The Stacks project

Lemma 108.11.3. Let $n \geq 1$ be an integer and let $P$ be a numerical polynomial. Let

\[ T \subset |\mathcal{P}\! \mathit{olarized}| \]

be a subset with the following property: for every $\xi \in T$ there exists a field $k$ and an object $(X, \mathcal{L})$ of $\mathcal{P}\! \mathit{olarized}$ over $k$ representing $\xi $ such that

  1. the Hilbert polynomial of $\mathcal{L}$ on $X$ is $P$, and

  2. there exists a closed immersion $i : X \to \mathbf{P}^ n_ k$ such that $i^*\mathcal{O}_{\mathbf{P}^ n}(1) \cong \mathcal{L}$.

Then $T$ is a Noetherian topological space, in particular quasi-compact.

Proof. Observe that $|\mathcal{P}\! \mathit{olarized}|$ is a locally Noetherian topological space, see Morphisms of Stacks, Lemma 101.8.3 (this also uses that $\mathop{\mathrm{Spec}}(\mathbf{Z})$ is Noetherian and hence $\mathcal{P}\! \mathit{olarized}$ is a locally Noetherian algebraic stack by Lemma 108.11.2 and Morphisms of Stacks, Lemma 101.17.5). Thus any quasi-compact subset of $|\mathcal{P}\! \mathit{olarized}|$ is a Noetherian topological space and any subset of such is also Noetherian, see Topology, Lemmas 5.9.4 and 5.9.2. Thus all we have to do is a find a quasi-compact subset containing $T$.

By Lemma 108.7.7 the algebraic space

\[ H = \mathrm{Hilb}^{P, \mathcal{O}(1)}_{\mathbf{P}^ n_\mathbf {Z}/\mathop{\mathrm{Spec}}(\mathbf{Z})} \]

is proper over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. By Quot, Lemma 99.9.31 the identity morphism of $H$ corresponds to a closed subspace

\[ Z \subset \mathbf{P}^ n_ H \]

which is proper, flat, and of finite presentation over $H$ and such that the restriction $\mathcal{N} = \mathcal{O}(1)|_ Z$ is relatively ample on $Z/H$ and has Hilbert polynomial $P$ on the fibres of $Z \to H$. In particular, the pair $(Z \to H, \mathcal{N})$ defines a morphism

\[ H \longrightarrow \mathcal{P}\! \mathit{olarized} \]

which sends a morphism of schemes $U \to H$ to the classifying morphism of the family $(Z_ U \to U, \mathcal{N}_ U)$, see Quot, Lemma 99.14.4. Since $H$ is a Noetherian algebraic space (as it is proper over $\mathbf{Z})$) we see that $|H|$ is Noetherian and hence quasi-compact. The map

\[ |H| \longrightarrow |\mathcal{P}\! \mathit{olarized}| \]

is continuous, hence the image is quasi-compact. Thus it suffices to prove $T$ is contained in the image of $|H| \to |\mathcal{P}\! \mathit{olarized}|$. However, assumptions (1) and (2) exactly express the fact that this is the case: any choice of a closed immersion $i : X \to \mathbf{P}^ n_ k$ with $i^*\mathcal{O}_{\mathbf{P}^ n}(1) \cong \mathcal{L}$ we get a $k$-valued point of $H$ by the moduli interpretation of $H$. This finishes the proof of the lemma. $\square$

[1] We will see later (insert future reference here) that $H$ is a scheme and hence the use of this lemma and Quot, Lemma 99.14.4 isn't necessary.

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