**Graduate Studies in Mathematics**

Volume: 55;
2003;
188 pp;
Hardcover

MSC: Primary 14; 30; 32;

Print ISBN: 978-0-8218-3307-0

Product Code: GSM/55

List Price: $48.00

Individual Member Price: $38.40

**Electronic ISBN: 978-1-4704-2102-1
Product Code: GSM/55.E**

List Price: $48.00

Individual Member Price: $38.40

# A Scrapbook of Complex Curve Theory: Second Edition

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*C. Herbert Clemens*

This fine book by Herb Clemens quickly became a favorite of many algebraic geometers when it was first published in 1980. It has been popular with novices and experts ever since. It is written as a book of “impressions” of a journey through the theory of complex algebraic curves. Many topics of compelling beauty occur along the way. A cursory glance at the subjects visited reveals a wonderfully eclectic selection, from conics and cubics to theta functions, Jacobians, and questions of moduli. By the end of the book, the theme of theta functions becomes clear, culminating in the Schottky problem.

The author's intent was to motivate further study and to stimulate mathematical activity. The attentive reader will learn much about complex algebraic curves and the tools used to study them. The book can be especially useful to anyone preparing a course on the topic of complex curves or anyone interested in supplementing his/her reading.

#### Table of Contents

# Table of Contents

## A Scrapbook of Complex Curve Theory: Second Edition

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Chapter One: Conics 114 free
- 1.1. Hyperbola Shadows 114
- 1.2. Real Projective Space, The "Unifier" 518
- 1.3. Complex Projective Space, The Great "Unifier" 720
- 1.4. Linear Families of Conies 922
- 1.5. The Mystic Hexagon 1124
- 1.6. The Cross Ratio 1326
- 1.7. Cayley's Way of Doing Geometries of Constant Curvature 1730
- 1.8. Through the Looking Glass 2033
- 1.9. The Polar Curve 2235
- 1.10. Perpendiculars in Hyperbolic Space 2639
- 1.11. Circles in the X-Geometry 3043
- 1.12. Rational Points on Conics 3346

- Chapter Two: Cubics 3750
- 2.1. Inflection Points 3750
- 2.2. Normal Form for a Cubic 3952
- 2.3. Cubics as Topological Groups 4255
- 2.4. The Group of Rational Points on a Cubic 4558
- 2.5. A Thought about Complex Conjugation 5063
- 2.6. Some Meromorphic Functions on Cubics 5164
- 2.7. Cross Ratio Revisited, A Moduli Space for Cubics 5265
- 2.8. The Abelian Differential on a Cubic 5366
- 2.9. The Elliptic Integral 5568
- 2.10. The Picard-Fuchs Equation 5871
- 2.11. Rational Points on Cubics over F[sub(p)] 6275
- 2.12. Manin's Result: The Unity of Mathematics 6578
- 2.13. Some Remarks on Serre Duality 6982

- Chapter Three: Theta Functions 7386
- 3.1. Back to the Group Law on Cubics 7386
- 3.2. You Can't Parametrize a Smooth Cubic Algebraically 7588
- 3.3. Meromorphic Functions on Elliptic Curves 7891
- 3.4. Meromorphic Functions on Plane Cubics 8295
- 3.5. The Weierstrass p-Function 8598
- 3.6. Theta-Null Values Give Moduli of Elliptic Curves 89102
- 3.7. The Moduli Space of "Level-Two Structures" on Elliptic Curves 92105
- 3.8. Automorphisms of Elliptic Curves 95108
- 3.9. The Moduli Space of Elliptic Curves 96109
- 3.10. And So, By the Way, We Get Picard's Theorem 98111
- 3.11. The Complex Structure of M 100113
- 3.12. The j-Invariant of an Elliptic Curve 102115
- 3.13. Theta-Nulls as Modular Forms 106119
- 3.14. A Fundamental Domain for Γ[sub(2)] 109122
- 3.15. Jacobis Identity 111124

- Chapter Four: The Jacobian Variety 113126
- 4.1. Cohomology of a Complex Curve 113126
- 4.2. Duality 116129
- 4.3. The Chern Class of a Holomorphic Line Bundle 118131
- 4.4. Abel's Theorem for Curves 122135
- 4.5. The Classical Version of Abel's Theorem 127140
- 4.6. The Jacobi Inversion Theorem 131144
- 4.7. Back to Theta Functions 132145
- 4.8. The Basic Computation 134147
- 4.9. Riemann's Theorem 136149
- 4.10. Linear Systems of Degree g 138151
- 4.11. Riemann's Constant 139152
- 4.12. Riemann's Singularities Theorem 142155

- Chapter Five: Quartics and Quintics 147160
- Chapter Six: The Schottky Relation 161174
- References 181194
- Additional References 183196
- Index 185198
- Back Cover Back Cover1202

#### Readership

Graduate students and others looking for inspiration in the theory of complex algebraic curves.