# The non-curvilinear locus in $\textrm{Hilb}^4(\mathbb C^2)$

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!

• If memory serves, Joel Briancon has an article about the natural stratification of the punctual Hilbert scheme. Mar 5, 2017 at 14:13
• Would inverse systems help? Your $A_1$ is the apolar algebra of the inverse system generated (redundantly) by $ab,a,b$ (where $a,b$ are dual variables to $x,y$), and $A_2$ is the apolar algebra of the inverse system generated by $a^2,a,b$. In general I think $Y_4$ should correspond to inverse systems generated by $a,b$, and a quadratic form. Mar 5, 2017 at 14:34
• @JasonStarr: Thanks, I looked at the paper, if I haven't misunderstood it there should be a $\mathbb P^1$ of subschemes isomorphic to $\textrm{Spec }A_2$. Mar 6, 2017 at 22:04