On K3 Surface Quotients of K3 or Abelian Surfaces
Printed: Apr 2017
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 Néron-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
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.
K3 surfaces, Kummer surfaces, Kummer type lattice, quotient surfaces
14J28 - $K3$ surfaces and Enriques surfaces
14J50 - Automorphisms of surfaces and higher-dimensional varieties
14J10 - Families, moduli, classification: algebraic theory