Log Canonical Minimal Model Program for corank one foliations on Threefolds

TL;DR

This paper develops a minimal model program for rank two foliations on threefolds, proving termination and connecting different models via flops, extending classical results to a foliated setting.

Contribution

It introduces a foliated MMP for rank two triples, proves its termination, and establishes relations between minimal models using flops.

Findings

The MMP terminates with minimal models or Mori fiber spaces.

Any two minimal models can be connected by flops.

In boundary polarized cases, minimal models are finite and well-behaved.

Abstract

If $(X, \mcF, \D)$ is a projective rank two foliated log canonical triple such that $(X,B)$ is klt for some $0 \leq B \leq \D$, we show that we can run a $(K_\mcF +\Delta)$-MMP and any such MMP terminates with either a minimal model or Mori fiber space. Next, we establish a Bertini type lemma and adjunction for generalized foliated quadruples. Using these, we extend the full log canonical MMP to the setting of rank two NQC generalized foliated quadruples. Finally, we apply the generalized MMP to study the relation between different minimal models, namely, any two minimal models of a given foliated log canonical triple can be connected by a sequence of flops and in the boundary polarized case, the minimal models are good and only finitely many in number.