Algebraic geometry : notes on a course

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. Cover Title page Copyright 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. Bézout’s Theorem 1.10. The Plücker Formulas Exercises Chapter 2. Affine Algebraic Geometry 2.1. The Zariski Topology 2.2. Some Affine Varieties 2.3. Hilbert’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. co -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. Bézout’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 Back Cover Provides an introduction to the geometry of complex algebraic varieties. The book is intended for students who have learned algebra, analysis, and topology, as taught in standard undergraduate courses. It is a suitable text for a beginning graduate course or an advanced undergraduate course.