  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.
