University Series in Mathematics A Scrapbook of Complex Curve Theory

This is a book of "impressions" of a journey through the theory of com­ plex algebraic curves. 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; 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.