Volume: 216; 2021; 259 pp; Softcover
MSC: Primary 14;
Print ISBN: 978-1-4704-6665-7
Product Code: GSM/216.S
List Price: $85.00
AMS Member Price: $68.00
MAA Member Price: $76.50
Electronic ISBN: 978-1-4704-6664-0
Product Code: GSM/216.E
List Price: $85.00
AMS Member Price: $68.00
MAA Member Price: $76.50
Supplemental Materials
A Concise Introduction to Algebraic Varieties
Share this pageBrian Osserman
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.
Readership
Undergraduate and graduate students interested in an introduction to fundamentals of algebraic geometry.
Table of Contents
Table of Contents
A Concise Introduction to Algebraic Varieties
- Cover Cover11
- Title page iii4
- Preface xi12
- Chapter 1. Introduction: An overview of algebraic geometry through the lens of plane curves 118
- Chapter 2. Affine algebraic varieties 1128
- Chapter 3. Regular functions and morphisms 3148
- Chapter 4. Singularities 5976
- Chapter 5. Abstract varieties via atlases 8198
- Chapter 6. Projective varieties 103120
- Chapter 7. Nonsingular curves and complete varieties 125142
- Chapter 8. Divisors on nonsingular curves 147164
- Chapter 9. Differential forms 169186
- Chapter 10. An invitation to the theory of algebraic curves 187204
- Appendix A. Complex varieties and the analytic topology 207224
- Appendix B. A roadmap through algebra 217234
- B.1. Field theory 218235
- B.2. Algebras 221238
- B.3. Noetherian rings 222239
- B.4. Rings of fractions 223240
- B.5. Nakayama’s lemma 225242
- B.6. Unique factorization 228245
- B.7. Integral extensions 229246
- B.8. Integral closure 231248
- B.9. The principal ideal theorem and regular local rings 232249
- B.10. Noether normalization and first applications 236253
- B.11. Dimension theory over fields 238255
- B.12. Extensions of Dedekind domains 241258
- B.13. Completion and power series 244261
- Bibliography 247264
- Index of Notation 251268
- Index 253270
- Back Cover Back Cover1279