On K3 surface quotients of K3 or Abelian surfaces

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 (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 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.