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Let C be a smooth projective curve over an algebraically closed field ofarbitrary characteristic. Let M_{r,L}^{ss} denote the projective coarse modulischeme of semistable rank r vector bundles over C with fixed determinant L. Weprove Pic(M_{r,L}^{ss}) = Z, identify the ample generator, and deduce thatM_{r,L}^{ss} is locally factorial. In characteristic zero, this has alreadybeen proved by Dr\'{e}zet and Narasimhan. The main point of the present note isto circumvent the usual problems with Geometric Invariant Theory in positivecaracteristic.

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