MMP for generalized foliated threefolds of rank one

TL;DR

This paper extends the minimal model program to generalized foliated threefolds of rank one, providing new results on base-point-freeness and the ACC for log canonical thresholds, advancing the understanding of foliated algebraic geometry.

Contribution

It establishes the MMP for generalized foliated threefolds of rank one, extending prior work and proving key theorems like base-point-freeness and ACC in this context.

Findings

Proved the MMP for generalized foliated threefolds of rank 1.

Established a base-point-free theorem for foliated triples.

Proved the ACC for log canonical thresholds in this setting.

Abstract

We establish the minimal model program (MMP) for generalized foliated threefolds $(X, \mathcal{F}, B, \mathbf{M})$ of rank 1, extending the result of Cascini and Spicer in [CS25d]. As an application of the generalized foliated MMP, we prove a base-point-free theorem for foliated triples on threefolds. We also prove the ACC for log canonical thresholds for generalized foliated threefolds of rank 1.