Algebraic Curves and Riemann Surfaces

by Miranda, Rick

• ISBN13: 9780821802687
• ISBN10: 0821802682
• Format: Hardcover
• Copyright: 1995-04-01
• Publisher: Amer Mathematical Society

Summary

In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking centre stage. But the main examples come fromprojective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Dualtiy Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves andcohomology are introduced as a unifying device in the later chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one term of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-term course in complex variables or a year-long course in algebraic geometry.

Table of Contents

Riemann surfaces: Basic definitions

Functions and maps More examples of Riemann surfaces

Integration on Riemann surfaces

Divisors and meromorphic functions

Algebraic curves and the Riemann-Roch theorem

Applications of Riemann-Roch Abel's theorem Sheaves and Cech cohomology

Algebraic sheaves Invertible sheaves, line bundles, and $\check H^1$

References

Index of notation

Index of terminology

Table of Contents provided by Publisher. All Rights Reserved.