Picard groups of the moduli spaces of semistable sheaves I

We compute the Picard group of the moduli space U' of semistable vector bundles of rank n and degree d on an irreducible nodal curve Y and show that U' is locally factorial. We determine the canonical line bundles of U' and U'L, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata in the compactification of U'