Lefschetz-type theorems for the effective cone on Hyperk\"ahler varieties

TL;DR

This paper establishes Lefschetz-type theorems for the effective cone of Hyperk"ahler varieties, showing how divisors influence the effective cone structure and enabling explicit computations for certain bundles.

Contribution

It proves that smooth ample divisors induce isomorphisms of effective cones and extends similar results to some effective exceptional divisors, facilitating cone computations.

Findings

Inclusion of smooth ample divisors induces isomorphism of effective cones.

Effective exceptional divisors also induce similar isomorphisms.

Computed effective cones for projectivized cotangent bundles and Lazarsfeld--Mukai bundles.

Abstract

In this paper we show some Lefschetz-type theorems for the effective cone of Hyperk\"ahler varieties. In particular we are able to show that the inclusion of any smooth ample divisor induces an isomorphism of effective cones. Moreover we deduce a similar statement for some effective exceptional divisors, which yields the computation of the effective cone of e.g. projectivized cotangent bundles and some projectivized Lazarsfeld--Mukai bundles.