Hilbert scheme of an infinitesimal neighborhood of a subvariety

Question

Let $X$ be a complex algebraic variety. Let $C\subset X$ be a compact (reduced) subvariety. Let $C^{(n)}$ denote the $n$th infinitesimal neighborhood of $C$ inside $X$. Let $Hilb(X)$ denote the Hilbert scheme of $X$. Thus $C$ defines the point $[C]\in Hilb(X)$.

In my situation one may assume that $X$ and $C$ are smooth varieties, and moreover $[C]$ is a smooth point of $Hilb(X)$.

I was told (here Basic questions on the Hilbert scheme/ Douady space) that $Hilb(C^{(n)})$ is a closed subscheme of $Hilb(X)$.

Question. Is it true that the $n$th infinitesimal neighborhood of the point $[C]$ in $Hilb(X)$ is equal to the connected component of $Hilb(C^{(n)})$ containing $[C]$ as a closed point?

I am particularly interested in a modification of this question when $X,C$ are complex analytic rather than algebraic manifolds; in that case Hilbert scheme is replaced by Douady space.

References are most welcome.

Answer 1

That already fails when $n$ equals $2$, $X$ equals $\mathbb{P}^3$ and $C$ is a line in $\mathbb{P}^3$. Choose homogeneous coordinates $[y_0,y_1,y_2,y_3]$ on $\mathbb{P}^3$ so that $C$ is $Z(y_2,y_3)$. Then an affine neighborhood $U$ of $[C]$ in $\text{Hilb}(X)$ is the affine $4$-space with affine coordinates $(a_{2,0},a_{2,1},a_{3,0},a_{3,1})$ associated to the closed subscheme $$Z(\widetilde{y}_2,\widetilde{y}_3), \ \ \widetilde{y}_2 := y_2-a_{2,0}y_0-a_{2,1}y_1, \ \ \widetilde{y}_3 := y_3-a_{3,0}y_0-a_{3,1}y_1.$$ The closed subscheme $C^{(2)}$ has homogeneous defining ideal $\langle y_2^2,y_2y_3,y_3^2\rangle$. Thus, the closed subscheme $\text{Hilb}(C^{(2)})\cap U$ of $U$ is the maximal closed subscheme such that each of the elements $$ y_2^2 = (\widetilde{y}_2+a_{2,0}y_0+a_{2,1}y_1)^2,y_2y_3 = (\widetilde{y}_2+a_{2,0}y_0+a_{2,1}y_1)(\widetilde{y}_3+a_{3,0}y_0+a_{3,1}y_1),y_3^2 =(\widetilde{y}_3+a_{3,0}y_0+a_{3,1}y_1)^2$$ is contained in the homogeneous ideal $\langle \widetilde{y}_2,\widetilde{y}_3 \rangle.$ Expanding out, this closed subscheme has defining ideal $$\langle a_{2,0}^2,a_{2,0}a_{2,1},a_{2,1}^2,a_{3,0}^2,a_{3,0}a_{3,1},a_{3,1}^2,a_{2,0}a_{3,0},a_{2,1}a_{3,1},a_{2,0}a_{3,1}+a_{2,1}a_{3,0} \rangle.$$ This ideal sits inside the full ideal $\langle a_{i,j} \rangle^2$, and the cokernel has length $1$. Thus, the component of $[C]$ in $\text{Hilb}(C^{(2)})$ contains the second order neighborhood of $[C]$ inside $\text{Hilb}(X)$, but the inclusion is strict.