  We study the Picard groups of moduli spaces in positive characteristics and
we give a "$p$-adic" proof that the Picard group of moduli of vector bundles of
fixed determinant is isomorphic to the group of integers. Along the way we
prove that the local fundamental group scheme of a normal unirational
projective variety is trivial. This is reminiscent of results of Serre and
Nygaard who studied the fundamental groups of smooth, projective, unirational
varieties.
