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3264 & All That - Intersection theory in algebraic geometry PDF

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This is page i Printer: Opaque this 3264 & All That Intersection Theory in Algebraic Geometry (cid:13)c David Eisenbud and Joe Harris July 11, 2011 This is page v Printer: Opaque this 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 viii Contents 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

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