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 result Kawamata, show that program (MMP) holds strictly semi-stable schemes over an excellent Dedekind scheme V V relative dimension two without any assumption on residue characteristics . We also prove can run alttext="left-parenthesis upper K Subscript X slash V Baseline plus normal Delta right-parenthesis"> ( K X / + Δ stretchy="false">) encoding="application/x-tex">(K_{X/V}+\Delta ) -MMP Z"> Z encoding="application/x-tex">Z , where alttext="pi colon right-arrow π stretchy="false">→<!-- → encoding="application/x-tex">\pi \colon \to Z is projective birational morphism alttext="double-struck Q"> mathvariant="double-struck">Q encoding="application/x-tex">\mathbb {Q} -factorial quasi-projective -schemes and comma , encoding="application/x-tex">(X,\Delta three-dimensional dlt pair with E x c left-parenthesis pi right-parenthesis subset-of left floor right floor"> E x c ⊂<!-- ⊂ fence="false" stretchy="false">⌊<!-- ⌊ stretchy="false">⌋<!-- ⌋ encoding="application/x-tex">Exc(\pi ) \subset \lfloor \Delta \rfloor