On the dual of the mobile cone

TL;DR

This paper establishes a duality between the cone of mobile divisors and a certain cone of curves in the context of klt pairs, providing structural and contraction theorems that advance the understanding of these geometric objects.

Contribution

It proves the duality of the mobile cone and the cone of birationally movable curves in codimension 1, along with structure and contraction theorems for these cones.

Findings

Duality of the cones in the $(K+B)$-negative part.

Structure theorem for the expanded cone of curves.

Contraction theorem for the cone of birationally movable curves.

Abstract

We prove that the cone of mobile divisors and the cone of curves birationally movable in codimension 1 are dual in the $(K+B)$-negative part for a klt pair $(X/Z,B)$. We also prove the structure theorem and the contraction theorem for the expanded cone of curves birationally movable in codimension 1. The duality of the cones gives a partial answer to the problem posed by Sam Payne.