3264 & All That: A second course in algebraic geometry.

Why you should read this book......Page 11
What\'s with the title?......Page 13
Second beginning......Page 14
Using the tools......Page 15
Further developments......Page 16
What you should know before you begin......Page 17
What\'s in a scheme?......Page 18
The Chow Group and the Intersection Product......Page 21
Cycles......Page 22
Rational equivalence......Page 23
First Chern class of a line bundle......Page 24
Intersection products......Page 26
Multiplicities......Page 28
A(X) is covariant for proper maps by pushforward......Page 29
Pullback......Page 30
Open Subsets of Affine space......Page 32
Affine stratifications......Page 33
The Chow ring of ¶n......Page 34
Degree of the dual of a hypersurface......Page 36
Products of projective spaces......Page 37
The class of the diagonal......Page 39
The graph of a morphism......Page 40
The blowup of ¶n at a point......Page 41
Loci of singular plane cubics......Page 44
The locus of reducible cubics......Page 47
Asterisks......Page 48
Tangent Bundle, Canonical Class and Adjunction......Page 49
Chern Classes......Page 51
Example: The topological Euler characteristic......Page 55
Exercises......Page 57
Keynote Questions:......Page 63
Chow groups and basic operations......Page 64
Examples......Page 67
Proper push-forward......Page 73
The Determinant of a Homomorphism......Page 76
Flat Pullback......Page 81
Affine Space Bundles......Page 83
The connection between An-1 and Pic.......Page 85
The Chow Ring......Page 89
Products via the Moving Lemma......Page 91
From Intersections to Pullbacks......Page 94
Proof of the Moving Lemma......Page 96
Kleiman\'s Transversality Theorem: The Case of a Transitive Group Action......Page 106
Intersections on singular varieties......Page 108
Intersection multiplicities......Page 110
Refined and Excess Intersection......Page 113
Reduction to the Diagonal and Deformation to the Normal Cone......Page 114
Pullbacks......Page 115
Introduction to Grassmann Varieties......Page 119
The Plücker embedding......Page 120
Covering by affine spaces; local coordinates......Page 125
Universal sub and quotient bundles......Page 127
The Tangent Bundle of the Grassmann Variety......Page 128
Schubert Cycles in G = G1mu(1,3)......Page 131
Equations of the Schubert Cycles......Page 134
Ring Structure......Page 135
A Specialization Argument......Page 139
Applications......Page 141
 How Many Lines Meet Four General Lines?......Page 142
Lines meeting a curve......Page 143
Chords to a space curve......Page 148
Tangent Lines to a Surface......Page 151
Schubert Cells and Schubert Cycles......Page 155
Intersections in complementary dimension......Page 161
Grassmannians of Lines......Page 163
Dynamic specialization......Page 166
How to Count Schubert Cycles with Young Diagrams......Page 168
Linear spaces on quadrics......Page 171
Intersecting with 1......Page 174
Pieri\'s formula in general......Page 176
Giambelli\'s formula......Page 177
Vector Bundles and Locally Free Sheaves......Page 179
Pullbacks......Page 181
Flat Families of Sheaves......Page 182
Direct Images......Page 183
Cohomology and Base Change......Page 187
Tools From Commutative Algebra......Page 189
Proof Of the Theorem On Cohomology and Base Change......Page 192
Double covers......Page 194
Ideals of points in projective space......Page 197
Symmetric powers of tautological bundles......Page 198
Vector bundles: linearizing nonlinear problems......Page 203
Definition of the Chern classes......Page 206
Basic properties......Page 209
Whitney sum......Page 210
The splitting principle......Page 214
Tensor products with line bundles......Page 216
Dual Bundles......Page 219
Chern Classes of Some Interesting Bundles......Page 220
Chern Classes of Tangent Bundles......Page 221
Tangent Bundles of Projective Spaces......Page 222
Tangent bundles to hypersurfaces......Page 224
Universal Bundles and Tangent Bundles on Grassmannians......Page 225
Generators and Relations for A*(G(k,n))......Page 227
The Broader Context of Chern Classes......Page 231
Chern classes of coherent sheaves......Page 232
The Chern character......Page 233
K-theory......Page 235
The topological Chern class......Page 236
Differential-geometric characterization of Chern classes......Page 237
A strong Bertini Theorem......Page 239
What to Expect......Page 243
Fano Schemes and Chern Classes......Page 245
Counting lines on cubics......Page 246
Normal Bundles and the Smoothness of the Fano Scheme......Page 248
First-order Deformations......Page 253
Lines on Quintic Threefolds and Beyond......Page 256
The Universal Fano Scheme and the Geometry of Families of Lines......Page 262
Lines on the quartic surfaces in a pencil......Page 267
Local Equations for......Page 269
Open problems......Page 272
Singular Elements of Linear Series......Page 275
General Singular Hypersurfaces and the Universal Singularity......Page 276
The bundles of principal parts......Page 278
From Pencils to Degeneracy Loci......Page 282
The Chern Class of a Bundle of Principal Parts......Page 283
Triple Points of Plane Curves......Page 287
Higher Dimensions and Other Line Bundles......Page 288
Class Of The Universal Singular Point......Page 290
Singular Points and Discriminant in a Net of Plane Curves......Page 291
