Nef cone of flag bundles over a curve

Abstract

Let $X$ be a smooth projective curve defined over an algebraically closed field $k$, and let $E$ be a vector bundle on $X$. Let ${\mathcal{O}}_{{Gr}_r\left(E\right)}\left(1\right)$ be the tautological line bundle over the Grassmann bundle ${Gr}_r\left(E\right)$ parameterizing all the $r$-dimensional quotients of the fibers of $E$. We give necessary and sufficient conditions for ${\mathcal{O}}_{{Gr}_r\left(E\right)}\left(1\right)$ to be ample and nef, respectively. As an application, we compute the nef cone of ${Gr}_r\left(E\right)$. This yields a description of the nef cone of any flag bundle over $X$ associated to $E$.