Plane Algebraic Curves

Plane Algebraic Curves

by C. Orzech, Morris Orzech
     
 

ISBN-10: 0824711599

ISBN-13: 9780824711597

Pub. Date: 01/01/1981

Publisher: Taylor & Francis

Plane Algebraic Curves is a classroom-tested textbook for advanced undergraduate and beginning graduate students in mathematics. The book introduces the contemporary notions of algebraic varieties, morphisms of varieties, and adeles to the classical subject of plane curves over algebraically closed fields. By restricting the rigorous development of these notions to a…  See more details below

Overview

Plane Algebraic Curves is a classroom-tested textbook for advanced undergraduate and beginning graduate students in mathematics. The book introduces the contemporary notions of algebraic varieties, morphisms of varieties, and adeles to the classical subject of plane curves over algebraically closed fields. By restricting the rigorous development of these notions to a traditional context the book makes its subject accessible without extensive algebraic prerequisites. Once the reader's intuition for plane curves has evolved, there is a discussion of how these objects can be generalized to higher dimensional settings. These features, as well as a proof of the Riemann-Roch Theorem based on a combination of geometric and algebraic considerations, make the book a good foundation for more specialized study in algebraic geometry, commutative algebra, and algebraic function fields. Plane Algebraic Curves is suitable for readers with a variety of backgrounds and interests. The book begins with a chapter outlining prerequisites, and contains informal discussions giving an overview of its material and relating it to non-algebraic topics which would be familiar to the general reader. There is an explanation of why the algebraic genus of a hyperelliptic curve agrees with its geometric genus as a compact Riemann surface, as well as a thorough description of how the classically important elliptic curves can be described in various normal forms. The book concludes with a bibliography which students can incorporate into their further studies.

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Product Details

ISBN-13:
9780824711597
Publisher:
Taylor & Francis
Publication date:
01/01/1981
Series:
Chapman & Hall/CRC Pure and Applied Mathematics Series, #61
Pages:
240
Product dimensions:
0.56(w) x 6.14(h) x 9.21(d)

Related Subjects

Table of Contents

Prefacev
Chapter 0Prerequisites1
Chapter 1Some Facts About Polynomials9
Chapter 2Affine Plane Curves17
Chapter 3Tangent Spaces32
Chapter 4The Local Ring at a Point43
Chapter 5Projective Plane Curves52
Chapter 6Rational Mappings, Birational Correspondences and Isomorphisms of Curves70
Chapter 7Examples of Rational Curves88
Chapter 8The Correspondence Between Valuations and Points93
Chapter 9An Overview and Sideways Glance110
Chapter 10Divisors122
Chapter 11The Divisor of a Function has Degree 0128
Chapter 12Riemann's Theorem133
Chapter 13The Genus of a Nonsingular Plane Curve137
Chapter 14Curves of Genus 0 and 1141
Chapter 15A Classification of Isomorphism Classes of Curves of Genus 1149
Chapter 16The Genus of a Singular Curve153
Chapter 17Inflection Points on Plane Curves163
Chapter 18Bezout's Theorem174
Chapter 19Addition on a Nonsingular Cubic179
Chapter 20Derivations, Differentials and the Canonical Class184
Chapter 21Adeles and the Riemann-Roch Theorem202
Bibliography217
Notation219
Index221

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