MMP for co-rank one foliations on threefolds

TL;DR

This paper establishes fundamental results such as the existence of flips, minimal models, and abundance for co-rank one foliations on threefolds, advancing the minimal model program in foliated geometry.

Contribution

It proves key theorems like flips and minimal models for F-dlt foliated pairs on threefolds, extending the minimal model program to foliations.

Findings

Existence of flips and minimal models for F-dlt foliated pairs.

Foliations with canonical or F-dlt singularities have non-dicritical singularities.

Abundance holds for numerically trivial foliated pairs.

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 pairs of co-rank one on a $\mathbb Q$-factorial projective threefold. As applications, we show the existence of F-dlt modifications and F-terminalisations for foliated 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 pairs.