Plane Algebraic Curves

Front Cover
CRC Press, Jan 1, 1981 - Mathematics - 240 pages
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.
 

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Contents

CONTENTS
1
Chapter
12
THE LOCAL RING AT A POINT
43
Chapter 5
52
Chapter 6
70
Chapter 7
78
EXAMPLES OF RATIONAL CURVES
88
Chapter 9
97
Chapter 10
122
THE DIVISOR OF A FUNCTION HAS DEGREE O
128
RIEMANNS THEOREM
133
Chapter 15
148
Chapter 17
163
Chapter 19
179
Chapter 21
202
Bibliography
217

AN OVERVIEW AND SIDEWAYS GLANCE
110

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