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Quotients by $(p-1)/p$-klt Foliations on Surfaces

Yutaro Hiroi

Abstract

We study the relation between birational singularities of 1-foliations and those of their quotients. We prove that the quotient $X/\mathcal{F}$ is log canonical (resp. klt) if and only if $\mathcal{F}$ is $\frac{p-1}{p}$-log canonical (resp. $\frac{p-1}{p}$-klt). Moreover, we obtain the classification of klt quotients by $1$-foliations on regular surfaces in the cases $p=2,3$ and $5$.

Quotients by $(p-1)/p$-klt Foliations on Surfaces

Abstract

We study the relation between birational singularities of 1-foliations and those of their quotients. We prove that the quotient is log canonical (resp. klt) if and only if is -log canonical (resp. -klt). Moreover, we obtain the classification of klt quotients by -foliations on regular surfaces in the cases and .
Paper Structure (13 sections, 24 theorems, 112 equations)

This paper contains 13 sections, 24 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Derivations
  4. Foliations
  5. Quotients by 1-foliations
  6. Birational singularities of 1-foliations
  7. Adjoint foliated structures and their quotients
  8. Adjoint foliated structures
  9. Birational singularities of quotients
  10. Classification of klt surface quotients
  11. Some auxiliary results
  12. Case $p=2,3$
  13. Case $p=5$

Key Result

Theorem 1.1.1

Let $S$ be a regular surface over $k$, and $\mathcal{F}$ a 1-foliation of rank 1 on $S$. Then $S/\mathcal{F}$ is F-regular if and only if $\mathcal{F}$ is log canonical.

Theorems & Definitions (50)

  • Theorem 1.1.1: Theorem \ref{['Posva']}
  • Definition 1.1.2: Definition \ref{['adjoint']}
  • Theorem 1.1.3: Theorem \ref{['(p-1)/p']}
  • Theorem 1.1.4: Theorem \ref{['p=2,3']} and Corollary \ref{['p=5']}
  • Definition 2.1.1
  • Lemma 2.1.2
  • proof
  • Definition 2.2.1
  • Definition 2.2.2
  • Example 2.2.3
  • ...and 40 more