Toric Foliated Minimal Model Program

TL;DR

This paper develops a combinatorial framework for analyzing toric foliations and their singularities, extending existing results and establishing the existence of a minimal model program for these structures.

Contribution

It introduces a combinatorial description of toric foliations using Klyachko filtrations and proves the existence of a toric foliated Minimal Model Program for all ranks.

Findings

Extended Spicer's results to higher-rank toric foliations

Provided a combinatorial description of singularities

Proved the existence of the toric foliated MMP

Abstract

Using the theory of Klyachko filtrations for reflexive sheaves on toric varieties, we give a description of toric foliations and their singularities in terms of combinatorial data. We extend Spicer's results about co-rank one toric foliations to those of any ranks on $\mathbb{Q}$-factorial toric varieties and prove that the toric foliated Minimal Model Program exists.