The cones of effective cycles on projective bundles over curves

Abstract

The study of the cones of curves or divisors on complete varieties is a classical subject in Algebraic Geometry (cf. [4,9,10]) and it still is an active research topic (cf. [1,11] or [2]). However, little is known if we pass to higher (co)dimension. In this paper we study this problem in the case of projective bundles over curves and describe the cones of effective cycles in terms of the numerical data appearing in a Harder–Narasimhan filtration. This generalizes to higher codimension results of Miyaoka and others ([3,15]) for the case of divisors. An application to projective bundles over a smooth base of arbitrary dimension is also given. Given a smooth complex projective variety X of dimension n, consider the vector spaces