The Picard group of a coarse moduli space of vector bundles in positive characteristic

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