1997/09/08 by Esteves, Eduardo
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.alg-geom/9709008
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.