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Adjoint log canonical foliated singularities on surfaces

Shi Xu

TL;DR

This work develops a birational classification of adjoint foliated singularities on complex surfaces through the lens of the adjoint divisor $K_{\mathcal F}+\epsilon K_X$. By analyzing the minimal resolution, negative-definite curve configurations, and special $K^{>0}$-chains, the authors obtain a finite, explicit list of possible exceptional divisors for $\epsilon$-adjoint lc singularities when $\epsilon\in(0,1/3)$, and show that for small $\epsilon$ (notably $\epsilon\in(0,1/5)$ or $\epsilon\in(0,1/4)$) the adjoint lc/canonical conditions imply corresponding lc/klt properties for the foliation and the underlying surface. They establish sharp bounds: for $\epsilon\in(0,1/5)$ every $\epsilon$-adjoint lc singularity is lc for the foliation and for $X$, and for $\epsilon\in(0,1/4)$ every $\epsilon$-adjoint canonical singularity is lc for the foliation and klt for the underlying surface. The paper also provides explicit examples showing the sharpness of these bounds and discusses the limitations by giving a counterexample demonstrating that adjoint canonical need not be canonical even for small $\epsilon$. Overall, the results significantly advance the understanding of foliated singularities via adjoint divisors, with direct implications for the birational geometry of foliations and the minimal model program in the foliated setting.

Abstract

Let $(X,\mathcal{F})$ be a foliated surface over $\mathbb{C}$. We study the singularities of the adjoint foliated divisor $K_{\mathcal{F}}+εK_X$. We provide a complete classification of $ε$-adjoint log canonical singularities of foliated surfaces for $ε\in(0,1/3)$. Moreover, we prove that for any $ε\in(0,1/5)$, every $ε$-adjoint log canonical singularity is log canonical for $\mathcal{F}$, and that for any $ε\in(0,1/4)$, every $ε$-adjoint canonical singularity is log canonical for $\mathcal{F}$. Finally, we present examples showing that both bounds are sharp.

Adjoint log canonical foliated singularities on surfaces

TL;DR

This work develops a birational classification of adjoint foliated singularities on complex surfaces through the lens of the adjoint divisor . By analyzing the minimal resolution, negative-definite curve configurations, and special -chains, the authors obtain a finite, explicit list of possible exceptional divisors for -adjoint lc singularities when , and show that for small (notably or ) the adjoint lc/canonical conditions imply corresponding lc/klt properties for the foliation and the underlying surface. They establish sharp bounds: for every -adjoint lc singularity is lc for the foliation and for , and for every -adjoint canonical singularity is lc for the foliation and klt for the underlying surface. The paper also provides explicit examples showing the sharpness of these bounds and discusses the limitations by giving a counterexample demonstrating that adjoint canonical need not be canonical even for small . Overall, the results significantly advance the understanding of foliated singularities via adjoint divisors, with direct implications for the birational geometry of foliations and the minimal model program in the foliated setting.

Abstract

Let be a foliated surface over . We study the singularities of the adjoint foliated divisor . We provide a complete classification of -adjoint log canonical singularities of foliated surfaces for . Moreover, we prove that for any , every -adjoint log canonical singularity is log canonical for , and that for any , every -adjoint canonical singularity is log canonical for . Finally, we present examples showing that both bounds are sharp.
Paper Structure (41 sections, 48 theorems, 124 equations, 8 figures)

This paper contains 41 sections, 48 theorems, 124 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Foliations and singularities
  4. Minimal resolution of foliated surfaces
  5. Index Theorems
  6. Non-invariant curves
  7. Invariant curves
  8. Pairs and Triples
  9. Log canonical foliated singularities
  10. $(\epsilon,\delta)$-adjoint log canonical foliated singularities
  11. Notation and basic configurations
  12. A negativity criterion on surfaces
  13. Negative definite configurations of curves
  14. Dual graphs and intersection equations
  15. $D^{>0}$-chains
  16. ...and 26 more sections

Key Result

Theorem 1.1

Fix $\epsilon \in (0,1/5)$. Let $(Y,\mathcal{G},p)$ be a germ of a foliated surface, and assume that $p$ is an $\epsilon$-adjoint log canonical singularity of $(Y,\mathcal{G})$. Let be the minimal resolution (cf. Definition def:min-resolution) of $(Y,\mathcal{G},p)$, with exceptional divisor $E=\bigcup_{i=1}^r E_i$. Then $E$ belongs to one of the following families: Moreover, configurations of t

Figures (8)

  • Figure 1:
  • Figure 2:
  • Figure 3: $\mathcal{F}$-chain with the first curve $\Gamma_1$
  • Figure 4: $C$ is a bad tail
  • Figure 5: $\mathcal{F}$-dihedral graph
  • ...and 3 more figures

Theorems & Definitions (120)

  • Theorem 1.1: cf. Theorem \ref{['thm:main-epsilon-ad-lc']}
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4
  • proof
  • Remark 1.5
  • Corollary 1.6
  • proof
  • Remark 1.7
  • Definition 2.1
  • ...and 110 more