Brauer group of punctual Quot scheme of points on a smooth projective surface

TL;DR

This paper investigates the Brauer group of punctual Quot schemes and related moduli spaces on smooth projective surfaces, providing explicit formulas and isomorphisms for their cohomological Brauer groups.

Contribution

It offers new formulas for the Brauer groups of punctual Quot schemes and establishes isomorphisms with Hilbert schemes and symmetric products for various degrees.

Findings

Explicit formula for the ll-primary part of the Brauer group of X^{[2]}

Isomorphism of cohomological Brauer groups induced by Hilbert to Chow morphism for d=2 and d  3

Natural morphism from Quot scheme to Hilbert scheme induces an isomorphism on Brauer groups

Abstract

Let $X$ be a smooth projective surface over an algebraically closed field $k$ such that $char(k) \neq 2$. Let $X^{[d]}$ denote the punctual Hilbert scheme of zero dimensional quotients of degree $d$ and $X^{(d)}$ denote the symmetric product of $X$. For $\ell \neq 2$, we give a formula for the $\ell$-primary part of the Brauer group of $X^{[2]}$. We show that the Hilbert to Chow morphism induces an isomorphism of cohomological Brauer groups for $d=2$ and a similar result for $d \geq 3$. Let $Q(r,d)$ denote the punctual Quot-scheme parametrising zero dimensional quotients of $\mathcal{O}_X^{ \oplus r}$ of degree $d$. We show that the natural morphism from $Q(r,d) \rightarrow X^{[d]}$ induces an isomorphism on cohomological Brauer groups.