Brauer groups and Picard groups of the moduli of parabolic vector bundles on a nodal curve

Abstract

We determine the Brauer groups and Picard groups of the moduli space $U^{' s}_{L,par}$ of stable parabolic vector bundles of rank r with determinant L on a complex nodal curve Y of arithmetic genus $g \ge 2$. We also compute the Picard group of the moduli stack for parabolic SL(r)-bundles on Y and use it to give another description of the Picard group of $U^{' s}_{L,par}$. For $g \ge 2$, we determine the Brauer group of the moduli space $U^{' s}_L$ of stable vector bundles on Y of rank r with determinant L, deduce that $U^{' s}_L$ is simply connected and show the non-existence of the universal bundle on $U^{' s}_L \times Y$ in the non-coprime case.