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 $X/\mathcal{F}$ 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 $5$.