A Concise Introduction to Algebraic Varieties

✍ Scribed by Brian Osserman

• Publisher: American Mathematical Society
• Year: 2021
• Tongue: English
• Leaves: 279
• Series: Graduate Studies in Mathematics 216
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover
Title page
Preface
Chapter 1. Introduction: An overview of algebraic geometry through the lens of plane curves
1.1. Plane curves
1.2. Elliptic curves
Chapter 2. Affine algebraic varieties
2.1. Zero sets and the Zariski topology
2.2. Zero sets and ideals
2.3. Noetherian spaces
2.4. Dimension
Chapter 3. Regular functions and morphisms
3.1. Regular functions
3.2. Morphisms
3.3. Rational maps
3.4. Chevalley’s theorem
3.A. Recovering geometry from categories
Chapter 4. Singularities
4.1. Tangent lines and singularities
4.2. Zariski cotangent spaces
4.3. The Jacobian criterion
4.4. Completions and power series
4.5. Normality and normalization
4.A. Local generation of ideals
Chapter 5. Abstract varieties via atlases
5.1. Prevarieties
5.2. Regular functions and morphisms
5.3. Abstract varieties
5.4. Normalization revisited
Chapter 6. Projective varieties
6.1. Projective space
6.2. Projective varieties and morphisms
6.3. Blowup of subvarieties
6.A. Homogeneous ideals and coordinate rings
Chapter 7. Nonsingular curves and complete varieties
7.1. Curves, regular functions, and morphisms
7.2. Quasiprojectivity
7.3. Projective curves
7.4. Completeness
7.5. A limit-based criterion
7.6. Irreducibility of polynomials in families
Chapter 8. Divisors on nonsingular curves
8.1. Morphisms of curves
8.2. Divisors on curves
8.3. Linear equivalence and morphisms to projective space
8.4. Embeddings of curves
8.5. Secant varieties and curves in projective space
Chapter 9. Differential forms
9.1. Differential forms
9.2. Differential forms on curves
9.3. Differential forms and ramification
9.A. Field automorphisms and Frobenius
Chapter 10. An invitation to the theory of algebraic curves
10.1. The Riemann-Roch theorem
10.2. The Riemann-Hurwitz theorem
10.3. Brill-Noether theory and moduli spaces of curves
10.A. Remarks on proofs of the Riemann-Roch theorem
Appendix A. Complex varieties and the analytic topology
A.1. Quasiaffine complex varieties
A.2. The analytic topology on prevarieties
A.3. Fundamental results
A.4. Nonsingularity and complex manifolds
A.5. Connectedness
Appendix B. A roadmap through algebra
B.1. Field theory
B.2. Algebras
B.3. Noetherian rings
B.4. Rings of fractions
B.5. Nakayama’s lemma
B.6. Unique factorization
B.7. Integral extensions
B.8. Integral closure
B.9. The principal ideal theorem and regular local rings
B.10. Noether normalization and first applications
B.11. Dimension theory over fields
B.12. Extensions of Dedekind domains
B.13. Completion and power series
Bibliography
Index of Notation
Index
Back Cover

✦ Subjects

Algebraic Varieties, Regular Functions, Singularities, Projective Varieties, Divisors, Differential Forms, Curves