Let $H_n=\textrm{Hilb}^n(\mathbb C^2)$ be the Hilbert scheme of $n$ points in $\mathbb C^2$ and let $H_n^0\subset H_n$ the punctual Hilbert scheme, parametrizing subschemes entirely supported at the origin. Then $H_n^0$ is irreducible of dimension $n-1$, in fact equal to the closure of the curvilinear locus $C_n^0$, which is open in $H_n^0$. I am interested in the complement of the curvilinear locus $$Y_n=H_n^0\setminus C_n^0.$$

We have $Y_n=\emptyset$ for $n=1,2$ and $Y_3$ is a single point, corresponding to the ideal $(x,y)^2$. I would like to describe $Y_4$. Up to isomorphism, we only have the algebras $$A_1=\mathbb C[x,y]/(x^2,y^2) \,\textrm{ and }\, A_2=\mathbb C[x,y]/(x^3,xy,y^2),$$ but how do the isomorphism classes of $\textrm{Spec }A_1$ and $\textrm{Spec }A_2$ "spread around" in $H_4^0$? I think the codimension of $Y_n\subset H_n^0$ is always $1$ for $n\geq 4$, so I would expect $Y_4$ to be two-dimensional. How to describe it explicitly? Is there a general description of $Y_n$ for arbitrary $n$?

thanks!