A Concise Introduction to Algebraic Varieties (Graduate Studies in Mathematics, 216)

Designed for a one-term introductory course on algebraic varieties over an algebraically closed field, this book prepares students to continue either with a course on schemes and cohomology, or to learn more specialized topics such as toric varieties and moduli spaces of curves. The book balances generality and accessibility by presenting local and global concepts, such as nonsingularity, normality, and completeness using the language of atlases, an approach that is most commonly associated with differential topology. The book concludes with a discussion of the Riemann-Roch theorem, the Brill-Noether theorem, and applications. The prerequisites for the book are a strong undergraduate algebra course and a working familiarity with basic point-set topology. A course in graduate algebra is helpful but not required. The book includes appendices presenting useful background in complex analytic topology and commutative algebra and provides plentiful examples and exercises that help build intuition and familiarity with algebraic varieties. Cover 1 Title page 4 Preface 12 Chapter 1. Introduction: An overview of algebraic geometry through the lens of plane curves 18 1.1. Plane curves 18 1.2. Elliptic curves 22 Chapter 2. Affine algebraic varieties 28 2.1. Zero sets and the Zariski topology 28 2.2. Zero sets and ideals 32 2.3. Noetherian spaces 39 2.4. Dimension 41 Chapter 3. Regular functions and morphisms 48 3.1. Regular functions 48 3.2. Morphisms 56 3.3. Rational maps 65 3.4. Chevalley’s theorem 69 3.A. Recovering geometry from categories 72 Chapter 4. Singularities 76 4.1. Tangent lines and singularities 77 4.2. Zariski cotangent spaces 79 4.3. The Jacobian criterion 82 4.4. Completions and power series 88 4.5. Normality and normalization 91 4.A. Local generation of ideals 94 Chapter 5. Abstract varieties via atlases 98 5.1. Prevarieties 98 5.2. Regular functions and morphisms 103 5.3. Abstract varieties 107 5.4. Normalization revisited 113 Chapter 6. Projective varieties 120 6.1. Projective space 120 6.2. Projective varieties and morphisms 125 6.3. Blowup of subvarieties 131 6.A. Homogeneous ideals and coordinate rings 139 Chapter 7. Nonsingular curves and complete varieties 142 7.1. Curves, regular functions, and morphisms 142 7.2. Quasiprojectivity 147 7.3. Projective curves 149 7.4. Completeness 152 7.5. A limit-based criterion 154 7.6. Irreducibility of polynomials in families 158 Chapter 8. Divisors on nonsingular curves 164 8.1. Morphisms of curves 164 8.2. Divisors on curves 167 8.3. Linear equivalence and morphisms to projective space 170 8.4. Embeddings of curves 177 8.5. Secant varieties and curves in projective space 180 Chapter 9. Differential forms 186 9.1. Differential forms 186 9.2. Differential forms on curves 190 9.3. Differential forms and ramification 193 9.A. Field automorphisms and Frobenius 198 Chapter 10. An invitation to the theory of algebraic curves 204 10.1. The Riemann-Roch theorem 204 10.2. The Riemann-Hurwitz theorem 207 10.3. Brill-Noether theory and moduli spaces of curves 212 10.A. Remarks on proofs of the Riemann-Roch theorem 216 Appendix A. Complex varieties and the analytic topology 224 A.1. Quasiaffine complex varieties 224 A.2. The analytic topology on prevarieties 226 A.3. Fundamental results 228 A.4. Nonsingularity and complex manifolds 229 A.5. Connectedness 230 Appendix B. A roadmap through algebra 234 B.1. Field theory 235 B.2. Algebras 238 B.3. Noetherian rings 239 B.4. Rings of fractions 240 B.5. Nakayama’s lemma 242 B.6. Unique factorization 245 B.7. Integral extensions 246 B.8. Integral closure 248 B.9. The principal ideal theorem and regular local rings 249 B.10. Noether normalization and first applications 253 B.11. Dimension theory over fields 255 B.12. Extensions of Dedekind domains 258 B.13. Completion and power series 261 Bibliography 264 Index of Notation 268 Index 270 Back Cover 279 "'A Concise Introduction to Algebraic Varieties' is designed for a one-term introductory course on algebraic varieties over an algebraically closed field, and it provides a solid basis for a course on schemes and cohomology or on specialized topics, such as toric varieties and moduli spaces of curves. The book balances generality and accessibility by presenting local and global concepts, such as nonsingularity, normality, and completeness using the language of atlases, an approach that is most commonly associated with differential topology. The book concludes with a discussion of the Riemann-Roch theorem, the Brill-Noether theorem, and applications."--Back cover