Let $[x,y,z,w]$ be homogeneous coordinates on $\mathbb{P}^3$ so that $\Gamma_*(\mathcal{O}_{\mathbb{P}^3})$ equals $k[x,y,z,w]$. Let $p$ be the point $[0,0,0,1]$ in these coordinates, whose associated homogeneous ideal $\Gamma_*(\mathcal{I}_{p/\mathbb{P}^3})$ is $\langle x,y,z \rangle$. Let $A\subset \mathbb{P}^3$ be the nonreduced, local Artin scheme supported at $p$ whose associated homogeneous ideal $I=\Gamma_*(\mathcal{I}_{A/\mathbb{P}^3})$ is $\langle x,y^2,yz,z^2 \rangle$.

For a given pair of positive integers, $(d,e)$, one parameter space (not *the* Hilbert scheme) for complete intersection curves is the open subscheme $L'$ of $$M':=\mathbb{P}k[x,y,z,w]_d \times_{\text{Spec}(k)} \mathbb{P}k[x,y,z,w]_e,$$ parameterizing pairs $([E(x,y,z,w)],[F(x,y,z,w)])$ of homogeneous polynomials of degree $d$, resp. $e$, such that the zero scheme $C:=\text{Zero}(E,F)\subset \mathbb{P}^3$ is a complete intersection curve. There is a dominant morphism,
$$ \pi: L'\to L, \ ([E(x,y,z,w)],[F(x,y,z,w)]) \mapsto [C].$$
The fiber over a point $[C]$ is a dense open subset of the product of projective spaces $$\mathbb{P} \Gamma(\mathcal{I}_{C/\mathbb{P}^3}(d))\times_{\text{Spec}(k)}\mathbb{P}\Gamma(\mathcal{I}_{C/\mathbb{P}^3}(e)).$$

With this description, the subscheme of $K'$ parameterizing curves $C$ that contain $A$ is the intersection of the open subset $L'$ with the subvariety
$$ N' = \mathbb{P}I_d \times_{\text{Spec}(k)}\mathbb{P}I_e.$$ In particular, $K'$ is a dense open subset of a product of projective spaces, hence $K'$ is rational.

Finally, there is a natural action of the automorphism scheme $\text{Aut}(\mathbb{P}^3,\mathcal{O}(1)) = \textbf{GL}_{4,k}$ on $\Gamma_*(\mathcal{O}_{\mathbb{P}^3})$ (in the sense of GIT, $G$-linearizations, etc.) That action induces an action on every projective space $\mathbb{P}k[x,y,z,w]_m$, and thus also an action on $M'$. The open subset $L'$ of $M'$ is invariant for this action. Denote the action morphism as follows, $$m:\textbf{GL}_{4,k} \times_{\text{Spec}(k)} L' \to L'.$$ There is an induced composition morphism,
$$ \textbf{GL}_{4,k}\times_{\text{Spec}(k)} K' \xrightarrow{m} L' \xrightarrow{\pi} L.$$ I claim that the closure of the image is $V$. Thus, since $V$ is dominated by a rational variety, $V$ is unirational.

singularcomplete intersection curve is singular at some point with a given 2-dimensional subspace of the Zariski tangent space. So you can form the incidence variety of a complete intersection curve together with a singular point and a 2-dimensional tangent space. There is a forgetful morphism from the incidence correspondence to the dual projectivized tangent bundle of $\mathbb{P}^3$ that only remembers the singular point and tangent space (a Cheshire cat leaving behind its grin) . . . $\endgroup$ – Jason Starr Sep 23 '15 at 10:32