# 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)?

Actually, the answers to 1 and 3 are mentioned in Nakajima's book on Hilbert Schemes, in Exercise 5.15, and later proved in the paper https://arxiv.org/pdf/math/0311058.pdf by Nakajima and Yoshioka. Thus, the core $$\Lambda = \pi^{-1}(0)$$ of $$\pi: \mathcal{M}(n,r) \rightarrow \mathcal{M}_0(n,r)$$ is
3. of complex dimension $$nr−1={1 \over 2} (dim(\mathcal{M}_θ(n,r))−1)$$
Though, as far as I can see the smoothness of core (also called the punctual Quot scheme) is not discussed in the paper. Though, one can look at the Theorem 3.8. which gives the Poincare polynomial - if it is not symmetric, for some $$n$$ and $$r,$$ then the corresponding core is not smooth. Though, that would not be enough - the example $$n=3, r=1$$ gives you symmetric Poincare polynomial though the corresponding core is singular.