Unramified Brauer group of the moduli spaces of $PGL_r({\mathbb C})$-bundles over curves

TL;DR

This paper proves that the Brauer group of a desingularization of the moduli space of semistable PGL_r(C)-bundles over a high-genus curve is trivial, advancing understanding of the space's geometric properties.

Contribution

It establishes the triviality of the Brauer group for desingularizations of these moduli spaces, a new result in the study of their geometric and arithmetic structure.

Findings

Brauer group of desingularized moduli space is trivial

Moduli space is unirational and normal

Results apply to curves of genus at least two

Abstract

Let $X$ be an irreducible smooth complex projective curve of genus at least two. Let $N$ be a connected component of the moduli space of semistable principal ${\rm PGL}_r({\mathbb C})$- bundles over $X$; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of $N$ is trivial.