MMP for co-rank one foliations on threefolds

Abstract

We prove existence of flips, special termination, the base point free theorem and, in the case of log general type, the existence of minimal models for F-dlt foliated log pairs of co-rank one on a projective threefold.As applications, we show the existence of F-dlt modifications and F-terminalisations for foliated log pairs and we show that foliations with canonical or F-dlt singularities admit non-dicritical singularities. Finally, we show abundance in the case of numerically trivial foliated log pairs.

Log canonical models of foliated surfaces

“…Next, for a fixed function P : Z ≥0 → Z, we use the techniques from [Spi20] and [CS18], to establish the semi-stable (K F + ∆)-minimal model program for a special type of families of foliated surfaces. (See Section 4 and Theorem 4.25.)…”

On the MMP for rank one foliations on threefolds

“…One can ask more generally if there is a similar way to control the singularities of the underlying variety in higher dimensions and higher ranks, and if such a bound holds if F has only canonical singularities. For foliations of co-rank one on a normal threefold, some of these questions were addressed in [CS18]. We will approach some cases of this problem in the rank one case in dimension three (cf.…”

“…see [Bru00,McQ08,Men00]). For foliations of rank two on a threefold, the program was carried out in [Spi20,CS18,SS19].…”

Codimension one regular foliations on rationally connected threefolds

“…Conjecturally, the numerical properties of the canonical sheaf of a foliation F governs the geometry of F , just like the canonical sheaf of a projective variety X governs the geometry of X. In particular, there is a conjectural MMP for foliations (using K F in the place of K X ), currently established if dim(X) ≤ 3 ( [Bru99], [McQ08], [Spi20], [CS18], [SS19], [CS20]).…”