Let $H$ be a hyperplane in $\mathbb P^n$, and $X \subseteq H$ be a subscheme. Let $CX \subseteq \mathbb P^n$ be the cone on $X$ from a point $p \notin H$.

Let $Hilb_{CX}$ be the Hilbert scheme whose points correspond to subschemes of $\mathbb P^{n}$ with the same Hilbert polynomial as $CX$. Let $U \subseteq Hilb_{CX}$ denote the open set of subschemes transversely intersecting $H$. (So $U \ni CX$.)

I'm interested in the subscheme of $U$ consisting of those $Y \subseteq \mathbb P^{n}$ such that $Y \cap H = X$. So, projective schemes with a fixed hyperplane section.

Has this space been studied before?