Hilbert scheme of a plane conic union a point

Question

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?

Answer 1

I think this paper 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.

Answer 2

I think this is correct. Moreover, I think one can describe explicitly $H''$ as follows.

First, consider $H_0$ the Hilbert scheme of conics. It is a $P^5$-bundle over $P^2$, in particular is smooth. Further, let $Z_0 \subset H_0 \times P^3$ be the universal conic. The natural map $Z_0 \to P^3$ is smooth (it is a fibration with fiber being $P^4$-bundle over $P^2$), hence $Z_0$ is smooth. And the claim is that $H''$ is isomorphic to the blowup of $H_0 \times P^3$ with center in $Z_0$, hence is smooth.