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Algebraic Curves and One-Dimensional Fields
Fedor Bogomolov and Tihomir Petrov, New York University - Courant Institute of Mathematical Sciences, NY
A co-publication of the AMS and Courant Institute of Mathematical Sciences at New York University.
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Courant Lecture Notes
2002; 214 pp; softcover
Volume: 8
ISBN-10: 0-8218-2862-2
ISBN-13: 978-0-8218-2862-5
List Price: US$29
Member Price: US$23.20
Order Code: CLN/8
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Algebraic curves have many special properties that make their study particularly rewarding. As a result, curves provide a natural introduction to algebraic geometry. In this book, the authors also bring out aspects of curves that are unique to them and emphasize connections with algebra.

This text covers the essential topics in the geometry of algebraic curves, such as line and vector bundles, the Riemann-Roch Theorem, divisors, coherent sheaves, and zeroth and first cohomology groups. The authors make a point of using concrete examples and explicit methods to ensure that the style is clear and understandable.

Several chapters develop the connections between the geometry of algebraic curves and the algebra of one-dimensional fields. This is an interesting topic that is rarely found in introductory texts on algebraic geometry.

This book makes an excellent text for a first course for graduate students.

Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

Readership

Advanced undergraduates, graduate students, and research mathematicians interested in algebra and algebraic geometry.

Reviews

"A highly interesting, individual and remarkably profound first introduction to the theory of algebraic curves ... the authors ... in a stunning way ... use comparatively elementary ad-hoc approaches ... no doubt to the benefit of the inexperienced reader, as it leads to some of the deep results on curves, grants an enlightening glimpse to what algebraic geometry is all about, and encourages the beginner to continue his/her studies at a more general and advanced level ... precisely the goal that the authors are aspiring to ... and they achieve it in a very original, careful thought out and well done manner ... presented in great detail, rigor, lucidity ... seasoned and self-supporting graduate students will find these original lecture notes very useful, inspiring, pleasantly challenging and profitable ... highly original and unconventional introduction to algebraic curves is certainly a delight ... unique and outstanding flavour ... an interesting change and a great enhancement within the existing introductory literature on the subject."

-- Zentralblatt MATH

Table of Contents

  • Algebraic Preliminaries
  • From algebra to geometry
  • Geometry of dimension one
  • Divisors and line bundles
  • Vector bundles, coherent sheaves, and cohomology
  • Vector bundles on \(\mathbb{P}^{1}\)
  • General theory of curves
  • Elliptic curves
  • The Riemann-Roch theorem
  • Curves over arithmetic fields
  • Bibliography
  • Index
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