**Graduate Studies in Mathematics**

Volume: 188;
2018;
484 pp;
Hardcover

MSC: Primary 14;

**Print ISBN: 978-1-4704-3518-9
Product Code: GSM/188**

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

**Electronic ISBN: 978-1-4704-4670-3
Product Code: GSM/188.E**

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

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#### Supplemental Materials

# Introduction to Algebraic Geometry

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*Steven Dale Cutkosky*

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.

#### Readership

Graduate students and researchers interested in algebraic geometry.

#### Reviews & Endorsements

The book is well written and self-contained; it contains both an introduction to the basics of the field and numerous advanced topics.

-- Luca Ugaglia, Mathematical Reviews

#### Table of Contents

# Table of Contents

## Introduction to Algebraic Geometry

- Cover Cover11
- Title page i2
- Contents v6
- Preface xi12
- Chapter 1. A Crash Course in Commutative Algebra 114
- Chapter 2. Affine Varieties 2740
- Chapter 3. Projective Varieties 6376
- Chapter 4. Regular and Rational Maps of Quasi-projective Varieties 87100
- Chapter 5. Products 99112
- Chapter 6. The Blow-up of an Ideal 111124
- Chapter 7. Finite Maps of Quasi-projective Varieties 127140
- Chapter 8. Dimension of Quasi-projective Algebraic Sets 139152
- Chapter 9. Zariski’s Main Theorem 147160
- Chapter 10. Nonsingularity 153166
- 10.1. Regular parameters 153166
- 10.2. Local equations 155168
- 10.3. The tangent space 156169
- 10.4. Nonsingularity and the singular locus 159172
- 10.5. Applications to rational maps 165178
- 10.6. Factorization of birational regular maps of nonsingular surfaces 168181
- 10.7. Projective embedding of nonsingular varieties 170183
- 10.8. Complex manifolds 175188

- Chapter 11. Sheaves 181194
- Chapter 12. Applications to Regular and Rational Maps 221234
- Chapter 13. Divisors 239252
- 13.1. Divisors and the class group 240253
- 13.2. The sheaf associated to a divisor 242255
- 13.3. Divisors associated to forms 249262
- 13.4. Calculation of some class groups 249262
- 13.5. The class group of a curve 254267
- 13.6. Divisors, rational maps, and linear systems 259272
- 13.7. Criteria for closed embeddings 264277
- 13.8. Invertible sheaves 269282
- 13.9. Transition functions 271284

- Chapter 14. Differential Forms and the Canonical Divisor 279292
- Chapter 15. Schemes 289302
- Chapter 16. The Degree of a Projective Variety 299312
- Chapter 17. Cohomology 307320
- Chapter 18. Curves 333346
- 18.1. The Riemann-Roch inequality 334347
- 18.2. Serre duality 335348
- 18.3. The Riemann-Roch theorem 340353
- 18.4. The Riemann-Roch problem on varieties 343356
- 18.5. The Hurwitz theorem 345358
- 18.6. Inseparable maps of curves 348361
- 18.7. Elliptic curves 351364
- 18.8. Complex curves 358371
- 18.9. Abelian varieties and Jacobians of curves 360373

- Chapter 19. An Introduction to Intersection Theory 365378
- Chapter 20. Surfaces 379392
- Chapter 21. Ramification and Étale Maps 391404
- 21.1. Norms and Traces 392405
- 21.2. Integral extensions 393406
- 21.3. Discriminants and ramification 398411
- 21.4. Ramification of regular maps of varieties 406419
- 21.5. Completion 408421
- 21.6. Zariski’s main theorem and Zariski’s subspace theorem 413426
- 21.7. Galois theory of varieties 421434
- 21.8. Derivations and Kähler differentials redux 424437
- 21.9. Étale maps and uniformizing parameters 426439
- 21.10. Purity of the branch locus and the Abhyankar-Jung theorem 433446
- 21.11. Galois theory of local rings 438451
- 21.12. A proof of the Abhyankar-Jung theorem 441454

- Chapter 22. Bertini’s Theorems and General Fibers of Maps 451464
- Bibliography 469482
- Index 477490
- Back Cover Back Cover1498