . . . . . . . . . . . . . . . . 76
2.2.1 Schubert Cycles in G=G(1,3). . . . . . . . . . . . 77
2.2.2 Equations of the Schubert Cycles . . . . . . . . . . 79
2.2.3 Ring Structure . . . . . . . . . . . . . . . . . . . . . 80
Tangent spaces to Schubert cycles . . . . . . . . . . 83
2.2.4 A Specialization Argument . . . . . . . . . . . . . . 84
2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.3.1 How Many Lines Meet Four General Lines?. . . . . 87
Contents vii
2.3.2 Lines meeting a curve . . . . . . . . . . . . . . . . . 88
2.3.3 Lines meeting a curve via specialization . . . . . . . 90
2.3.4 Chords to a space curve . . . . . . . . . . . . . . . . 91
2.3.5 Chords via specialization . . . . . . . . . . . . . . . 93
2.3.6 Lines on a quadric . . . . . . . . . . . . . . . . . . . 95
2.3.7 Tangent Lines to a Surface . . . . . . . . . . . . . . 95
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3 Grassmannians in General 105
3.1 Schubert Cells and Schubert Cycles . . . . . . . . . . . . . 106
3.2 Intersections in complementary dimension. . . . . . . . . . 111
3.2.1 Varieties swept out by linear spaces . . . . . . . . . 113
3.3 Grassmannians of Lines . . . . . . . . . . . . . . . . . . . . 114
3.3.1 Dynamic specialization . . . . . . . . . . . . . . . . 117
3.4 How to Count Schubert Cycles with Young Diagrams . . . 119
3.5 Linear spaces on quadrics . . . . . . . . . . . . . . . . . . . 121
3.6 Pieri and Giambelli . . . . . . . . . . . . . . . . . . . . . . 124
3.6.1 Intersecting with σ . . . . . . . . . . . . . . . . . . 124
1
3.6.2 Pieri’s formula in general . . . . . . . . . . . . . . . 126
3.6.3 Giambelli’s formula . . . . . . . . . . . . . . . . . . 126
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4 Chow Groups 133
Keynote Questions: . . . . . . . . . . . . . . . . . . 133
4.1 Chow groups and basic operations . . . . . . . . . . . . . . 134
4.1.1 Examples of Rational Equivalences . . . . . . . . . 138
Dynamic projection . . . . . . . . . . . . . . . . . . 139
4.2 Rational Equivalence through Divisors. . . . . . . . . . . . 143
4.3 Proper push-forward . . . . . . . . . . . . . . . . . . . . . . 146
4.3.1 The Determinant of a Homomorphism. . . . . . . . 149
4.4 Flat Pullback . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.4.1 Affine Space Bundles . . . . . . . . . . . . . . . . . 157
4.5 The connection between A and Pic. . . . . . . . . . . . 158
n−1
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5 Intersection Products and Pullbacks 163
5.1 The Chow Ring . . . . . . . . . . . . . . . . . . . . . . . . 163
5.1.1 Products via the Moving Lemma. . . . . . . . . . . 164
5.1.2 From Intersections to Pullbacks . . . . . . . . . . . 168
5.2 Proof of the Moving Lemma . . . . . . . . . . . . . . . . . 170
5.3 Kleiman’sTransversalityTheorem:TheCaseofaTransitive
Group Action . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.4 Intersections on singular varieties . . . . . . . . . . . . . . 183
5.5 Intersection multiplicities . . . . . . . . . . . . . . . . . . . 185
5.6 Refinements and Other Approaches . . . . . . . . . . . . . 188
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5.6.1 Refined and Excess Intersection . . . . . . . . . . . 188
5.6.2 Reduction to the Diagonal and Deformation to the
Normal Cone . . . . . . . . . . . . . . . . . . . . . . 189
5.7 Proofs of the results on Pullbacks . . . . . . . . . . . . . . 190
5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6 Interlude: Vector Bundles and Direct Images 195
6.1 Vector Bundles and Locally Free Sheaves . . . . . . . . . . 195
6.2 Pullbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.3 Flat Families of Sheaves . . . . . . . . . . . . . . . . . . . . 198
6.4 Direct Images . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.5 Cohomology and Base Change . . . . . . . . . . . . . . . . 203
