Introduction to Algebraic Geometry (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 188)

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. - Provided by publisher 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

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.

Introduction -- A Crash Course In Commutative Algebra -- Affine Varieties -- Projective Varieties -- Regular And Rational Maps Of Quasi Projective Varieties -- Products -- The Blow Up Of An Ideal -- Finite Maps Of Quasi Projective Varieties -- Dimension Of Quasi Projective Algebraic Sets -- Zariski's Main Theorem -- Nonsingularity -- Sheaves -- Applications To Regular And Rational Maps -- Divisors -- Dierential Forms And The Canonical Divisor -- Schemes -- The Degree Of A Projective Variety -- Cohomology -- Curves -- An Introduction To Intersection Theory -- Surfaces -- Ramication And Etale Maps -- Bertini's Theorems And General Fibers Of Maps. Steven Dale Cutkosky. Includes Bibliographical References And Index.