Seshadri constants on some Quot schemes

TL;DR

This paper computes Seshadri constants for ample line bundles on Quot schemes and Grassmann bundles associated with a vector bundle over projective space, providing insights into their geometric properties.

Contribution

It explicitly calculates Seshadri constants on Quot schemes and Grassmann bundles related to vector bundles over projective space, a novel contribution in this area.

Findings

Explicit formulas for Seshadri constants on Quot schemes.

Explicit formulas for Seshadri constants on Grassmann bundles.

Enhanced understanding of the positivity properties of line bundles on these schemes.

Abstract

Let $E$ be a vector bundle of rank $n$ on $\mathbb{P}^1$. Fix a positive integer $d$. Let $\mathcal{Q}(E,d)$ denote the Quot scheme of torsion quotients of $E$ of degree $d$ and let $Gr(E,d)$ denote the Grassmann bundle that parametrizes the $d$-dimensional quotients of the fibers of $E$. We compute Seshadri constants of ample line bundles on $\mathcal{Q}(E,d)$ and $Gr(E,d)$.