6.5.1 Tools From Commutative Algebra . . . . . . . . . . 205
6.5.2 Proof Of the Theorem On Cohomology and Base
Change . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.6.1 Applications to Line Bundles . . . . . . . . . . . . . 209
6.6.2 Double covers . . . . . . . . . . . . . . . . . . . . . 210
6.6.3 Projective bundles . . . . . . . . . . . . . . . . . . . 212
6.6.4 Ideals of points in projective space . . . . . . . . . . 213
6.7 Applications I: Bundles revisited . . . . . . . . . . . . . . . 214
6.7.1 Symmetric powers of tautological bundles . . . . . . 214
6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7 Vector Bundles and Chern Classes 219
7.1 Chern Classes . . . . . . . . . . . . . . . . . . . . . . . . . 220
7.1.1 Chern Classes as Dependency Loci. . . . . . . . . . 220
7.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . 223
7.2.1 Pullbacks . . . . . . . . . . . . . . . . . . . . . . . . 224
7.2.2 Line bundles . . . . . . . . . . . . . . . . . . . . . . 224
7.2.3 Whitney sum. . . . . . . . . . . . . . . . . . . . . . 224
7.2.4 The splitting principle. . . . . . . . . . . . . . . . . 228
7.2.5 Tensor products with line bundles . . . . . . . . . . 229
7.2.6 Definition of Chern Classes in General . . . . . . . 231
7.2.7 Dual Bundles . . . . . . . . . . . . . . . . . . . . . 233
7.2.8 Determinant of a Bundle . . . . . . . . . . . . . . . 233
7.3 Application: A strong Bertini Theorem . . . . . . . . . . . 234
7.4 Chern Classes of Some Interesting Bundles . . . . . . . . . 235
7.4.1 Universal bundles on projective space . . . . . . . . 236
7.4.2 Chern Classes of Tangent Bundles . . . . . . . . . . 236
7.4.3 Tangent Bundles of Projective Spaces . . . . . . . . 237
7.4.4 Tangent bundles to hypersurfaces . . . . . . . . . . 238
7.4.5 Bundles on Grassmannians . . . . . . . . . . . . . . 240
7.5 Generators and Relations for A∗(G(k,n)) . . . . . . . . . . 241
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Contents ix
8 Lines on Hypersurfaces 249
8.1 What to Expect . . . . . . . . . . . . . . . . . . . . . . . . 249
8.2 Fano Schemes and Chern Classes . . . . . . . . . . . . . . . 251
8.2.1 Counting lines on cubics . . . . . . . . . . . . . . . 253
8.3 Tangent spaces to Fano schemes . . . . . . . . . . . . . . . 254
8.3.1 Normal Bundles and the Smoothness of the Fano
Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 255
8.3.2 First-order Deformations . . . . . . . . . . . . . . . 260
8.3.3 Normal bundles of lines on hypersurfaces . . . . . . 263
8.4 Lines on Quintic Threefolds and Beyond . . . . . . . . . . 266
8.5 The Universal Fano Scheme and the Geometry of Families
of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
8.5.1 Lines on the quartic surfaces in a pencil . . . . . . . 274
8.6 Local Equations for F (X) . . . . . . . . . . . . . . . . . . 274
1
8.6.1 Tangent spaces to Fano schemes via local equations 275
8.6.2 Lines on singular cubics . . . . . . . . . . . . . . . . 277
8.7 The Debarre/de Jong Conjecture. . . . . . . . . . . . . . . 279
8.7.1 Further open problems . . . . . . . . . . . . . . . . 282
8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
9 Singular Elements of Linear Series 289
9.1 GeneralSingularHypersurfacesandtheUniversalSingularity290
9.2 The bundles of principal parts . . . . . . . . . . . . . . . . 293
9.3 Singular Elements of a Pencil . . . . . . . . . . . . . . . . . 296
9.3.1 From Pencils to Degeneracy Loci. . . . . . . . . . . 296
9.3.2 The Chern Class of a Bundle of Principal Parts . . 297
9.3.3 Triple Points of Plane Curves . . . . . . . . . . . . 301
9.3.4 Higher Dimensions and Other Line Bundles . . . . 302
9.4 Geometry Of Nets of Plane Curves. . . . . . . . . . . . . . 304
9.4.1 Class Of The Universal Singular Point . . . . . . . 304
9.4.2 Singular Points and Discriminant in a Net of Plane
