The geometric cone conjecture in relative dimension two

TL;DR

This paper proves finiteness of Mori chambers and faces in the movable cone for certain two-dimensional fibrations with klt pairs, advancing the understanding of the geometric structure in relative dimension two.

Contribution

It establishes the finiteness of Mori chambers and faces in the movable cone for relative dimension two fibrations with klt pairs, confirming a geometric conjecture in this setting.

Findings

Finiteness of Mori chambers in the movable cone

Finiteness of Mori faces in the movable cone

Results hold up to the action of relative pseudo-automorphisms

Abstract

Let $X\rightarrow S$ be a fibration of relative dimension at most two and let $(X,\Delta)$ be a klt pair for which $K_X+\Delta \equiv_S 0$. 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 $\Delta$.