On the Picard group of moduli spaces

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.