Curves . . . . . . . . . . . . . . . . . . . . . . . . . 305
9.5 Flexes, and inflection points in general . . . . . . . . . . . 308
9.5.1 Inflection points . . . . . . . . . . . . . . . . . . . . 308
9.5.2 The Plu¨cker formula. . . . . . . . . . . . . . . . . . 310
9.5.3 Consequences of the Plu¨cker formula . . . . . . . . 312
9.5.4 Curves with Hyperflexes . . . . . . . . . . . . . . . 314
9.6 The Topological Hurwitz Formula . . . . . . . . . . . . . . 314
9.6.1 Application to pencils . . . . . . . . . . . . . . . . . 316
9.6.2 Multiplicities of the discriminant hypersurface . . . 318
9.6.3 Tangent cones . . . . . . . . . . . . . . . . . . . . . 319
10 Compactifying Parameter Spaces 323
10.1 Approaches To The Five Conic Problem. . . . . . . . . . . 324
Blowing Up The Excess Locus . . . . . . . . . . . . 325
x Contents
Excess Intersection Formulas . . . . . . . . . . . . . 326
Changing the Parameter Space . . . . . . . . . . . . 326
10.2 Complete conics . . . . . . . . . . . . . . . . . . . . . . . . 327
10.2.1 Informal Description . . . . . . . . . . . . . . . . . 327
Degenerating the dual . . . . . . . . . . . . . . . . . 327
Types of Complete Conics . . . . . . . . . . . . . . 329
10.2.2 Rigorous Description . . . . . . . . . . . . . . . . . 329
Duals of Quadrics . . . . . . . . . . . . . . . . . . . 329
Equations For the Variety of Complete Conics . . . 331
10.2.3 Solution to the Five Conic Problem . . . . . . . . . 333
Outline . . . . . . . . . . . . . . . . . . . . . . . . . 334
Complete Conics Tangent To Five General Conics
are Smooth . . . . . . . . . . . . . . . . . . . . . 334
Transversality . . . . . . . . . . . . . . . . . . . . . 335
10.2.4 Chow Ring Of The Space Of Complete Conics . . . 337
TheClassoftheDivisorofCompleteConicsTangent
to C . . . . . . . . . . . . . . . . . . . . . . . . . 338
10.3 Parameter spaces of curves . . . . . . . . . . . . . . . . . . 340
10.3.1 Hilbert schemes . . . . . . . . . . . . . . . . . . . . 340
10.3.2 Examples of Hilbert Schemes . . . . . . . . . . . . . 340
Hypersurfaces . . . . . . . . . . . . . . . . . . . . . 340
Twisted Cubics . . . . . . . . . . . . . . . . . . . . 340
Report Card for the Hilbert Scheme . . . . . . . . . 341
10.3.3 The Kontsevich space . . . . . . . . . . . . . . . . . 342
10.3.4 Examples of Kontsevich spaces . . . . . . . . . . . . 343
Plane conics . . . . . . . . . . . . . . . . . . . . . . 343
Conics in space . . . . . . . . . . . . . . . . . . . . 343
Plane cubics . . . . . . . . . . . . . . . . . . . . . . 344
Twisted cubics . . . . . . . . . . . . . . . . . . . . . 344
Report Card for the Kontsevich Space. . . . . . . . 345
10.4 Rational plane curves . . . . . . . . . . . . . . . . . . . . . 346
11 Chow Rings of Projective Bundles 351
11.1 Projective bundles and the tautological divisor class . . . . 351
11.2 Chow ring of a projective bundle . . . . . . . . . . . . . . . 353
11.2.1 Example: the blow-up of Pn along a linear space . . 358
11.3 Chow ring of a Grassmannian bundle . . . . . . . . . . . . 358
11.4 Conics in P3 Meeting Eight Lines . . . . . . . . . . . . . . 360
11.4.1 The Parameter Space . . . . . . . . . . . . . . . . . 360
11.4.2 Transversality . . . . . . . . . . . . . . . . . . . . . 363
11.4.3 The Chow Ring . . . . . . . . . . . . . . . . . . . . 366
11.4.4 The Cycle of Plane Conics Meeting a Line . . . . . 367
12 Segre Classes and Varieties of Linear Spaces 371
12.1 Segre Classes . . . . . . . . . . . . . . . . . . . . . . . . . . 372
Contents xi
12.2 Universal hyperplane . . . . . . . . . . . . . . . . . . . . . 375
12.3 Varieties Swept Out by Linear Spaces . . . . . . . . . . . . 376
12.3.1 Lines on hypersurfaces revisited . . . . . . . . . . . 376
12.4 Secants of Rational Curves . . . . . . . . . . . . . . . . . . 378
12.4.1 Points on Secants and Sums of Powers . . . . . . . 378
