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

Abstract

Let ${\mathcal P}{\mathcal M}^\alpha_s$ be a moduli space of stable parabolic vector bundles of rank $n \geq 2$ and fixed determinant of degree $d$ over a compact connected Riemann surface $X$ of genus $g(X) \geq 2$. 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 ${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)$ is isomorphic to the cyclic group ${\mathbb Z}/ m{\mathbb Z}$. We also show that ${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)$ is generated by the Brauer class of the projective bundle over ${\mathcal P}{\mathcal M}^\alpha_s$ obtained by restricting the universal projective bundle over $X\times {\mathcal P}{\mathcal M}^\alpha_s$. We also prove that there is a universal vector bundle over $X\times {\mathcal P}{\mathcal M}^\alpha_s$ if and only if $m=1$.