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We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global F -regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita's conjecture to mixed characteristic.

We prove that many of the results of the LMMP hold for 3-folds over fields of characteristic p > 5 which are not necessarily perfect. In particular, the existence of flips, the cone theorem, the contraction theorem for birational extremal rays, and the existence of log minimal models. As well as pertaining to the geometry of fibrations of relative dimension 3 over algebraically closed fields, they have applications to tight closure in dimension 4.

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We prove that the log canonical ring of a projective klt pair of dimension 3 with Q-boundary over an algebraically closed field of characteristic p > 5 is finitely generated. In the process we prove log abundance for such pairs in the case κ = 2.Proof. [15, A018] Lemma 2.2 (Zariski's Main Theorem I). Let f : X → Y be a quasi-finite and separated map of algebraic spaces over a scheme S, such that Y is quasicompact and quasi-separated. Then there exists a factorisation X → T → Y so that X → T is a quasi-compact open immersion and T → Y is finite.