Brauer group of a moduli space of parabolic vector bundles over a curve

Abstract

Let $\mathcal{P}{\mathcal{M}}_s^{\alpha }$ be a moduli space of stable parabolic vector bundles of rank $n\ge 2$ and fixed determinant of degree $d$ over a compact connected Riemann surface $X$ of genus $g(X)\ge 2$. If $g(X)=2$, then we assume that $n>2$. Let $m$ denote the greatest common divisor of $d$, $n$ and the dimensions of all the successive quotients of the quasi-parabolic filtrations. We prove that the cohomological Brauer group $\mathrm{Br}(\mathcal{P}{\mathcal{M}}_s^{\alpha })$ is isomorphic to the cyclic group $\mathbb{Z}/m\mathbb{Z}$. We also show that $\mathrm{Br}(\mathcal{P}{\mathcal{M}}_s^{\alpha })$ is generated by the Brauer class of the projective bundle over $\mathcal{P}{\mathcal{M}}_s^{\alpha }$ obtained by restricting the universal projective bundle over $X\times \mathcal{P}{\mathcal{M}}_s^{\alpha }$. We also prove that there is a universal vector bundle over $X\times \mathcal{P}{\mathcal{M}}_s^{\alpha }$ if and only if $m=1$.

arXiv commentJournal of K-Theory (to appear)