Effective cone of a Grassmann bundle over a curve defined over ${\bar{\pmb {\mathbb {F}}}}_{{\varvec{p}}}$

Abstract

Let X be an irreducible smooth projective curve defined over $\bar{{\mathbb {F}}}_p$ and E a vector bundle on X of rank at least two. For any $1 \le r < \textrm{rank}(E)$, let $\textrm{Gr}_r(E)$ be the Grassmann bundle over X parametrizing all the r dimensional quotients of the fibers of E. We prove that the effective cone in $\textrm{NS}(\textrm{Gr}_r(E))\otimes _{{\mathbb {Z}}} {{\mathbb {R}}}$ coincides with the pseudo-effective cone in $\textrm{NS}(\textrm{Gr}_r(E))\otimes _{{\mathbb {Z}}} {{\mathbb {R}}}$. When $r=1$ or $\textrm{rank}(E)-1$, this was proved in [4].