Formula omitted] double structures on Enriques surfaces and their smoothings

Abstract

Let Y be a smooth Enriques surface. A K 3 carpet on Y is a double structure on Y with the same invariants as a smooth K 3 surface (i.e., regular and with trivial canonical sheaf). The surface Y possesses an étale K 3 double cover X ⟶ π Y . We prove that π can be deformed to a family X ⟶ P T ∗ N of projective embeddings of K 3 surfaces and that any projective K 3 carpet on Y arises from such a family as the flat limit of smooth, embedded K 3 surfaces.