The Picard group of a coarse moduli space of vector bundles in positive characteristic (Pre-published version)

Abstract

Let C be a smooth projective curve over an algebraically closed field of arbitrary characteristic. Let Mss r,L denote the projective coarse moduli scheme of semistable rank r vector bundles over C with fixed determinant L. We prove Pic(Mss r,L) = Z, identify the ample generator, and deduce that Mss r,L is locally factorial. In characteristic zero, this has already been proved by Dr´ezet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant Theory in positive caracteristic.