  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 $2$.
