The cones of effective cycles on projective bundles over curves

TL;DR

This paper computes the cones of effective cycles of any dimension on projective bundles over curves using Harder-Narasimhan filtrations, extending previous work on divisors and curves to higher-dimensional cycles.

Contribution

It generalizes the description of effective cones to all cycle dimensions on projective bundles over curves, linking them to Harder-Narasimhan filtrations.

Findings

Effective cones are explicitly described in terms of numerical data.

The results extend to projective bundles over higher-dimensional bases.

Applications to cycles on bundles over arbitrary smooth projective bases.

Abstract

Generalizing work done by Miyaoka and others in the case of divisors and of curves, we compute the cones of effective cycles of arbitrary dimension on a projective bundle over a complex projective curve in terms of the numerical data in an associated Harder-Narasimhan filtration. An application to cycles on projective bundles over a smooth complex projective base of arbitrary dimension is also given.