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MMP for generalized foliated threefolds of rank one

Mengchu Li

TL;DR

This work extends the minimal model program to lc generalized foliated triples of rank one on normal projective threefolds, eliminating the need for algebraic integrability in key steps. It proves a cone theorem with length bound $2$ and an MMP that terminates in a minimal model or Mori fiber space, without assuming $K_ ext{F}+B+oldsymbol{M}_X$ is $oldsymbol{R}$-Cartier. The paper also establishes a base-point-free theorem for lc foliated triples in this generalized setting and proves ACC for log canonical thresholds, using a blend of foliated adjunction, surface techniques, and b-divisor formalism. The approach leverages a surface-focused reduction (Mumford intersection theory, surface num-gfq) and precise adjunction to control coefficients, enabling robust sector-by-sector analysis of extremal rays and lc-centers. Overall, the results deepen the birational geometry of foliations by extending generalized-pair techniques to rank-1 foliations on threefolds, with implications for canonical bundle computations and singularity theory in the foliated context.

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.

MMP for generalized foliated threefolds of rank one

TL;DR

This work extends the minimal model program to lc generalized foliated triples of rank one on normal projective threefolds, eliminating the need for algebraic integrability in key steps. It proves a cone theorem with length bound and an MMP that terminates in a minimal model or Mori fiber space, without assuming is -Cartier. The paper also establishes a base-point-free theorem for lc foliated triples in this generalized setting and proves ACC for log canonical thresholds, using a blend of foliated adjunction, surface techniques, and b-divisor formalism. The approach leverages a surface-focused reduction (Mumford intersection theory, surface num-gfq) and precise adjunction to control coefficients, enabling robust sector-by-sector analysis of extremal rays and lc-centers. Overall, the results deepen the birational geometry of foliations by extending generalized-pair techniques to rank-1 foliations on threefolds, with implications for canonical bundle computations and singularity theory in the foliated context.

Abstract

We establish the minimal model program (MMP) for generalized foliated threefolds 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.

Paper Structure

This paper contains 22 sections, 32 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Idea of the proof
  4. Preliminaries
  5. Basic definitions
  6. Finite morphism
  7. Divisorial adjunction
  8. Sets
  9. Convex geometry
  10. Bend and break
  11. Singularities in the sense of McQuillan
  12. Simple singularities
  13. Normal surfaces
  14. Cone theorem
  15. Invariant curves
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $X$ be a normal projective threefold, and let $(X,\mathcal{F},B,\mathbf{M})$ be an lc generalized foliated quadruple of rank $1$ on $X$. Assume that $X$ is $\mathbb{Q}$-factorial klt. Then

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Foliations
  • Definition 2.2: Pullbacks
  • Definition 2.3: Invariance
  • Definition 2.4: b-divisors
  • Definition 2.5: Restricted b-divisors
  • Definition 2.6
  • ...and 75 more