The Geometric Theory of Riemann’s Theta Function

Abstract

The results of the Brill-Noether theory stated in the preceding chapter have been proved in full generality only fairly recently. However, special but important cases of them were classically known and, in a sense, provided a motivation for the entire theory. What we have in mind here are the classical theorems concerning the geometry of W _{g − 1}(C), that is, the geometry of Riemann’s theta function. Of course, these results are more than mere exemplifications of the general ones of Chapter V. Rather, they are to be viewed as illustrations of how those general results can be used in the study of concrete geometrical problems. Our analysis will be carried out partly by means of classical methods, and partly in the language of Chapter IV.