Minimal model program for semi-stable threefolds in mixed characteristic

Abstract

In this paper, we study the minimal model theory for threefolds in mixed characteristic. As a generalization of a result of Kawamata, we show that the MMP holds for strictly semi-stable schemes over an excellent Dedekind scheme $V$ of relative dimension two without any assumption on the residue characteristics of $V$. We also prove that we can run a $(K_{X/V}+\Delta )$-MMP over $Z$, where $\pi :X\to Z$ is a projective birational morphism of $\mathbb{Q}$-factorial quasi-projective $V$-schemes and $(X,\Delta )$ is a three-dimensional dlt pair with $\mathrm{Exc}(\pi )\subset \lfloor \Delta \rfloor$.

42 pages. We revised the assumption of Theorem 6.3 and the statement of Theorem 5.12. We also revised Definition 2.15 and the proofs of Proposition 2.16 and Proposition 2.17. To appear Journal of Algebraic Geometry