# Core of the Jordan quiver variety

It is known that, given the Jordan quiver, dimension vectors $$\textbf{v}=n,\textbf{w}=1$$ and a stability condition $$\theta<0,$$ the corresponding quiver variety $$\mathcal{M}_{\theta}(n,1)\cong \text{Hilb}^n(\mathbb{C}^2)$$ is isomorphic to the Hilbert scheme of n points on $$\mathbb{C}^2.$$ Furthermore, the canonical morphism $$\pi:\mathcal{M}_{\theta}(n,1)\rightarrow \mathcal{M}_{0}(n,1)$$ is identified with the Hilbert Chow morphism $$\text{Hilb}^n(\mathbb{C}^2)\rightarrow Sym^n(\mathbb{C}^2).$$ Under this morphism, the central fiber (also called the punctual Hibert scheme) $$\Lambda=\pi^{-1}(0)$$ is known to be:

1. Irreducible
2. Smooth for $$n=2$$ and singular for $$n\geq 3$$
3. Of complex dimension $$n-1={1\over 2} dim(\mathcal{M}_{\theta}(n,1))-1$$

The question is, whether any of 1-3 is known for Jordan quiver varieties $$\mathcal{M}_{\theta}(n,r)$$ with framing $$\mathbf{w}=r,r>1$$ (also known as Gieseker moduli spaces)?