Separation properties of theta functions

Abstract

In a 1993 article, G. Faltings gave a new construction of the moduli space $U$ of semistable vector bundles on a smooth curve $X$, avoiding geometric invariant theory. Roughly speaking, Faltings showed that the normalisation $B$ of the ring $A$ of theta functions (associated with vector bundles on $X$) suffices to realize $U$ as a projective variety. Describing Faltings' work, C.S. Seshadri asked how close $A$ is to $B$. In this article, we address this question from a geometric point of view. We consider the rational map, $\pi : U @>>> Proj(A)$, and show that, not only is $\pi$ defined everywhere, but also $\pi$ is bijective, and is an isomorphism over the stable locus of $U$, if the characteristic of the ground field is 0. Moreover, we give a direct local construction of $U$ as a fine moduli space, when the rank and degree are coprime, in any characteristic. The methods in the article apply to singular curves as well.