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 Dr\'{e}zet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant Theory in positive caracteristic.