Is there some standard technique or approach to determine when a (irreducible) subvariety of a rationally connected variety is again rationally connected? Any reference/text dealing with this kind of question will be most welcome.


The example that I have in mind is the following: Let $P$ be the Hilbert polynomial of a complete intersection curve in $\mathbb{P}^3$ and $L$ an irreducible component of the corresponding Hilbert scheme parametrizing complete intersection curves in $\mathbb{P}^3$ with Hilbert polynomial $P$. Denote by $V \subset L$ the sublocus parametrizing curves which are not smooth. As far as I understand $L$ is rationally connected. Am I right? Then, I want to understand when is a connected component of $V$ again rationally connected.



Standard techniques on rationally connected varieties
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.