12.4.2 Degrees of secant varieties to rational curves . . . . 382
12.4.3 Chern classes of E . . . . . . . . . . . . . . . . . . . 383
12.5 Subbundles of Projective Bundles . . . . . . . . . . . . . . 386
12.5.1 The Class of a Subbundle . . . . . . . . . . . . . . . 386
12.5.2 Ruled surfaces . . . . . . . . . . . . . . . . . . . . . 387
12.5.3 Self-intersection of the zero section of a vector bundle388
12.6 Dual varieties and conormal varieties . . . . . . . . . . . . 390
12.6.1 The universal hyperplane as a projective bundle . . 390
12.6.2 Conormal and dual varieties . . . . . . . . . . . . . 391
12.6.3 Proof of the duality theorem . . . . . . . . . . . . . 393
12.6.4 Degree of the dual . . . . . . . . . . . . . . . . . . . 397
12.6.5 Pencils of hypersurfaces . . . . . . . . . . . . . . . . 399
13 The Tangent Bundle of a Projective Bundle and Relative
Principal Parts 401
13.0.6 Chern Classes of Relative Tangent Bundles . . . . . 402
13.1 Application: Contact problems . . . . . . . . . . . . . . . . 403
13.1.1 Contact bundles . . . . . . . . . . . . . . . . . . . . 405
13.1.2 Chern classes of contact bundles . . . . . . . . . . . 406
13.1.3 Counting lines with contact of order 5 . . . . . . . . 407
13.2 The Case of Negative Expected Dimension . . . . . . . . . 409
13.2.1 Predestination versus Free Will . . . . . . . . . . . 409
13.2.2 Lines on surfaces of degree d≥4 in P3 . . . . . . . 410
13.2.3 Otherconfigurationswithnegativeexpecteddimen-
sion . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
13.3 Flexes, again . . . . . . . . . . . . . . . . . . . . . . . . . . 413
13.3.1 The Cartesian view of flexes . . . . . . . . . . . . . 413
13.4 Cusps of plane curves . . . . . . . . . . . . . . . . . . . . . 420
13.4.1 Another approach to the cusp problem . . . . . . . 423
14 Porteous’ Formula 429
14.1 Porteous’ formula . . . . . . . . . . . . . . . . . . . . . . . 429
14.2 The top Chern class of a tensor product . . . . . . . . . . . 432
14.2.1 Sylvester’s determinant . . . . . . . . . . . . . . . . 432
14.3 Back to our original problem . . . . . . . . . . . . . . . . . 436
14.3.1 One last wrinkle . . . . . . . . . . . . . . . . . . . . 439
14.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 440
14.4.1 Degrees of determinantal varieties . . . . . . . . . . 440
14.4.2 Pinch points of surfaces . . . . . . . . . . . . . . . . 443
14.4.3 Quadrisecant lines to a rational space curve . . . . 445
xii Contents
Other proofs . . . . . . . . . . . . . . . . . . . . . . 449
14.5 Miscellaneous. . . . . . . . . . . . . . . . . . . . . . . . . . 450
14.5.1 Porteousforsymmetricandskew-symmetricbundle
maps . . . . . . . . . . . . . . . . . . . . . . . . . . 450
14.5.2 Positivevectorbundles;StatementofFulton-Lazarsfeld
theorem. . . . . . . . . . . . . . . . . . . . . . . . . 450
14.5.3 Webs of quadrics. . . . . . . . . . . . . . . . . . . . 450
15 The Chow Ring of a Blowup 451
15.1 The Chow Ring of a Blowup . . . . . . . . . . . . . . . . . 452
15.2 The blow-up of P3 along a curve . . . . . . . . . . . . . . . 455
15.2.1 TheintersectionofthreesurfacesinP3 containinga
curve . . . . . . . . . . . . . . . . . . . . . . . . . . 456
15.3 Intersections in a Subvariety . . . . . . . . . . . . . . . . . 459
15.3.1 Deformation to the normal cone . . . . . . . . . . . 460
16 Excess Intersection Formulas 465
16.1 Introduction to the Excess Intersection Formula . . . . . . 467
16.1.1 First Examples. . . . . . . . . . . . . . . . . . . . . 469
16.1.2 Intersection of two surfaces in P4 containing a curve 470
16.2 Justification of the Formula . . . . . . . . . . . . . . . . . . 471
16.3 Avoiding Smoothness Hypotheses . . . . . . . . . . . . . . 473
16.3.1 Computing the Segre class of a Cone . . . . . . . . 475
16.3.2 How Many Conics are Tangent to Five Lines? . . . 476
16.4 Pullbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
