This book presents a readable and accessible introductory course in algebraic geometry, with most of the fundamental classical results presented with complete proofs. An emphasis is placed on developing connections between geometric and algebraic aspects of the theory. Differences between the theory in characteristic 0 and positive characteristic are emphasized. The basic tools of classical and modern algebraic geometry are introduced, including varieties, schemes, singularities, sheaves, sheaf cohomology, and intersection theory. Basic classical results on curves and surfaces are proved. More advanced topics such as ramification theory, Zariski's main theorem, and Bertini's theorems for general linear systems are presented, with proofs, in the final chapters.
With more than 200 exercises, the book is an excellent resource for teaching and learning introductory algebraic geometry.
Author(s): Steven Dale Cutkosky
Series: Graduate Studies in Mathematics 188
Publisher: AMS
Year: 2018
Language: English
Pages: 498
Tags: Algebraic Geometry
Cover......Page 1
Title page......Page 2
Contents......Page 6
Preface......Page 12
1.1. Basic algebra......Page 14
1.2. Field extensions......Page 19
1.3. Modules......Page 21
1.4. Localization......Page 22
1.5. Noetherian rings and factorization......Page 23
1.6. Primary decomposition......Page 26
1.7. Integral extensions......Page 29
1.8. Dimension......Page 32
1.9. Depth......Page 33
1.10. Normal rings and regular rings......Page 35
2.1. Affine space and algebraic sets......Page 40
2.2. Regular functions and regular maps of affine algebraic sets......Page 46
2.3. Finite maps......Page 53
2.4. Dimension of algebraic sets......Page 55
2.5. Regular functions and regular maps of quasi-affine varieties......Page 61
2.6. Rational maps of affine varieties......Page 71
3.1. Standard graded algebras......Page 76
3.2. Projective varieties......Page 80
3.3. Grassmann varieties......Page 86
3.4. Regular functions and regular maps of quasi-projective varieties......Page 87
4.1. Criteria for regular maps......Page 100
4.2. Linear isomorphisms of projective space......Page 103
4.3. The Veronese embedding......Page 104
4.4. Rational maps of quasi-projective varieties......Page 106
4.5. Projection from a linear subspace......Page 108
5.1. Tensor products......Page 112
5.2. Products of varieties......Page 114
5.3. The Segre embedding......Page 118
5.4. Graphs of regular and rational maps......Page 119
6.1. The blow-up of an ideal in an affine variety......Page 124
6.2. The blow-up of an ideal in a projective variety......Page 133
7.1. Affine and finite maps......Page 140
7.2. Finite maps......Page 144
7.3. Construction of the normalization......Page 148
8.1. Properties of dimension......Page 152
8.2. The theorem on dimension of fibers......Page 154
Chapter 9. Zariski’s Main Theorem......Page 160
10.1. Regular parameters......Page 166
10.2. Local equations......Page 168
10.3. The tangent space......Page 169
10.4. Nonsingularity and the singular locus......Page 172
10.5. Applications to rational maps......Page 178
10.6. Factorization of birational regular maps of nonsingular surfaces......Page 181
10.7. Projective embedding of nonsingular varieties......Page 183
10.8. Complex manifolds......Page 188
11.1. Limits......Page 194
11.2. Presheaves and sheaves......Page 198
11.3. Some sheaves associated to modules......Page 209
11.4. Quasi-coherent and coherent sheaves......Page 213
11.5. Constructions of sheaves from sheaves of modules......Page 217
11.6. Some theorems about coherent sheaves......Page 222
12.1. Blow-ups of ideal sheaves......Page 234
12.2. Resolution of singularities......Page 238
12.3. Valuations in algebraic geometry......Page 241
12.4. Factorization of birational maps......Page 245
12.5. Monomialization of maps......Page 249
Chapter 13. Divisors......Page 252
13.1. Divisors and the class group......Page 253
13.2. The sheaf associated to a divisor......Page 255
13.4. Calculation of some class groups......Page 262
13.5. The class group of a curve......Page 267
13.6. Divisors, rational maps, and linear systems......Page 272
13.7. Criteria for closed embeddings......Page 277
13.8. Invertible sheaves......Page 282
13.9. Transition functions......Page 284
14.1. Derivations and Kähler differentials......Page 292
14.2. Differentials on varieties......Page 296
14.3. ��-forms and canonical divisors......Page 299
15.1. Subschemes of varieties, schemes, and Cartier divisors......Page 302
15.2. Blow-ups of ideals and associated graded rings of ideals......Page 306
15.3. Abstract algebraic varieties......Page 308
15.5. General schemes......Page 309
Chapter 16. The Degree of a Projective Variety......Page 312
17.1. Complexes......Page 320
17.2. Sheaf cohomology......Page 321
17.3. Čech cohomology......Page 323
17.4. Applications......Page 325
17.5. Higher direct images of sheaves......Page 333
17.6. Local cohomology and regularity......Page 338
Chapter 18. Curves......Page 346
18.1. The Riemann-Roch inequality......Page 347
18.2. Serre duality......Page 348
18.3. The Riemann-Roch theorem......Page 353
18.4. The Riemann-Roch problem on varieties......Page 356
18.5. The Hurwitz theorem......Page 358
18.6. Inseparable maps of curves......Page 361
18.7. Elliptic curves......Page 364
18.8. Complex curves......Page 371
18.9. Abelian varieties and Jacobians of curves......Page 373
Chapter 19. An Introduction to Intersection Theory......Page 378
19.1. Definition, properties, and some examples of intersection numbers......Page 379
19.2. Applications to degree and multiplicity......Page 388
20.1. The Riemann-Roch theorem and the Hodge index theorem on a surface......Page 392
20.2. Contractions and linear systems......Page 396
Chapter 21. Ramification and Étale Maps......Page 404
21.1. Norms and Traces......Page 405
21.2. Integral extensions......Page 406
21.3. Discriminants and ramification......Page 411
21.4. Ramification of regular maps of varieties......Page 419
21.5. Completion......Page 421
21.6. Zariski’s main theorem and Zariski’s subspace theorem......Page 426
21.7. Galois theory of varieties......Page 434
21.8. Derivations and Kähler differentials redux......Page 437
21.9. Étale maps and uniformizing parameters......Page 439
21.10. Purity of the branch locus and the Abhyankar-Jung theorem......Page 446
21.11. Galois theory of local rings......Page 451
21.12. A proof of the Abhyankar-Jung theorem......Page 454
Chapter 22. Bertini’s Theorems and General Fibers of Maps......Page 464
22.1. Geometric integrality......Page 465
22.2. Nonsingularity of the general fiber......Page 467
22.3. Bertini’s second theorem......Page 470
22.4. Bertini’s first theorem......Page 471
Bibliography......Page 482
Index......Page 490
Back Cover......Page 498
