A Scrapbook of Complex Curve Theory

This is a book of "impressions" of a journey through the theory of com­ plex algebraic curves. It is neither self-contained, balanced, nor particularly tightly organized. As with any notebook made on a journey, what appears is that which strikes the writer's fancy. Some topics appear because of their compelling intrinsic beauty. Others are left out because, for all their impor­ tance, the traveler found them boring or was too dull or lazy to give them their due. Looking back at the end of the journey, one can see that a common theme in fact does emerge, as is so often the case; that theme is the theory of theta functions. In fact very much of the material in the book is prepara­ tion for our study of the final topic, the so-called Schottky problem. More than once, in fact, we tear ourselves away from interesting topics leading elsewhere and return to our main route.

Contenu

· One Conics.- 1.1. Hyperbola Shadows.- 1.2. Real Projective Space, The "Unifier".- 1.3. Complex Projective Space, The Great "Unifier".- 1.4. Linear Families of Conics.- 1.5. The Mystic Hexagon.- 1.6. The Cross Ratio.- 1.7. Cayley's Way of Doing Geometries of Constant Curvature.- 1.8. Through the Looking Glass.- 1.9. The Polar Curve.- 1.10. Perpendiculars in Hyperbolic Space.- 1.11. Circles in the K-Geometry.- 1.12. Rational Points on Conics.- Two · Cubics.- 2.1. Inflection Points.- 2.2. Normal Form for a Cubic.- 2.3. Cubics as Topological Groups.- 2.4. The Group of Rational Points on a Cubic.- 2.5. A Thought about Complex Conjugation.- 2.6. Some Meromorphic Functions on Cubics.- 2.7. Cross Ratio Revisited, A Moduli Space for Cubics.- 2.8. The Abelian Differential on a Cubic.- 2.9. The Elliptic Integral.- 2.10. The Picard-Fuchs Equation.- 2.11. Rational Points on Cubics over Fp.- 2.12. Manin's Result: The Unity of Mathematics.- 2.13. Some Remarks on Serre Duality.- Three · Theta Functions.- 3.1. Back to the Group Law on Cubics.- 3.2. You Can't Parametrize a Smooth Cubic Algebraically.- 3.3. Meromorphic Functions on Elliptic Curves.- 3.4. Meromorphic Functions on Plane Cubics.- 3.5. The Weierstrass p-Function.- 3.6. Theta-Null Values Give Moduli of Elliptic Curves.- 3.7. The Moduli Space of "Level-Two Structures" on Elliptic Curves.- 3.8. Automorphisms of Elliptic Curves.- 3.9. The Moduli Space of Elliptic Curves.- 3.10. And So, By the Way, We Get Picard's Theorem.- 3.11. The Complex Structure of M.- 3.12. The j-Invariant of an Elliptic Curve.- 3.13. Theta-Nulls as Modular Forms.- 3.14. A Fundamental Domain for ?2.- 3.15. Jacobi's Identity.- Four · The Jacobian Variety.- 4.1. Cohomology of a Complex Curve.- 4.2. Duality.- 4.3. The Chern Class of a Holomorphic Line Bundle.- 4.4. Abel's Theorem for Curves.- 4.5. The Classical Version of Abel's Theorem.- 4.6. The Jacobi Inversion Theorem.- 4.7. Back to Theta Functions.- 4.8. The Basic Computation.- 4.9. Riemann's Theorem.- 4.10. Linear Systems of Degree g.- 4.11. Riemann's Constant.- 4.12. Riemann's Singularities Theorem.- Five · Quartics and Quintics.- 5.1. Topology of Plane Quartics.- 5.2. The Twenty-Eight Bitangents.- 5.3. Where Are the Hyperelliptic Curves of Genus 3?.- 5.4. Quintics.- Six · The Schottky Relation.- 6.1. Prym Varieties.- 6.2. Riemann's Theta Relation.- 6.3. Products of Pairs of Theta Functions.- 6.4. A Proportionality Theorem Relating Jacobians and Pryms.- 6.5. The Proportionality Theorem of Schottky-Jung.- 6.6. The Schottky Relation.- References.