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

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