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$\DeclareMathOperator\Hilb{Hilb}$It is well known that the Hilbert scheme $\Hilb^n(\mathbb C^3)$ has a (symmetric) perfect obstruction theory.
Consider the punctual part at $0 \in \mathbb C^3$, which I denote by $\Hilb_0^n(\mathbb C^3)$.

Is there an induced (symmetric?) perfect obstruction theory here giving rise to a virtual fundamental class $[\Hilb_0^n(\mathbb C^3)]^{\mathrm{vir}}$?
In the affirmative, is this cycle in the punctual part known?

My thoughts
Since a perfect obstruction theory is equivalent to locally having Kuranishi structures (at least this is how I understand it), I thought it should somehow be possible to take a neighborhood $0 \in U \subset \mathbb C^3$ and pull back via the Hilbert-Chow morphism $\pi : \Hilb^n(\mathbb C^3) \to \mathbb C^3$ to a neighborhood $\pi^{-1}(U)$ of $\Hilb_0^n(\mathbb C^3)$ in $\Hilb^n(\mathbb C^3)$. Now, having local Kuranishi structures is certainly not enough a priori to have a bundle $E \to \pi^{-1}(U)$ with at section $s$ cutting out the punctual part. But I thought this might be possible...
But maybe an induced obstruction theory comes automatically via the theory?

Moreover, the expected dimension of the punctual part is $(3-1)(n-1)$, and so I thought – in the affirmative – that this must be the virtual dimension.
There is a certain irreducible component of $\Hilb_0^n(\mathbb C^3)$ called the curvilinear part, and I write $\operatorname{CHilb}_0^n(\mathbb C^3)$. It is the closure of the locus of $n$ points comming together along a smooth curve on $\mathbb C^3$. The dimension of this component is the expected dimension $(3-1)(n-1)$. Possibly it is known whether this is actually the virtual fundamental class – if/when it exists.

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