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)?


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