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Introduction
CHAPTER I
Varieties
1 Affine Varieties
2 Projective Varieties
3 Morphisms
4 Rational Maps
5 Nonsingular Varieties
6 Nonsingular Curves
7 Intersections in Projective Space
8 What Is Algebraic Geometry
CHAPTER II
Schemes
1 Sheaves
2 Schemes
3 First Properties of Schemes
4 Separated and Proper Morphisms
5 Sheaves of Modules
6 Divisors
7 Projective Morphisms
8 Differentials
9 Formal Schemes
CHAPTER III
Cohomology
1 Derived Functors
2 Cohomology of Sheaves
3 Cohomology of a Noetherian Affine Scheme
4 Cech Cohomology
5 The Cohomology of Projective Space
6 Ext Groups and Sheaves
7 The Serre Duality Theorem
8 Higher Direct images of Sheaves
9 Flat Morphisms
10 Smooth Morphisms
11 The Theorem on Formal Functions
12 The Semicontinuity Theorem
CHAPTER IV
Curves
1 Riemann-Roch Theorem
2 Hurwitz''s Theorem
3 Embeddings in Projective Space
4 Elliptic Carves
5 The Canonical Embedding
6 Classification of Curves in P3
CHAPTER V
Surfaces
1 Geometry on a Surface
2 Ruled Surfaces
3 Monoidal Transformations
4 The Cubic Surface in P3
5 Birational Transformations
6 Classification of Surfaces
APPENDIX A
Intersection Theory
1 Intersection Theory
2 Properties of the Chow Ring
3 Chern Classes
4 The Riemann-Roch Theorem
5 Complements and Generalizations
APPENDIX B
Transcendental Methods
1 The Associated Complex Analytic Space
2 Comparison of the Algebraic and Analytic Categories
3 When is a Compact Complex Manifold Algebraic
4 Kahler Manifolds
5 The Exponential Sequence
APPENDIX C
The Weil Conjectures
1 The Zeta Function and the Well Conjectures
2 History of Work on the Weil Conjectures
3 The I-adic Cohomology
4 Cohomological Interpretation of the Well Conjectures
Bibliography
Results from Algebra
Glossary of Notations
Index