  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 $U'$.
