Degenerations of K3 surfaces in projective space

Abstract

A K3 carpet S is a double structure on a rational normal scroll such that its dualizing sheaf is trivial and h1(OS) = 0. In this note the authors show that every K3 carpet S can be smoothed, i.e. there exists a flat family over a smooth curve with smooth generic fiber and with a special closed fiber isomorphic top S. Moreover, they study the Hilbert scheme of numerical K3 surfaces at the locus parametrizing K3 carpets, characterizing those K3 carpets whose corresponding Hilbert point is smooth. The proof is based on the properties of the hyperelliptic linear systems on K3 surfaces.