Inflection points......Page 294
The Plücker formula......Page 296
Consequences of the Plücker formula......Page 298
The Topological Hurwitz Formula......Page 300
Application to pencils......Page 302
Multiplicities of the discriminant hypersurface......Page 304
Tangent cones......Page 305
Compactifying Parameter Spaces......Page 309
Approaches To The Five Conic Problem......Page 310
Blowing Up The Excess Locus......Page 311
Changing the Parameter Space......Page 312
Types of Complete Conics......Page 313
Duals of Quadrics......Page 315
Equations For the Variety of Complete Conics......Page 317
Solution to the Five Conic Problem......Page 319
Transversality......Page 320
The Class of the Divisor of Complete Conics Tangent to C......Page 323
Report Card for the Hilbert Scheme......Page 326
The Kontsevich space......Page 328
Twisted cubics......Page 329
Report Card for the Kontsevich Space......Page 331
Rational plane curves......Page 332
Projective bundles and the tautological divisor class......Page 337
Chow ring of a projective bundle......Page 339
Chow ring of a Grassmannian bundle......Page 344
The Parameter Space......Page 346
Transversality......Page 349
The Chow Ring......Page 352
The Cycle of Plane Conics Meeting a Line......Page 353
Segre Classes and Varieties of Linear Spaces......Page 357
Segre Classes......Page 358
Universal hyperplane......Page 361
Varieties Swept Out by Linear Spaces......Page 362
Points on Secants and Sums of Powers......Page 363
Degrees of secant varieties to rational curves......Page 367
Chern classes of E......Page 368
The Class of a Subbundle......Page 371
Ruled surfaces......Page 372
Self-intersection of the zero section of a vector bundle......Page 373
The universal hyperplane as a projective bundle......Page 375
Conormal and dual varieties......Page 376
Proof of the duality theorem......Page 378
Degree of the dual......Page 382
Pencils of hypersurfaces......Page 384
The Tangent Bundle of a Projective Bundle and Relative Principal Parts......Page 387
Chern Classes of Relative Tangent Bundles......Page 388
Application: Contact problems......Page 389
Contact bundles......Page 391
Chern classes of contact bundles......Page 392
Counting lines with contact of order 5......Page 393
Predestination versus Free Will......Page 395
Lines on surfaces of degree d 4 in ¶3......Page 396
Other configurations with negative expected dimension......Page 398
The Cartesian view of flexes......Page 399
Cusps of plane curves......Page 406
Another approach to the cusp problem......Page 409
Porteous\' formula......Page 415
Sylvester\'s determinant......Page 418
Back to our original problem......Page 422
One last wrinkle......Page 425
Degrees of determinantal varieties......Page 426
Pinch points of surfaces......Page 429
Quadrisecant lines to a rational space curve......Page 431
Miscellaneous......Page 435
Webs of quadrics......Page 436
The Chow Ring of a Blowup......Page 437
The Chow Ring of a Blowup......Page 438
The blow-up of ¶3 along a curve......Page 441
The intersection of three surfaces in ¶3 containing a curve......Page 442
Intersections in a Subvariety......Page 445
Deformation to the normal cone......Page 446
Excess Intersection Formulas......Page 451
Introduction to the Excess Intersection Formula......Page 453
First Examples......Page 455
Intersection of two surfaces in ¶4 containing a curve......Page 456
Justification of the Formula......Page 457
Avoiding Smoothness Hypotheses......Page 459
Computing the Segre class of a Cone......Page 461
How Many Conics are Tangent to Five Lines?......Page 462
Pullbacks......Page 464
The double point formula......Page 466
Apparent Double Points of a Surface......Page 467
Generalizations......Page 470
Nineteenth Century Riemann-Roch......Page 471
The Hirzebruch Riemann-Roch formula......Page 473
Higher Direct Images of sheaves......Page 476
Grothendieck Riemann-Roch......Page 479
Application: Jumping lines......Page 480
Vector bundles with even first Chern class......Page 482
Vector bundles with odd first Chern class......Page 484
Examples and Generalizations......Page 486
The Jacobian......Page 489
Riemann-Roch......Page 491
Behavior on special curves: Clifford and Castelnuovo Theorems......Page 493
How you might be led to the Brill-Noether Theorem......Page 494
Description of various methods of proof......Page 496
Chow rings of Jacobians and Symmetric Products of Curves......Page 498
Poincaré\'s formula......Page 500
Symmetric products as projective bundles......Page 501
Chern classes of tautological bundles on Jacobians......Page 503
Construction of the Poincaré bundle and the bundle E......Page 504
Calculation of the class of Wrd......Page 505
Invariants of families of curves......Page 509
Families of hyperelliptic curves......Page 511
The Mumford Relation......Page 516
Moduli spaces in general; examples in genus 2......Page 518
Stable curves, and the Deligne-Mumford compactification......Page 519
Unirationality in low genus......Page 520
The Brill-Noether divisor......Page 521
Introduction......Page 523
Curves and adjunction......Page 524
Genus of a Curve in the Plane or on a Quadric Surface......Page 526
Self-Intersection of a Curve on a Surface......Page 527
Blow-ups of surfaces......Page 528
The Riemann-Roch Theorem for Surfaces......Page 529
References......Page 531