Pseudo-effective cone of Grassmann bundles over a curve

Abstract

Let $E$ be a vector bundle over a smooth projective curve $X$ defined over an algebraically closed field $k$. For any integer $1\,\le \, r\, <\, \mathrm{rank}(E)$, let $\mathrm{Gr}_r(E)\,\longrightarrow \, X$ be a Grassmann bundle parametrizing all $r$ dimensional quotients of the fibers of $E$. We compute the pseudo-effective cone in the real Néron–Severi group $\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }$. We prove that this cone coincides with the nef cone in $\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }$ if and only if the vector bundle $E$ is semistable (respectively, strongly semistable) when the characteristic of $k$ is zero (respectively, positive). Examples are given to show that this characterization of (strong) semistability is not true for vector bundles on higher dimensional projective varieties.