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,
π:U@>>>Proj(A)
, and show that, not only is
π
defined everywhere, but also
π
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.