In Alex Lee's undergraduate thesis (2000), it was said that the Hilbert scheme $H\_{2m+2}(\mathbb{P}^3)$ has two components $\mathcal{H}',\mathcal{H}''$, where a general point of $\mathcal{H}'$ corresponds to a pair of skew lines and a general point of $\mathcal{H}''$ corresponds to a conic union a point. At the time of the thesis, the smoothness of neither of the two components were known.


Since then, it was shown in the paper *Hilbert scheme of a pair of codimension two subspaces* (2011) that $\mathcal{H}'$ is smooth. Is it known by now whether $\mathcal{H}''$ is smooth?



Hilbert scheme of a plane conic union a point
I think [this paper](https://arxiv.org/pdf/0911.2221v3.pdf) by Chen and Nollet is relevant. Theorem 1.9 (proved as Theorem 4.3 on p. 16) shows the following about plane curves in $\mathbb P^3$.



> 
> The component $H\_d\subset \textrm{Hilb}^{dz+2-g}(\mathbb P^3)$ whose
>  general point is a degree $d$ plane curve union an isolated point is
>  smooth for all $d\geq 1$.
> 
> 
> 


Since there is a bijective morphism $f:\textrm{Bl}\_\Sigma(\textrm{Hilb}^{dz+1-g}(\mathbb P^3)\times \mathbb P^3)\to H\_d$, where $\Sigma$ is the incidence correspondence, $f$ is an isomorphism by Zariski's main theorem.


