Brauer Groups of Quot Schemes

TL;DR

This paper investigates the Brauer groups of Quot schemes over complex curves, establishing isomorphisms with Picard groups for degrees greater than or equal to 2, thereby linking their algebraic structures.

Contribution

It proves that for degrees d ≥ 2, the Brauer group of the Quot scheme is isomorphic to that of the Picard variety, revealing a new connection between these geometric objects.

Findings

Brauer group of Quot scheme is isomorphic to that of Picard variety for d ≥ 2.

Natural morphisms induce isomorphisms of Brauer groups.

Results connect algebraic properties of Quot schemes and Picard varieties.

Abstract

Let $X$ be an irreducible smooth complex projective curve. Let ${\mathcal Q}(r,d)$ be the Quot scheme parametrizing all coherent subsheaves of ${\mathcal O}^{\oplus r}_X$ 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 $d \geq 2$.