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](https://mathoverflow.net/questions/176662/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.



Hilbert scheme of an infinitesimal neighborhood of a subvariety
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.


