Plane Algebraic Curves

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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

Table of Contents

Preface v
Chapter 0 Prerequisites 1
Chapter 1 Some Facts About Polynomials 9
Chapter 2 Affine Plane Curves 17
Chapter 3 Tangent Spaces 32
Chapter 4 The Local Ring at a Point 43
Chapter 5 Projective Plane Curves 52
Chapter 6 Rational Mappings, Birational Correspondences and Isomorphisms of Curves 70
Chapter 7 Examples of Rational Curves 88
Chapter 8 The Correspondence Between Valuations and Points 93
Chapter 9 An Overview and Sideways Glance 110
Chapter 10 Divisors 122
Chapter 11 The Divisor of a Function has Degree 0 128
Chapter 12 Riemann's Theorem 133
Chapter 13 The Genus of a Nonsingular Plane Curve 137
Chapter 14 Curves of Genus 0 and 1 141
Chapter 15 A Classification of Isomorphism Classes of Curves of Genus 1 149
Chapter 16 The Genus of a Singular Curve 153
Chapter 17 Inflection Points on Plane Curves 163
Chapter 18 Bezout's Theorem 174
Chapter 19 Addition on a Nonsingular Cubic 179
Chapter 20 Derivations, Differentials and the Canonical Class 184
Chapter 21 Adeles and the Riemann-Roch Theorem 202
Bibliography 217
Notation 219
Index 221
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