Brauer groups of resolved quiver moduli via gerbes

Abstract

We show that the Brauer group of any resolution of singularities of the moduli space of semistable quiver representations is trivial. We do this by extending the quiver-curve dictionary, translating a proof of the analogous result by Biswas-Hogadi-Holla for moduli of vector bundles on a curve to the setting of moduli of quiver representations, giving an algebro-geometric proof. This gives a new proof of this triviality, first proved by Le Bruyn-Schofield, building on algebraic (resp. cohomological) vanishing results due to Saltman (resp. Colliot-Thélène-Sansuc). Reversing the logic, our approach gives a new algebro-geometric proof of these vanishing results.