**Courant Lecture Notes**

Volume: 8;
2002;
214 pp;
Softcover

MSC: Primary 13; 14; 32;

Print ISBN: 978-0-8218-2862-5

Product Code: CLN/8

List Price: $31.00

Individual Member Price: $24.80

**Electronic ISBN: 978-1-4704-3109-9
Product Code: CLN/8.E**

List Price: $31.00

Individual Member Price: $24.80

# Algebraic Curves and One-Dimensional Fields

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*Fedor Bogomolov; Tihomir Petrov*

A co-publication of the AMS and the Courant Institute of Mathematical Sciences at New York University

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 & Endorsements

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

# Table of Contents

## Algebraic Curves and One-Dimensional Fields

- Cover Cover11
- General editor’s page ii3
- Title page iii4
- Dedication v6
- Contents vii8
- Preface ix10
- Algebraic Preliminaries 114
- From algebra to geometry 4356
- Geometry of dimension one 6174
- Divisors and line bundles 95108
- Vector bundles, coherent sheaves, and cohomology 119132
- Vector bundles on ℙ¹ 149162
- General theory of curves 163176
- Elliptic curves 173186
- The Riemann-Roch theorem 183196
- Curves over arithmetic fields 197210
- Bibliography 209222
- Index 211224
- Back Cover Back Cover1229