Separation Properties of Theta Functions

0. Introduction. Let X be a non-singular, connected, projective curve defined over an algebraically closed field k. Let U denote the set of isomorphism classes of stable vector bundles on X with given degree d and rank r. In the sixties, C.S. Seshadri and D. Mumford ([14], [17] and [18]) supplied U with a natural structure of quasi-projective variety, together with a natural compactification U , by adding semistable vector bundles at the boundary. The method used in the construction of such a structure was Mumford’s then recently developed Geometric Invariant Theory [8]. Roughly, the method consists of producing a variety R, and an action of a reductive group G on X , linearized at some ample invertible sheaf L on R, such that U = R/G set-theoretically. Then, Geometric Invariant Theory (G.I.T.) tells us how to supply R/G with a natural scheme structure, obtained from the G-invariant sections of tensor powers of L. Up until recently, Seshadri’s and Mumford’s construction was the only purely algebraic construction available. In 1993, Faltings [7] showed how to construct U, and its compactification U , avoiding G.I.T.. His method, described also in [20], consisted in considering the so-called theta functions on R, naturally defined provided R admits a family with the so-called local universal property. (We observe that the theta functions considered in this article are just those associated with vector bundles on X , as it will be clear from our definition in Sect. 2. Beauville [3, Sect. 2] has a more encompassing definition of theta functions than ours.) The theta functions are in fact G-invariant sections of tensor powers of a certain G-linear invertible sheaf L′θ on R. Roughly speaking, using his first main lemma [20, Lemma 3.1, p. 166], Faltings showed that there are enough theta functions to produce a G-invariant morphism, θ : R −→ P . By semistable reduction, the image,