The geometric cone conjecture in relative dimension two

Abstract

Let $X\to S$ be a fibration of relative dimension at most two and let $(X,\hat{I}”)$ be a klt pair for which $K_X+\hat{I}”{\equiv }_{S}0$. We show that there are only finitely many Mori chambers and Mori faces in the movable effective cone ${\mathcal{M}}^e(X/S)$ up to the action of relative pseudo-automorphisms of $X/S$ preserving $\hat{I}”$.

35 pages