On the dual of the mobile cone

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.