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?

  • 1
    $\begingroup$ Previous post: mathoverflow.net/questions/152217/… $\endgroup$
    – S. Carnahan
    Commented Jan 2, 2014 at 19:03
  • $\begingroup$ @S.Carnahan: Sorry, I had not realized this was a repeated question when I answered. $\endgroup$ Commented Jan 2, 2014 at 19:20
  • $\begingroup$ @JasonStarr No problem. I had intended to work out the details eventually, but I did not have a ready answer. $\endgroup$
    – S. Carnahan
    Commented Jan 2, 2014 at 19:24

1 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 $\text{Coker}(r)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.