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Brauer Groups of Quot Schemes

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