Plane Algebraic Curves

by Orzech,C.

• ISBN13: 9780824711597
• ISBN10: 0824711599
• eBook ISBN(s): 9781040280027
• Format: Hardcover
• Copyright: 1981-01-01
• Publisher: CRC Press
• More Book Details

Summary

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. Book jacket.

Author Biography

Grace Orzech is Assistant Professor of Mathematics at Queen's University in Kingston, Canada Morris Orzech is Associate Professor of Mathematics at Queen's University in Kingston, Canada

Table of Contents

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