On K3 surface quotients of K3 or Abelian surfaces

TL;DR

This paper characterizes when a K3 surface can be realized as a minimal model of quotients of Abelian or K3 surfaces by groups, using lattice embeddings and rational curve configurations, extending known results to new group cases.

Contribution

It extends the classification of K3 surfaces as quotients of Abelian or K3 surfaces by groups, beyond previously known cases, via lattice and rational curve criteria.

Findings

Characterizes K3 surfaces as quotients of Abelian or K3 surfaces using lattice embeddings.

Provides conditions involving rational curves for such quotient structures.

Extends known results to new group cases beyond order 2 or 3.

Abstract

The aim of this paper is to prove that a K3 surface is the minimal model of the quotient of an Abelian surface by a group $G$ (respectively of a K3 surface by an Abelian group $G$) if and only if a certain lattice is primitively embedded in its N\'eron--Severi group. This allows one to describe the coarse moduli space of the K3 surfaces which are (rationally) $G$-covered by Abelian or K3 surfaces (in the latter case $G$ is an Abelian group). If either $G$ has order 2 or $G$ is cyclic and acts on an Abelian surface, this result was already known, so we extend it to the other cases. Moreover, we prove that a K3 surface $X_G$ is the minimal model of the quotient of an Abelian surface by a group $G$ if and only if a certain configuration of rational curves is present on $X_G$. Again this result was known only in some special cases, in particular if $G$ has order 2 or 3.