Minimal model program for semi-stable threefolds in mixed characteristic

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 minimal model program (MMP) holds for strictly semi-stable schemes over an excellent Dedekind scheme V V of relative dimension two without any assumption on the residue characteristics of V V . We also prove that we can run a ( K X / V + Δ ) (K_{X/V}+\Delta ) -MMP over Z Z , where π : X → Z \pi \colon X \to Z is a projective birational morphism of Q \mathbb {Q} -factorial quasi-projective V V -schemes and ( X , Δ ) (X,\Delta ) is a three-dimensional dlt pair with E x c ( π ) ⊂ ⌊ Δ ⌋ Exc(\pi ) \subset \lfloor \Delta \rfloor .