Algebraic geometry

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 Afine 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 Curves 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