Hilbert schemes of 8 points

Cartwright, Dustin; Erman, Daniel; Velasco, Mauricio; Viray, Bianca

doi:10.2140/ant.2009.3.763

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Hilbert schemes of 8 points

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

Vol. 3 (2009), No. 7, 763–795

DOI: 10.2140/ant.2009.3.763

Abstract

The Hilbert scheme $H_{n}^{d}$ of $n$ points in $A^{d}$ contains an irreducible component $R_{n}^{d}$ which generically represents $n$ distinct points in $A^{d}$ . We show that when $n$ is at most $8$ , the Hilbert scheme $H_{n}^{d}$ is reducible if and only if $n = 8$ and $d \geq 4$ . In the simplest case of reducibility, the component $R_{8}^{4} \subset H_{8}^{4}$ 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 $H_{n}^{d}$ 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

Vol. 3, No. 7, 2009