Singular points of theta functions, quadric relations and holomorphic differentials with prescribed zeros

Abstract

Introduction. In this, for the most part, expository paper we are concerned with the connection between singular points of the Riemann theta function, relations among holomorphic muadratic differentials and representations of holomorphic differentials with prescribed zeros. The main tools used are the Riemann-Roch theorem, Abel's theorem and the theory of theta functions on a Riemann surface. These are all treated in [RF]. The fact that there are connections between the objects in the title is not a new fact. The ideas are contained implicitly in many papers. It is our purpose here to make explicit many of these ideas and to highlight the role of the Riemann theta function in the solution of function theoretic problems on a Riemann surface. It is our hope to convince the reader that the subject of theta functions can still be an important tool in solving function theory problems on compact Riemann surfaces. The two main function theoretic problems we deal with here are the construction of explicit rank four (three) relations between the holomorphic quadratic differentials on a compact surface and explicit representations of holomorphic differentials with prescribed zeros. The first problem is treated in [AM] and the second problem is a generalization of a very classical idea. Some of the ideas contained in this paper are also contained in the papers [Ma I, II]. Part of the motivation for this work was also to understand a recent theorem of Mumford [Mu] on singular points for the theta function associated with the Prym variety of a Riemann surface. The Prym variety and the theta functions on it have been a useful tool in the investigations of the Schottky problem for compact Riemann surfaces [F, B, FR l, RF]. Andreotti and Mayer [AM] characterized Jacobian varieties in terms of the singular sets of the theta functions on them. One of the by products of this investigation will be that there is probably no analogue of their result for the Prym varieties.