Brauer group of the moduli spaces of stable vector bundles of fixed determinant over a smooth curve

TL;DR

This paper determines the Brauer group of moduli spaces of stable vector bundles with fixed determinant over a smooth curve, showing it is cyclic of a specific order and generated by a universal projective bundle class.

Contribution

It extends previous results by proving the Brauer group is cyclic of order gcd(r, degree(L)) and identifying its generator over an algebraically closed field of any characteristic.

Findings

Brauer group is cyclic of order gcd(r, degree(L))

Generated by the class of a universal projective bundle

Results hold over any algebraically closed field

Abstract

Let $X$ be an irreducible smooth projective curve, defined over an algebraically closed field $k$, of genus at least three and $L$ a line bundle on $X$. Let ${\mathcal M}_X(r,L)$ be the moduli space of stable vector bundles on $X$ of rank $r$ and determinant $L$ with $r\geq 2$. We prove that the Brauer group ${\rm Br}(\mathcal{M}_X(r,L))$ is cyclic of order ${\rm g.c.d.}(r,{\rm degree}(L))$. We also prove that ${\rm Br}(\mathcal{M}_X(r,L))$ is generated by the class of the projective bundle obtained by restricting the universal projective bundle. These results were proved earlier in \cite{BBGN} under the assumption that $k=\mathbb C$.