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Boundedness of toric foliations

Chih-Wei Chang, Yen-An Chen

TL;DR

The paper develops a toric approach to the boundedness and singularity theory of foliations in the adjoint setting. It proves a boundedness result for toric Fano adjoint foliated structures under $\delta$-lc assumptions and ampleness, with finiteness of isomorphism classes and potential klt-ness of the ambient variety. It also establishes structural results on the dicritical and singular loci for Fano toric foliations, including connectedness, and analyzes interpolated $\delta$-log canonical thresholds, showing density phenomena and DCC/ACC behavior in the toric context. These results advance the minimal model program for foliations in the toric category and provide finite-type classification tools for adjoint foliated structures.

Abstract

We discuss boundedness of toric Fano foliations and connectedness of its dicritical and singular loci. Moreover, we show the set of interpolated $δ$-lcts for the toric foliations satisfies the descending chain condition.

Boundedness of toric foliations

TL;DR

The paper develops a toric approach to the boundedness and singularity theory of foliations in the adjoint setting. It proves a boundedness result for toric Fano adjoint foliated structures under -lc assumptions and ampleness, with finiteness of isomorphism classes and potential klt-ness of the ambient variety. It also establishes structural results on the dicritical and singular loci for Fano toric foliations, including connectedness, and analyzes interpolated -log canonical thresholds, showing density phenomena and DCC/ACC behavior in the toric context. These results advance the minimal model program for foliations in the toric category and provide finite-type classification tools for adjoint foliated structures.

Abstract

We discuss boundedness of toric Fano foliations and connectedness of its dicritical and singular loci. Moreover, we show the set of interpolated -lcts for the toric foliations satisfies the descending chain condition.

Paper Structure

This paper contains 13 sections, 24 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Foliations
  4. Toric varieties
  5. Toric foliations
  6. Adjoint foliated structure
  7. Boundedness of Fano ajoint foliated structures
  8. Dicritical locus
  9. Interpolated delta-lct
  10. Toroidal foliations
  11. delta-log canonical
  12. Dicritical locus
  13. Dicritical locus of Fano foliation of rank one

Key Result

Theorem 1.3

Fix a positive integer $n$, a non-negative integer $r\leq n$, a positive real number $\delta$, and two real numbers $t_i\in [0,1)$ for $i=1$, $2$. Let ${\mathcal{F}}={\mathcal{F}}_W$ be a toric foliation of rank $r$ on a complete toric variety $X=X_\Sigma$ of dimension $n$, $\Delta$ be an effective

Theorems & Definitions (63)

  • Theorem 1.3: Theorem \ref{['thm:bounded_adjoint_Fano']}
  • Proposition 1.4: Proposition \ref{['prop:loci_connected']}
  • Theorem 1.5: Theorem \ref{['thm:upper_lct_dense']}
  • Theorem 1.6: Theorem \ref{['thm:lower_delta_lct_dcc']}
  • Definition 2.1: Singular locus
  • Definition 2.2: Invariance
  • Definition 2.3: Dicriticality, CC
  • proof
  • Theorem 2.5
  • proof
  • ...and 53 more