Brauer Groups of Quot Schemes

Abstract

Let $X$ be an irreducible smooth complex projective curve. Let $\mathcal{Q}(r,d)$ be the Quot scheme parametrizing all coherent subsheaves of ${\mathcal{O}}_X^{\oplus r}$ of rank $r$ and degree $-d$. There are natural morphisms $\mathcal{Q}(r,d)⟶{\text{Sym}}^d(X)$ and ${\text{Sym}}^d(X)⟶{\text{Pic}}^d(X)$. We prove that both these morphisms induce isomorphism of Brauer groups if $d\ge 2$. Consequently, the Brauer group of $\mathcal{Q}(r,d)$ is identified with the Brauer group of ${\text{Pic}}^d(X)$ if $d\ge 2$.