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alg-geo.md

Algebraic Geometry

Introduction

1.1. Overview of algebraic geometry

1.2. Historical background

1.3. Basic concepts and notation

Affine Varieties

2.1. Affine algebraic sets

2.2. The coordinate ring

2.3. Regular functions

2.4. Nullstellensatz

2.5. Morphisms and rational maps

2.6. Zariski topology

2.7. Dimension

Projective Varieties

3.1. Projective space

3.2. Projective algebraic sets

3.3. Homogeneous coordinate rings

3.4. Sheaves on projective varieties

3.5. Divisors and linear systems

3.6. Bezout's theorem

3.7. Grassmannians

Schemes

4.1. Commutative algebra

4.2. Local rings and maximal ideals

4.3. Affine schemes

4.4. Gluing schemes

4.5. Projective schemes

4.6. Properties of schemes

4.7. Cohomology

Varieties and Schemes

5.1. Morphisms of varieties

5.2. Separated and proper morphisms

5.3. Flat morphisms

5.4. Fibered products

5.5. Nonsingular varieties

5.6. Intersection theory

5.7. Moduli spaces

Curves

6.1. Curves over algebraically closed fields

6.2. Riemann-Roch theorem

6.3. Elliptic curves

6.4. The group law on an elliptic curve

6.5. Jacobians

6.6. Genus 0 and 1 curves

6.7. Moduli spaces of curves

Surfaces

7.1. Rational surfaces

7.2. Projective planes

7.3. Enriques-Kodaira classification

7.4. Ruled surfaces

7.5. Elliptic surfaces

7.6. Minimal models

7.7. Picard group

Higher-dimensional Varieties

8.1. Rational and ruled varieties

8.2. Fano varieties

8.3. Complete intersections

8.4. Calabi-Yau varieties

8.5. Mori theory

8.6. Hodge theory

8.7. Mixed Hodge structures

Algebraic Groups

9.1. Linear algebraic groups

9.2. Representations of algebraic groups

9.3. Quotient varieties

9.4. Torus actions and fans

9.5. Borel-Weil-Bott theorem

9.6. Geometric invariant theory

9.7. Flag varieties

Geometric Invariant Theory

10.1. Reductive groups

10.2. Basic constructions

10.3. The Hilbert-Mumford criterion

Intersection Theory

11.1. Chow groups and cycles

11.2. Intersection products

11.3. Riemann-Roch for surfaces

11.4. Grothendieck-Riemann-Roch

11.5. Hodge index theorem

11.6. Bloch's conjecture

11.7. K-theory

Analytic Methods

12.1. Complex manifolds

12.2. Sheaves and cohomology

12.3. Dolbeault cohomology

12.4. Hodge decomposition

12.5. The Kodaira embedding theorem

12.6. The Riemann-Roch theorem for curves

12.7. Compactifications

Arithmetic Geometry

13.1. Diophantine equations

13.2. The Mordell-Weil theorem

13.3. Elliptic curves over number fields

13.4. Modular forms

13.5. The Shimura-Taniyama-Weil conjecture

13.6. Rational points on varieties

13.7. Arithmetic intersection theory

Topics in Algebraic Geometry

14.1. D-modules

14.2. Derived categories

14.3. Mirror symmetry

14.4. Tropical geometry

14.5. Noncommutative algebraic geometry

14.6. A-infinity algebras

14.7. Topological field theories

Appendix

15.1. Algebraic background

15.2. Analytic background

15.3. Category theory

15.4. Homological algebra

15.5. Computational algebraic geometry

15.6. Glossary of notation and terminology

15.7. Bibliography