Lemma 108.7.7. Let $f : X \to B$ be a proper morphism of finite presentation of algebraic spaces. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module ample on $X/B$, see Divisors on Spaces, Definition 71.14.1. The algebraic space $\mathrm{Hilb}^ P_{X/B}$ parametrizing closed subschemes having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is proper over $B$.
Proof. Recall that $\mathrm{Hilb}_{X/B} = \mathrm{Quot}_{\mathcal{O}_ X/X/B}$, see Quot, Lemma 99.9.2. Thus this lemma is an immediate consequence of Lemma 108.6.3. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$
). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)