On the termination of the MMP for semi-stable fourfolds in mixed characteristic

Abstract

We improve on the result of Hacon and Witaszek by showing that the MMP for semi-stable fourfolds in mixed characteristic terminates in several new situations. In particular, we show the validity of the MMP for strictly semi-stable fourfolds over excellent Dedekind schemes globally when the residue fields are perfect and have characteristics > 5.

On the relative minimal model program for fourfolds in positive and mixed characteristic

“…For the sake of readability and in order to avoid dealing with unnecessary technicalities, we do not strive to provide the most general version. Note, however, that there is a fundamental obstacle to generalising Theorem 5.1 to the case of Dedekind domains such as Spec Z (or Spec 𝑘 [𝑡])) due to the issues with termination of flips; indeed, over such a base it could a priori happen that there is an infinite sequence of flipping curves contained in fibres over different prime numbers (fortunately, this has been now resolved in [XX21a]). Furthermore, Theorem 5.1 needs the residue field to be perfect so that we can invoke the three-dimensional base point free theorem as well as [NT20], and the positive characteristic case thereof requires essentially that the base spreads out over an algebraically closed field so that we can apply positive characteristic Bertini theorems from [SZ13].…”

“…The assumption on the residue field being perfect is necessary as the three-dimensional base point free theorem is not known in full generality in a more general setting (for example, when the residue field is F-finite). Note that the assumptions on the Kodaira dimension and the base of the MMP have been weakened after our article was made available by improving on our termination results (see [XX21a]).…”

Algebraic geometry in mixed characteristic

“…Theorem 4.10 uses ideas from [46,178] and extends [220,142]. Global geometric applications of (the ideas going into) Theorem 4.10 can be found in [35,219,112,216,231].…”

Termination of threefold flips in mixed characteristic

“…Subsequent work in [12] establishes analogous results in the 𝑝 = 5 case. For the special case of semi-stable fibrations, [11] covers the case of semi-stable threefolds without constraint on the characteristic, while [4] and [13] show MMP's exist over Dedekind domains with perfect residue fields of characteristic 𝑝 > 5.…”