Seshadri constants and Grassmann bundles over curves

TL;DR

This paper investigates Seshadri constants on Grassmann bundles over curves, providing explicit values under certain conditions, thus extending known results from rank 2 vector bundles to higher ranks.

Contribution

It offers new explicit calculations of Seshadri constants for Grassmann bundles over curves with non-semistable vector bundles, generalizing previous rank 2 results.

Findings

Explicit Seshadri constants for many cases

Generalization from rank 2 to higher ranks

Conditions based on Harder-Narasimhan filtration

Abstract

Let $X$ be a smooth complex projective curve, and let $E$ be a vector bundle on $X$ which is not semistable. For a suitably chosen integer $r$, let $\text{Gr}(E)$ be the Grassmann bundle over $X$ that parametrizes the quotients of the fibers of $E$ of dimension $r$. Assuming some numerical conditions on the Harder-Narasimhan filtration of $E$, we study Seshadri constants of ample line bundles on $\text{Gr}(E)$. In many cases, we give the precise value of Seshadri constant. Our results generalize various known results for ${\rm rank}(E)=2$.