A few general points of the Hilbert scheme of K3 surfaces

1995/11/23 by Francisco Javier Gallego, Francisco Gallego, Gallego, Francisco +3
Mathematics · #14 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #alg-geom #math.AG #msc:14

paper · pdf · doi:10.48550/arxiv.alg-geom/9511014

AmsTeX 2.1 with Amsppt. 19 pages

arxiv created 1995/11/23 · openalex publication_date 1995/11/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 I² = 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.

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