On the threefold minimal model program in positive and mixed characteristic

This dissertation explores the Minimal Model Program (MMP) in positive and mixed characteristic in dimension three with a particular focus on outputs of the program. In purely positive characteristic we combine the program with a detailed study of conic bundles to prove a birational boundedness result. We show that given a suitable set of log Calabi-Yau varieties, we can construct a bounded family containing bres birational to any member of the chosen set. For threefolds over a base of dimension at least one, we resolve the Abundance Conjecture for klt pairs in joint work with F. Bernasconi and I. Brivio. Showing in particular that every klt minimal model in mixed characteristic admits an Iitaka Fibration. This is then applied to prove an Invariance of Plurigenera result for suitable families of surfaces. Finally we consider outstanding questions around Mori brations in mixed characteristic. We show that every klt threefold MMP terminates and that any two Mori bre space outputs of an MMP from the same starting pair are connected by a series of Sarkisov links. As part of this we prove a mixed characteristic Finiteness of Minimal Models result. While the proof is focused in dimension three, the arguments work in any generality given that the requisite MMP results are known.Open Acces