Generalization of Hilbert/Quot schemes

Question

For some positive integer $n$, recall that the Quot scheme $Quot(\mathcal{O}_{\mathbb{P}^n})$ parametrizes ideal sheaves of subschemes in $\mathbb{P}^n$. As far as I understand (from a previous post) that the subset of $Quot(\mathcal{O}_{\mathbb{P}^n}) \times Quot(\mathcal{O}_{\mathbb{P}^n})$ consisting of pairs $(\mathcal{I}_1, \mathcal{I}_2)$ such that $\mathcal{I}_1 \hookrightarrow \mathcal{I}_2$, is infact a scheme (hence a closed subscheme of the above mentioned product of Quot schemes). Can somebody please explain/suggest reference why this is the case?

Answer 1

For convenience, denote $\text{Quot}_{\mathcal{O}_{\mathbb{P}^n}/\mathbb{P}^n/\mathbb{Z}}$ by $H$; of course this is the same as the Hilbert scheme of $\mathbb{P}^n$, but it is indeed a Quot scheme as well. On the scheme $H\times \mathbb{P}^n_{\mathbb{Z}}$, there is a universal surjective homomorphism of quasi-coherent sheaves, $$p:\mathcal{O}_{H\times \mathbb{P}^n} \to \mathcal{O}_Z,$$ such that $\mathcal{O}_Z$ is locally finitely presented, has proper support over $H$, and is $H$-flat. Denote the kernel of $p$ by $$q:\mathcal{I} \to \mathcal{O}_{H\times \mathbb{P}^n}.$$ If you choose, you can formulate the universal property of $H$ in terms of $q$ instead of $p$. Altogether, we have a short exact sequence of $H$-flat, quasi-coherent sheaves, $$ 0 \to \mathcal{I} \xrightarrow{q} \mathcal{O}_{H\times \mathbb{P}^n} \xrightarrow{p} \mathcal{O}_Z \to 0.$$

Thus, on the scheme $H\times H \times \mathbb{P}^n_{\mathbb{Z}}$, there are two short exact sequences, $$ 0 \to \text{pr}_{1,3}^*\mathcal{I} \xrightarrow{\text{pr}_{1,3}^*q} \mathcal{O}_{H\times H\times \mathbb{P}^n} \xrightarrow{\text{pr}_{1,3}^*p} \text{pr}_{1,3}^*\mathcal{O}_Z \to 0,$$ and $$ 0 \to \text{pr}_{2,3}^*\mathcal{I} \xrightarrow{\text{pr}_{2,3}^*q} \mathcal{O}_{H\times H\times \mathbb{P}^n} \xrightarrow{\text{pr}_{2,3}^*p} \text{pr}_{2,3}^*\mathcal{O}_Z \to 0.$$ Thus there is a composition, $$\text{pr}_{2,3}^*p\circ \text{pr}_{1,3}^*q:\text{pr}_{1,3}^*\mathcal{I} \to \text{pr}_{2,3}^*\mathcal{O}_Z.$$

Both the domain and target are quasi-coherent sheaves that are locally finitely presented. Moreover, the target is $H\times H$-flat and has proper support over $H\times H$. Thus, by Théorème 7.7.6 and Corollaire 7.7.8 of EGA IV, there is a quasi-coherent sheaf $\mathcal{N}$ on $H\times H$ that is locally finitely presented, as well as a homomorphism of quasi-coherent sheaves, $$ r:\mathcal{N}\to \mathcal{O}_{H\times H},$$ such that the zero scheme of $r$, as an $H\times H$-scheme, represents the functor of $H\times H$-schemes on which the pullback of $\text{pr}_{2,3}^*p\circ \text{pr}_{1,3}^*q$ is zero. Thus, the flag Hilbert scheme is the closed subscheme of $H\times H$ whose structure sheaf is $\text{Coker}(r)$.