K3K3 double structures on Enriques surfaces and their smoothings

Let YY be a smooth Enriques surface. A K3K3 carpet on YY is a double structure on YY with the same invariants as a smooth K3K3 surface (i.e., regular and with trivial canonical sheaf). The surface YY possesses an étale K3K3 double cover X⟶πY. We prove that ππ can be deformed to a family X⟶PT∗N of projective embeddings of K3K3 surfaces and that any projective K3K3 carpet on YY arises from such a family as the flat limit of smooth, embedded K3K3 surfaces.