Extra components of Hilbert schemes of points on affine three-space

Let $\Bbbk$ be an algebraically closed field of characteristic zero, and write

$$H_d=\operatorname{Hilb}^d(\mathbb A^3_{\Bbbk})$$

for the Hilbert scheme of length-$d$ zero-dimensional subschemes of affine three-space. Its smoothable component is the closure of the locus of $d$ distinct points and has dimension $3d$. Let

$$H_{d,0}=\operatorname{Hilb}^d(\mathbb A^3_{\Bbbk},0)$$

denote the punctual Hilbert scheme parametrizing subschemes supported at the origin.

Open problems. Determine the smallest $d$ for which $H_{d,0}$ is reducible, and the smallest $d$ for which $H_d$ is reducible. The presently known bounds are

$$12\leq d_{\mathrm{punctual}}\leq18, \qquad 12\leq d_{\mathrm{full}}\leq78.$$

Indeed, both spaces are irreducible through length $11$; reducibility is known for the punctual space at length $18$ and for the full Hilbert scheme at length $78$.

A related explicit challenge is to describe a component other than the smoothable component as concretely as possible. One would like an explicit family of finite local algebras

$$A=\Bbbk[x,y,z]/I, \qquad \dim_{\Bbbk}A=d,$$

or an equivalent family of Macaulay inverse systems, together with a deformation-theoretic certificate that its closure is an irreducible component and is not contained in the smoothable component.

A computational approach could enumerate local Hilbert functions and inverse systems, calculate

$$T_{[I]}H_d\cong\operatorname{Hom}_{\Bbbk[x,y,z]}(I,\Bbbk[x,y,z]/I),$$

study obstruction spaces, and search for a parameter family of the dimension required to force a new component. Even a single explicit component in the unresolved range $12\leq d<78$, or a sharper threshold for either Hilbert scheme, would be substantial progress.