Algebraic surfaces with quotient singularities - including some discussion on automorphisms and fundamental groups

TL;DR

This paper surveys recent advances in the study of algebraic surfaces with quotient singularities, focusing on their automorphisms, fundamental groups, and related conjectures in the context of log terminal singularities.

Contribution

It compiles recent results on the structure of algebraic surfaces with quotient singularities, including automorphism groups, fundamental groups, and conjectures on their properties.

Findings

Automorphism groups of certain K3 surfaces are described.

Conjecture: Q-Fano surfaces have finite fundamental groups of their smooth locus.

Conjecture: log Enriques surfaces have either finite fundamental group or an abelian surface cover.

Abstract

We survey some recent progress in the study of algebraic varieties X with log terminal singularities, especially, the uni-ruledness of the smooth locus X^0 of X, the fundamental group of X^0 and the automorphisms group on (smooth or singular) X when dim X = 2. The full automorphism groups of a few interesting types of K3 surfaces are described, mainly by Keum-Kondo. We conjecture that when X is Q-Fano then X^0 has a finite fundamental group, which had been proved if either dim X < 3 or the Fano index is bigger than dim X - 2. We also conjecture that when X is a log Enriques (e.g. a normal K3 or a normal Enriques) surface then either pi_1(X^0) is finite or X has an abelian surface as its quasi-etale cover, which has been proved by Catanese-Keum-Oguiso under some extra conditions.