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 $Z_1 \subset Z_2$. Does it have an induced scheme structure? Can we look at it as a relative Hilbert scheme?

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?