Effective cone of a Grassmann bundle over a curve defined over ${\overline{\mathbb{F}}}_p$

Abstract

Let $X$ be an irreducible smooth projective curve defined over ${\overline{\mathbb{F}}}_p$ and $E$ a vector bundle on $X$ of rank at least two. For any $1\le r<\mathrm{rank}(E)$, let ${\mathrm{Gr}}_r(E)$ be the Grassmann bundle over $X$ parametrizing all the $r$ dimensional quotients of the fibers of $E$. We prove that the effective cone in $\mathrm{NS}({\mathrm{Gr}}_r(E)){\otimes }_{\mathbb{Z}}\mathbb{R}$ coincides with the pseudo-effective cone in $\mathrm{NS}({\mathrm{Gr}}_r(E)){\otimes }_{\mathbb{Z}}\mathbb{R}$. When $r=1$ or $\mathrm{rank}(E)-1$, this was proved by A. Moriwaki.

arXiv commentFinal version; 5 pages; to appear in Proc. Indian Acad. Sci. Math. Sci