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 $\textrm{Hilb}^d(\mathbb{A}^n)$ 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 $2$.