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

TL;DR

This paper explores the relationship between singularities of 1-foliations on surfaces and their quotients, establishing conditions for log canonical and klt singularities, and classifying certain klt quotients for specific primes.

Contribution

It introduces a precise correspondence between foliation singularities and quotient singularities, and classifies klt quotients on surfaces for p=2,3,5.

Findings

Quotient X/𝓕 is log canonical if and only if 𝓕 is (p-1)/p-log canonical.

Quotient X/𝓕 is klt if and only if 𝓕 is (p-1)/p-klt.

Classification of klt quotients on regular surfaces for p=2,3,5.

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 $5$.