Algebraic Geometry: Notes on a Course

✍ Scribed by Michael Artin

• Publisher: American Mathematical Society
• Year: 2022
• Tongue: English
• Leaves: 331
• Category: Library

✦ Synopsis

This book is an introduction to the geometry of complex algebraic varieties. It is intended for students who have learned algebra, analysis, and topology, as taught in standard undergraduate courses. So it is a suitable text for a beginning graduate course or an advanced undergraduate course. The book begins with a study of plane algebraic curves, then introduces affine and projective varieties, going on to dimension and constructibility. $mathcal{O}$-modules (quasicoherent sheaves) are defined without reference to sheaf theory, and their cohomology is defined axiomatically. The Riemann-Roch Theorem for curves is proved using projection to the projective line. Some of the points that aren't always treated in beginning courses are Hensel's Lemma, Chevalley's Finiteness Theorem, and the Birkhoff-Grothendieck Theorem. The book contains extensive discussions of finite group actions, lines in $mathbb{P}^3$, and double planes, and it ends with applications of the Riemann-Roch Theorem.

✦ Table of Contents

Cover
Title page
Library of Congress Cataloging-in-Publication Data
Contents
Preface
A Note for the Student
Chapter 1. Plane Curves
1.1. The Affine Plane
1.2. The Projective Plane
1.3. Plane Projective Curves
1.4. Tangent Lines
1.5. The Dual Curve
1.6. Resultants and Discriminants
1.7. Nodes and Cusps
1.8. Hensel's Lemma
1.9. Bezout's Theorem
1.10. The Plucker Formulas
Exercises
Chapter 2. Affine Algebraic Geometry
2.1. The Zariski Topology
2.2. Some Affine Varieties
2.3. Hubert's Nullstellensatz
2.4. The Spectrum
2.5. Morphisms of Affine Varieties
2.6. Localization
2.7. Finite Group Actions
Exercises
Chapter 3. Projective Algebraic Geometry
3.1. Projective Varieties
3.2. Homogeneous Ideals
3.3. Product Varieties
3.4. Rational Functions
3.5. Morphisms
3.6. Affine Varieties
3.7. Lines in Three-Space
Exercises
Chapter 4. Integral Morphisms
4.1. The Nakayama Lemma
4.2. Integral Extensions
4.3. Normalization
4.4. Geometry of Integral Morphisms
4.5. Dimension
4.6. Chevalley's Finiteness Theorem
4.7. Double Planes
Exercises
Chapter 5. Structure of Varieties in the Zariski Topology
5.1. Local Rings
5.2. Smooth Curves
5.3. Constructible Sets
5.4. Closed Sets
5.5. Projective Varieties Are Proper
5.6. Fibre Dimension
Exercises
Chapter 6. Modules
6.1. The Structure Sheaf
6.2. O-Modules
6.3. The Sheaf Property
6.4. More Modules
6.5. Direct Image
6.6. Support
6.7. Twisting
6.8. Extending a Module: Proof
Exercises
Chapter 7. Cohomology
7.1. Cohomology
7.2. Complexes
7.3. Characteristic Properties
7.4. Existence of Cohomology
7.5. Cohomology of the Twisting Modules
7.6. Cohomology of Hypersurfaces
7.7. Three Theorems about Cohomology
7.8. Bezout's Theorem
7.9. Uniqueness of the Coboundary Maps
Exercises
Chapter 8. The Riemann-Roch Theorem for Curves
8.1. Divisors
8.2. The Riemann-Roch Theorem I
8.3. The Birkhoff-Grothendieck Theorem
8.4. Differentials
8.5. Branched Coverings
8.6. Trace of a Differential
8.7. The Riemann-Roch Theorem II
8.8. Using Riemann-Roch
8.9. What Is Next
Exercises
Chapter 9. Background
9.1. Rings and Modules
9.2. The Implicit Function Theorem
9.3. Transcendence Degree
Glossary
Index of Notation
Bibliography
Index