Slope semistability and positive cones of Grassmann bundles

Abstract

Let [Formula: see text] be a vector bundle of rank [Formula: see text] on a smooth complex projective variety [Formula: see text]. In this paper, we compute the nef and pseudoeffective cones of divisors on the Grassmann bundle [Formula: see text] parametrizing [Formula: see text]-dimensional subspaces of the fibers of [Formula: see text], where [Formula: see text], under assumptions on [Formula: see text] as well as on the vector bundle [Formula: see text]. In particular, we show that nef cone and the pseudoeffective cone of [Formula: see text] coincide if and only if nef cone and pseudoeffective cone of [Formula: see text] coincide under the assumption that [Formula: see text] is a slope semistable bundle on [Formula: see text] with [Formula: see text]. We also discuss about the nefness and ampleness of the universal quotient bundle [Formula: see text] on [Formula: see text].