Let $G$ be a finite subgroup of $\text{SL}(2,\mathbb{C})$ and let $Y=\text{GHilb}(\mathbb{C}^2)$ be the minimal resolution of $X=\mathbb{C}^2/G$ where $\text{GHilb}(\mathbb{C}^2)$ is the Nakamura $G$-Hilbert scheme. According to *Bridgeland-King-Reid* there is a universal closed subscheme $\mathfrak{Z} \subset Y \times \mathbb{C}^2$ which makes a certain diagram commutative. What is really known about $\mathfrak{Z},$ for example

Is $\mathfrak{Z}$ reduced?

I was thinking, if one can prove that $Z \subset \text{Hilb}^{|G|}(\mathbb{C}^2) \times \mathbb{C}^2$ where $Z$ is the universal closed subscheme associated to the Hilbert scheme of $|G|$ points on $\mathbb{C}^2,$ is *reduced*, because $Y \times \mathbb{C}^2 \hookrightarrow \text{Hilb}^{|G|}(\mathbb{C}^2) \times \mathbb{C}^2$ we would then know that $\mathfrak{Z}$ is reduced. So does this hold?