Punctual Hilbert scheme of four points

Question

I am looking at $\text{Hilb}^4(\mathbb{C}^2)$, which is the Hilbert scheme of four points on $\mathbb{C}^2$. In particular, I am just looking at four points collided (at the origin), and want to know what all the non-isomorphic ideal representations are. For example, $I = \langle x, y^4 \rangle$ has colength 4, thus is an element in $\text{Hilb}^4(\mathbb{C}^2)$. I'm curious to know what are the non-isomorphic ideals like this of colength four. Thanks for the help.

Answer 1

The $T=(\mathbb C^\times)^2$-action on $\textrm{Hilb}^n\mathbb C^2$ (lifted from the natural one on $\mathbb C^2$) has, as fixed points, the monomial ideals, which correspond to (one-dimensional) partitions of $n$. Hence if you can list the $p(n)$ partitions (Young tableaux if you prefer!) of $n$ and you write down the corresponding monomial ideals, you are done. In the case of $n=4$, you get: $$(x,y^4),(y^4,x),(x^2,y^2),(x^3,xy,y^2),(x^2,xy,y^3).$$

NB. (In case you wonder how to find the generators in general.) Represent the polynomial ring $\mathbb C[x,y]$ by listing all possible monomials in $x$ and $y$, along two axes. A Young tableaux shapes for you a "staircase" (called Hilbert staircase), and since the Young tableaux (or partition) is the complement of the corresponding ideal, you can read the generators of the ideal "under the stairs"!