Smooth double covers of K3 surfaces

In this paper we classify the topological invariants of the possible branch loci of a smooth double cover f : X → Y of a K3 surface Y . We describe some geometric properties of X which depend on the properties of the branch locus. We give explicit examples of surfaces X with Kodaira dimension 1 and 2 obtained as double cover of K3 surfaces and we describe some of them as bidouble cover of rational surfaces. Then, we classify the K3 surfaces which admit smooth double covers X satisfying certain conditions; under these conditions the surface X is of general type, h^{1,0}(X)=0 and h^{2,0}(X)=2. We discuss the variation of the Hodge structure of H^2(X, Z) for some of these surfaces X.