Hilbert schemes of 8 points

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.