16.5 The double point formula . . . . . . . . . . . . . . . . . . . 480
16.5.1 Apparent Double Points of a Surface . . . . . . . . 481
16.5.2 Generalizations . . . . . . . . . . . . . . . . . . . . 484
17 The Grothendieck-Riemann-Roch Theorem 485
17.1 The Riemann-Roch Formula for Curves and Surfaces . . . 485
17.1.1 Nineteenth Century Riemann-Roch . . . . . . . . . 485
17.2 The Chern character. . . . . . . . . . . . . . . . . . . . . . 487
17.2.1 K-theory . . . . . . . . . . . . . . . . . . . . . . . . 490
17.2.2 Chern classes of coherent sheaves . . . . . . . . . . 490
17.3 Hirzebruch-Riemann-Roch . . . . . . . . . . . . . . . . . . 491
17.4 Higher Direct Images of sheaves . . . . . . . . . . . . . . . 494
17.5 Grothendieck-Riemann-Roch . . . . . . . . . . . . . . . . . 497
17.6 Application: Jumping lines . . . . . . . . . . . . . . . . . . 499
17.6.1 Vector bundles with even first Chern class . . . . . 500
17.6.2 Vector bundles with odd first Chern class . . . . . . 502
17.6.3 Examples and Generalizations . . . . . . . . . . . . 504
18 Brill-Noether 507
18.1 What maps to projective space “should” a curve have? . . 507
Contents xiii
The Jacobian. . . . . . . . . . . . . . . . . . . . . . 508
Abel’s Theorem . . . . . . . . . . . . . . . . . . . . 509
Riemann-Roch . . . . . . . . . . . . . . . . . . . . . 510
18.1.1 Behavioronspecialcurves:CliffordandCastelnuovo
Theorems . . . . . . . . . . . . . . . . . . . . . . . . 511
18.1.2 Behavior on general curves: naive dimension counts
leading to statement of Brill-Noether . . . . . . . . 512
How you might be led to the Brill-Noether Theorem 513
18.1.3 Description of various methods of proof . . . . . . . 514
18.2 Chow rings of Jacobians and Symmetric Products of Curves 516
18.2.1 Poincar´e’s formula . . . . . . . . . . . . . . . . . . . 518
18.2.2 Symmetric products as projective bundles . . . . . 519
18.2.3 Chern classes of tautological bundles on Jacobians . 521
18.3 Application of Porteous; calculation of class of Wr . . . . . 522
d
18.3.1 ConstructionofthePoincar´ebundleandthebundle
E . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
18.3.2 Calculation of the class of Wr . . . . . . . . . . . . 523
d
19 Families of Curves 527
19.1 Invariants of families of curves . . . . . . . . . . . . . . . . 527
19.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . 529
Pencils of quartics in the plane . . . . . . . . . . . . 529
Pencils of plane sections of a quartic surface . . . . 531
Families of hyperelliptic curves . . . . . . . . . . . . 532
19.1.2 The Mumford Relation . . . . . . . . . . . . . . . . 534
19.1.3 A final word . . . . . . . . . . . . . . . . . . . . . . 536
19.2 The moduli space of curves . . . . . . . . . . . . . . . . . . 536
19.2.1 Moduli spaces in general; examples in genus 2 . . . 536
19.2.2 Stacks. . . . . . . . . . . . . . . . . . . . . . . . . . 537
19.2.3 Stable curves, and the Deligne-Mumford compacti-
fication . . . . . . . . . . . . . . . . . . . . . . . . . 537
19.2.4 Divisor classes on M . . . . . . . . . . . . . . . . . 538
g
19.2.5 Recasting the Mumford relation . . . . . . . . . . . 538
19.3 The Kodaira dimension of M . . . . . . . . . . . . . . . . 538
g
19.3.1 Unirationality in low genus . . . . . . . . . . . . . . 538
19.3.2 The canonical class of M . . . . . . . . . . . . . . 539
g
The Reid-Tai criterion . . . . . . . . . . . . . . . . 539
19.3.3 The Brill-Noether divisor . . . . . . . . . . . . . . . 539
20 Appendix: Intersection Theory on Surfaces 541
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 541
20.2 Riemann-Roch and Serre Duality for Curves . . . . . . . . 542
20.3 Curves and adjunction . . . . . . . . . . . . . . . . . . . . 542
20.3.1 GenusofaCurveinthePlaneoronaQuadricSurface544
20.3.2 Self-Intersection of a Curve on a Surface . . . . . . 545