ag.algebraic geometry - Hilbert scheme of points on a complex surface

Here is a geometric description in the case of $H_n(mathbb{C}^2)$. This is meant to be a geometric rewrite of Proposition 2.6 in Mark Haiman's "(t,q)-Catalan numbers and the Hilbert scheme",
Discrete Math. 193 (1998), 201-224.

Let $S= (mathbb{C}^2)^n/S_n$; notice that this is an orbifold. Let $S_0$ be the open dense set where the $n$ points are distinct. For $D$ an $n$-element subset of $mathbb{Z}_{geq 0}^2$, let $A_{D}$ be the polynomial $det( x_i^{a} y_i^{b})$, where $(a, b)$ ranges over the elements of $D$ and $i$ runs from $1$ to $n$. For any $D$ and $D'$, the ratio $A_D/A_{D'}$ is a meromorphic function on $S$, and is well defined on $S_0$.

Map $S_0$ into $S_0 times mathbb{CP}^{infty}$ where the homogenous coordinates on $ mathbb{CP}^{infty}$ are the $A_{D}$'s. (Only finitely many of the $A_D$'s are needed, but it would be a little time consuming to say which ones.) The Hilbert scheme is the closure of $S_0$ in $S times mathbb{CP}^{infty}$.

Algebraically, we can describe this as the blow up of $S$ along the ideal generated by all products $A_D A_{D'}$. Haiman points out that the reduction of this ideal is the locus where two of the points collide and speculates that this ideal may be reduced. If his speculation is correct, then we can describe $H_n(mathbb{C}^2)$ geometrically as the blow up of $(mathbb{C}^2)^n/S_n$ along the reduced locus where at least two of the points are equal.