Small elementary components of Hilbert schemes of points

TL;DR

This paper constructs an infinite class of elementary components in the Hilbert scheme of points in four-dimensional affine space, answering a long-standing open problem and providing explicit examples with special algebraic properties.

Contribution

It introduces new elementary components of Hilbert schemes with smaller dimension than previously known, and constructs explicit local Artinian rings with unique tangent and obstruction properties.

Findings

Constructed infinite classes of elementary components in Hilb^d(ℝ^4).

Provided explicit examples of local Artinian rings with trivial negative tangents.

Demonstrated vanishing of nonnegative obstruction space.

Abstract

We answer an open problem posed by Iarrobino in the '80s: is there an elementary component of the Hilbert scheme of points $\textrm{Hilb}^d(\mathbb{A}^n)$ with dimension less than $(n-1)(d-1)$? We construct an infinite class of such components in $\textrm{Hilb}^d(\mathbb{A}^4)$. 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 $2$.