K3 double structures on Enriques surfaces and their smoothings

Abstract

Let $Y$ be a smooth Enriques surface. A $K3$ carpet on $Y$ is a locally Cohen-Macaulay double structure on $Y$ with the same invariants as a smooth $K3$ surface (i.e., regular and with trivial canonical sheaf). The surface $Y$ possesses an étale $K3$ double cover $X\stackrel{\pi }{⟶}Y$. We prove that $\pi$ can be deformed to a family $\SX \longrightarrow \mathbf P^N_{T^*}$ of projective embeddings of $K3$ surfaces and that any projective $K3$ carpet on $Y$ arises from such a family as the flat limit of smooth, embedded $K3$ surfaces.

New title (old title:"Smoothing of $K3$ carpets on Enriques surfaces"). Improved Section 1. Simplified step 2 of proof of theorem 3.2