A few general points of the Hilbert scheme of K3 surfaces

Abstract

A ribbon D over a variety C is a scheme such that D_{red} = C, the ideal I in O_D of Cis a line bundle on C and I^2 = 0. A two dimensional ribbon is called a carpet. In this article we show that if D is a K3 carpet, that is a ribbon associated to a rational normal scroll with numerical invariants that of a K3 surface,then it can be smoothed to a K3 surface. We also show that not all K3 carpets are smooth points of the Hilbert scheme unlike in the dimension one case as shown by Bayer and Eisenbud. We prove that only those K3 carpets which are supported on "balanced" scrolls are smooth points of the Hilbert scheme.