Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More precisely, it represents the functor $F$ from schemes (resp. complex analytic spaces) to sets given by $$F(S)=\{Y\subset X\times S|\, Y \mbox{- closed subscheme (resp. anal. subspace), flat proper over } S\}.$$

**Question 1. Let $X_1\subset X$ be a closed subscheme. Is it true that the Hilbert scheme of $X_1$ is a closed subscheme of the Hilbert scheme of $X$? The same question for the Douady spaces.**

**Question 2.** I would like to have a modification of the Hilbert scheme/ Douady space which parameterizes closed subschemes of $X$ with a marked point. I guess one should consider the functor
$$G(S)=\{Y\subset X\times S \mbox{ as above and a morphism } f\colon S\to Y|\, p\circ f=Id_S\},$$
where $p\colon Y\to S$ is the natural projection. **Is the functor $G$ representable?**

**Question 3.** Assume that Questoin 2 has a positive answer, namely the functor $G$ is representable by a scheme (resp. complex analytic space) $\mathcal{F}$. Obviously we have the morphisms $\mathcal{F}\to X$ and $\mathcal{F}\to Hilb(X)$ which forget either the subscheme or the marked point. **Is it true that the induced morphism $\mathcal{F}\to X\times Hilb(X)$ is a closed imbedding?** The same question for Douady space.

I believe answers to all these questions are well known to experts. **Is there a standard reference to this material?** The complex analytic case is more important for me, but the algebraic case is also of interest.