Picard Groups of Some Quot Schemes

TL;DR

This paper computes the Picard group of certain Quot schemes over complex curves, providing explicit results for cases where the quotient rank is positive and the degree is sufficiently large.

Contribution

It offers the first explicit computation of the Picard group for Quot schemes of locally free sheaves on complex curves under specified conditions.

Findings

Picard group computed for Quot schemes with positive quotient rank.

Results applicable when the degree is sufficiently large.

Provides new insights into the line bundle structure of Quot schemes.

Abstract

Let $C$ be a smooth projective curve over the field of complex numbers $\mathbb{C}$ of genus $g(C)>0$. Let $E$ be a locally free sheaf on $C$ of rank $r$ and degree $e$. Let $\mathcal{Q}:={\rm Quot}_{C/\mathbb{C}}(E,k,d)$ denote the Quot scheme of quotients of $E$ of rank $k$ and degree $d$. For $k>0$ and $d\gg 0$ we compute the Picard group of $\mathcal{Q}$.