Small elementary components of Hilbert schemes of points

Abstract

We answer an open problem posed by Iarrobino in the '80s: is there an elementary component of the Hilbert scheme of points ${\text{Hilb}}^d({\mathbb{A}}^n)$ with dimension less than $(n-1)(d-1)$? We construct an infinite class of such components in ${\text{Hilb}}^d({\mathbb{A}}^4)$. Our techniques also allow us to construct an explicit example of a local Artinian ring with trivial negative tangents, vanishing nonnegative obstruction space, and socle-dimension $2$.