#### Vol. 3, No. 7, 2009

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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
##### Abstract

The Hilbert scheme ${H}_{n}^{d}$ of $n$ points in ${\mathbb{A}}^{d}$ contains an irreducible component ${R}_{n}^{d}$ which generically represents $n$ distinct points in ${\mathbb{A}}^{d}$. We show that when $n$ is at most $8$, the Hilbert scheme ${H}_{n}^{d}$ is reducible if and only if $n=8$ and $d\ge 4$. 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 $\left(1,3,2,1\right)$ is the minimal reducible example.

##### Keywords
Hilbert scheme, zero-dimensional ideal, smoothable
Primary: 14C05
Secondary: 13E10