A Scrapbook of Complex Curve Theory [Paperback]

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.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.? 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. Cayleys Way of DoinlC5