Vol. 3, No. 7, 2009

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Hilbert schemes of 8 points

Dustin A. Cartwright, Daniel Erman, Mauricio Velasco and Bianca Viray

Vol. 3 (2009), No. 7, 763–795
Abstract

The Hilbert scheme Hnd of n points in Ad contains an irreducible component Rnd which generically represents n distinct points in Ad. We show that when n is at most 8, the Hilbert scheme Hnd is reducible if and only if n = 8 and d 4. In the simplest case of reducibility, the component R84 H84 is defined by a single explicit equation, which serves as a criterion for deciding whether a given ideal is a limit of distinct points.

To understand the components of the Hilbert scheme, we study the closed subschemes of Hnd which parametrize those ideals which are homogeneous and have a fixed Hilbert function. These subschemes are a special case of multigraded Hilbert schemes, and we describe their components when the colength is at most 8. In particular, we show that the scheme corresponding to the Hilbert function (1,3,2,1) is the minimal reducible example.

Keywords
Hilbert scheme, zero-dimensional ideal, smoothable
Mathematical Subject Classification 2000
Primary: 14C05
Secondary: 13E10
Milestones
Received: 27 June 2008
Revised: 23 April 2009
Accepted: 30 June 2009
Published: 29 November 2009
Authors
Dustin A. Cartwright
Department of Mathematics
University of California
Berkeley, CA 94720
United States
Daniel Erman
Department of Mathematics
University of California
Berkeley, CA 94720
United States
Mauricio Velasco
Department of Mathematics
University of California
Berkeley, CA 94720
United States
Bianca Viray
Department of Mathematics
University of California
Berkeley, CA 94720
United States