Hilbert schemes of 8 points

Abstract

The Hilbert scheme Hdn of n points in Ad contains an irreducible component Rdn which genetically represents n distinct points in Ad. We show that when n is at most 8, the Hilbert scheme Hdn 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 Hdn 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.