The minimal model program for threefolds in characteristic 5

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

“…Since is birational, is relatively globally F -regular ([DW19a, Theorem 5.1], cf. [HX15, Theorem 3.1], [HW22, Proposition 2.9]) or completely relatively globally -regular ([BMPSTWW20, Theorem 7.14]), respectively. Lemma 2.9 implies that is relatively purely F-regular (resp.…”

“…Since the log resolution was an isomorphism at the generic points of strata of , it is easy to see that this is the required flip (cf. [HW22, Section 7.1]).…”

“…As before, by [HW21, Lemma 3.1] and Theorem 4.3, both Y and are relatively F-regular over Z . In particular, and (see [SS10, Theorem 6.8] and the proof of [HW22, Proposition 3.4]). By [NT20, Lemma 2.17] and [GNH19, Lemma 2.19 and Lemma 2.21], and for .…”

“…In recent years, there has been much progress in understanding the birational geometry of threefolds over algebraically closed fields of characteristic by using methods inspired by the theory of F-singularities. One of the main achievements in this area is the proof of the minimal model program (MMP) in dimension three in wide generality (see, e.g., [HX15], [CTX15], [Bir16], [BW17], [HW21], [HW22]).…”

On the log minimal model program for threefolds over imperfect fields of characteristic p>5$p>5$

“…Recently, Hacon and Witaszek have extended parts of the LMMP for threefolds over perfect fields to low characteristics by giving new arguments for the existence of flips. It is therefore likely that all of our results can be extended to characteristic 5 using the construction of flips in [21], and partially to characteristic 2 and 3 using that in [22]. However, we have not attempted to verify this.…”