The construction of the Hilbert scheme of a projective scheme $X$ requires us to fix an ample line bundle $L$ on $X$ in order to define the Hilbert polynomial $P$.
Suppose that $S \subset X$ is a closed subscheme of $X$. Suppose $L_1$ and $L_2$ are two different ample line bundles on $X$, and say that $S$ has Hilbert polynomial $P_1$ with respect to $L_1$ and $P_2$ with respect to $L_2$.
Then we can construct the Hilbert scheme $\operatorname{Hilb}^1(X)$ using $L_1$, and in here $S$ will give a point in the component $\operatorname{Hilb}^1_{P_1}(X)$. Alternatively we can construct $\operatorname{Hilb}^2(X)$ using $L_2$, and here $S$ is a point in $\operatorname{Hilb}^2_{P_2}(X)$.
Do we then have an isomorphism $\operatorname{Hilb}^1_{P_1}(X) \cong \operatorname{Hilb}^2_{P_2}(X)$?
I think the answer must be yes, but I would like to have a reference.
Thanks!