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On The Log Sarkisov Program For Foliations On Projective 3-Folds

Roktim Mascharak

TL;DR

The work extends the Sarkisov program to foliations on normal projective 3-folds with F-dlt singularities by developing a foliated MMP framework and log geography, proving that maps between foliated Mori fiber spaces decompose into foliated Sarkisov links of types 1-4. It constructs polyhedral structures in the divisor space (BCHM-style) and uses them to control MMP steps and conclude finiteness and factorization results. A parallel theory for rank one foliations is developed with a distinct method to obtain common open foliated Mori fiber spaces, while highlighting limitations when singularities are canonical. Together, these results provide a foliated birational toolkit for analyzing birational maps and automorphisms of foliated 3-folds, bridging foliations and classical birational geometry.

Abstract

In this article we prove the Sarkisov Program for co-rank one foliation with suitable singularities on normal projective threefolds. We also relate two foliated Mori fiber spaces with rank one foliations on normal projective threefolds in the spirit of the Sarkisov Program.

On The Log Sarkisov Program For Foliations On Projective 3-Folds

TL;DR

The work extends the Sarkisov program to foliations on normal projective 3-folds with F-dlt singularities by developing a foliated MMP framework and log geography, proving that maps between foliated Mori fiber spaces decompose into foliated Sarkisov links of types 1-4. It constructs polyhedral structures in the divisor space (BCHM-style) and uses them to control MMP steps and conclude finiteness and factorization results. A parallel theory for rank one foliations is developed with a distinct method to obtain common open foliated Mori fiber spaces, while highlighting limitations when singularities are canonical. Together, these results provide a foliated birational toolkit for analyzing birational maps and automorphisms of foliated 3-folds, bridging foliations and classical birational geometry.

Abstract

In this article we prove the Sarkisov Program for co-rank one foliation with suitable singularities on normal projective threefolds. We also relate two foliated Mori fiber spaces with rank one foliations on normal projective threefolds in the spirit of the Sarkisov Program.
Paper Structure (7 sections, 13 theorems, 8 equations)

This paper contains 7 sections, 13 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Log Geography of log terminal models
  4. Sarkisov Program
  5. Proof of Main Theorem
  6. Examples
  7. Sarkisov for rank one Foliation

Key Result

Theorem 1.1

Let $\phi:X\rightarrow S$ and $\psi: Y\rightarrow T$ be two foliated Mori fiber spaces which are end products of two MMPs starting from $(Z,\mathcal{F},\Phi)$, where $Z$ is a normal projective $\mathbb{Q}$-factorial threefold and $(\mathcal{F},\Phi)$ is a co-rank one foliated F-dlt pair on $Z$ with

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2: Singularities of the foliated pairs
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 31 more