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.

Texte du rabat

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 Classof 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.