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 G (respectively of a K3 surface by an Abelian group G) if and only if a certain lattice is primitively embedded in its Neron-Severi group. This allows one to describe the coarse moduli space of the K3 surfaces that are (rationally) G-covered by Abelian or K3 surfaces (in the latter case G is an Abelian group). When G has order 2 or G is cyclic and acts on an Abelian surface, this result is already known; 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.