Slope Semistability and Positive cones of Grassmann bundles

Abstract: Let $E$ be a vector bundle of rank $r$ on a smooth complex projective variety $X$. In this article, we compute the nef and pseudoeffective cones of divisors in the Grassmann bundle $Gr_X(k,E)$ parametrizing $k$-dimensional subspaces of the fibers of $E$, where $1\leq k \leq rank(E)$, under assumptions on $X$ as well as on the vector bundle $E$. In particular, we show that nef cone and the pseudoeffective cone of $Gr_X(k,E)$ coincide if and only if $E$ is a slope semistable bundle on $X$ with $c_2(End(E))=0$. We also discuss about the nefness and ampleness of the universal quotient bundle $Q_k$ on $Gr_X(k,E)$.