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 Neron–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.