This book provides an introduction to abstract algebraic geometry using the methods of schemes and cohomology. The main objects of study are algebraic varieties in an affine or projective space over an algebraically closed field; these are introduced in Chapter I, to establish a number of basic concepts and examples. Then the methods of schemes and cohomology are developed in Chapters II and III, with emphasis on applications rather than excessive generality. The last two chapters of the book (IV and V) use these methods to study topics in the classical theory of algebraic curves and surfaces.
目录
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 APPENDIX B APPENDIX C Bibliography Results from Algebra Glossary of Notations Index
