Hilbert schemes of 8 points

The Hilbert scheme H^d_n of n points in A^d contains an irreducible component R^d_n which genetically represents n distinct points in A^d. We show that when n is at most 8, the Hilbert scheme H^d_n is reducible if and only if n = 8 and d ≥ 4. In the simplest case of reducibility, the component R ⁴₈ ⊂ H⁴₈ 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^d_n which parametrize those ideals which are homogeneous and have a fixed Hilbert function. These subschemes are a special case of multi-graded 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.