A question on nested Hilbert scheme

Question

Let $H_1, H_2$ be two Hilbert schemes parametrizing subschemes in $\mathbb{P}^{n_1}, \mathbb{P}^{n_2}$ with Hilbert polynomials $P_1, P_2$, respectively. Given a pair $(Z_1, Z_2)$ of subschemes in $\mathbb{P}^{n_1}, \mathbb{P}^{n_2}$, respectively, we say that $Z_1 \subset Z_2$ if there exists a closed immersion $i$ of $\mathbb{P}^{n_1}$ into $\mathbb{P}^{n_2}$ such that $i(Z_1)$ is a closed subscheme of $Z_2$. Denote by $\mathcal{D}$ the subet of $H_1 \times H_2$ consisting of all pairs $(Z_1, Z_2)$ such that $Z_i \in H_i$ and $Z_1 \subset Z_2$. Can we look at it as a scheme?

Answer 1

I guess you are looking for a construction that gives an object over $H_2$, that parametrizes closed immersions $Z_1 \to Z_2$ for each $Z_2 \in H_2$. If I'm not mistaken, it is the closed subscheme of $Quot(\mathscr{O}_{\mathbb{P}^{n_2}}) \times Quot(\mathscr{O}_{\mathbb{P}^{n_2}})$ (which parametrizes pairs of ideals) defined by containment of one ideal in the other. Let's call this subscheme $H_3$. It is a special case of a generalized Hilbert scheme construction whose name I don't know, for any commutative diagram of closed immersions in a projective scheme.

Let $H_4 = \underline{\operatorname{Imm}}(\mathbb{P}^{n_1}/\mathbb{P}^{n_2})$ denote the scheme of closed immersions $\mathbb{P}^{n_1} \to \mathbb{P}^{n_2}$ (see page 268 of Grothendieck's Bourbaki 221 - this is an open subscheme of the Hom scheme). Then we can make a subscheme $X$ of $H_1 \times H_4 \times H_3$ defined by commutativity of the compositions $Z_1 \to \mathbb{P}^{n_1} \to \mathbb{P}^{n_2}$ and $Z_1 \to Z_2 \to \mathbb{P}^{n_2}$.

It seems that the object you want is the image of $X$ under the natural map $H_1 \times H_4 \times H_3 \to H_1 \times H_2$ taking $(i_1, i_4, Z_1 \subset Z_2 \subset \mathbb{P}^{n_2})$ to $(i_1, Z_2 \subset \mathbb{P}^{n_2})$.