The Stacks project

107.5 Properties of the stack of curves

The following lemma isn't true for moduli of surfaces, see Remark 107.5.2.

Lemma 107.5.1. The diagonal of $\mathcal{C}\! \mathit{urves}$ is separated and of finite presentation.

Proof. Recall that $\mathcal{C}\! \mathit{urves}$ is a limit preserving algebraic stack, see Quot, Lemma 97.15.6. By Limits of Stacks, Lemma 100.3.6 this implies that $\Delta : \mathcal{P}\! \mathit{olarized}\to \mathcal{P}\! \mathit{olarized}\times \mathcal{P}\! \mathit{olarized}$ is limit preserving. Hence $\Delta $ is locally of finite presentation by Limits of Stacks, Proposition 100.3.8.

Let us prove that $\Delta $ is separated. To see this, it suffices to show that given a scheme $U$ and two objects $Y \to U$ and $X \to U$ of $\mathcal{C}\! \mathit{urves}$ over $U$, the algebraic space

\[ \mathit{Isom}_ U(Y, X) \]

is separated. This we have seen in Moduli Stacks, Lemmas 106.10.2 and 106.10.3 that the target is a separated algebraic space.

To finish the proof we show that $\Delta $ is quasi-compact. Since $\Delta $ is representable by algebraic spaces, it suffices to check the base change of $\Delta $ by a surjective smooth morphism $U \to \mathcal{C}\! \mathit{urves}\times \mathcal{C}\! \mathit{urves}$ is quasi-compact (see for example Properties of Stacks, Lemma 98.3.3). We choose $U = \coprod U_ i$ to be a disjoint union of affine opens with a surjective smooth morphism

\[ U \longrightarrow \textit{PolarizedCurves} \times \textit{PolarizedCurves} \]

Then $U \to \mathcal{C}\! \mathit{urves}\times \mathcal{C}\! \mathit{urves}$ will be surjective and smooth since $\textit{PolarizedCurves} \to \mathcal{C}\! \mathit{urves}$ is surjective and smooth by Lemma 107.4.2. Since $\textit{PolarizedCurves}$ is limit preserving (by Artin's Axioms, Lemma 96.11.2 and Quot, Lemmas 97.15.6, 97.14.8, and 97.13.6), we see that $\textit{PolarizedCurves} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite presentation, hence $U_ i \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite presentation (Limits of Stacks, Proposition 100.3.8 and Morphisms of Stacks, Lemmas 99.26.2 and 99.32.5). In particular, $U_ i$ is Noetherian affine. This reduces us to the case discussed in the next paragraph.

In this paragraph, given a Noetherian affine scheme $U$ and two objects $(Y, \mathcal{N})$ and $(X, \mathcal{L})$ of $\textit{PolarizedCurves}$ over $U$, we show the algebraic space

\[ \mathit{Isom}_ U(Y, X) \]

is quasi-compact. Since the connected components of $U$ are open and closed we may replace $U$ by these. Thus we may and do assume $U$ is connected. Let $u \in U$ be a point. Let $Q$, $P$ be the Hilbert polynomials of these families, i.e.,

\[ Q(n) = \chi (Y_ u, \mathcal{N}_ u^{\otimes n}) \quad \text{and}\quad P(n) = \chi (X_ u, \mathcal{L}_ u^{\otimes n}) \]

see Varieties, Lemma 33.44.1. Since $U$ is connected and since the functions $u \mapsto \chi (Y_ u, \mathcal{N}_ u^{\otimes n})$ and $u \mapsto \chi (X_ u, \mathcal{L}_ u^{\otimes n})$ are locally constant (see Derived Categories of Schemes, Lemma 36.30.2) we see that we get the same Hilbert polynomial in every point of $U$. Set

\[ \mathcal{M} = \text{pr}_1^*\mathcal{N} \otimes _{\mathcal{O}_{Y \times _ U X}} \text{pr}_2^*\mathcal{L} \]

on $Y \times _ U X$. Given $(f, \varphi ) \in \mathit{Isom}_ U(Y, X)(T)$ for some scheme $T$ over $U$ then for every $t \in T$ we have

\begin{align*} \chi (Y_ t, (\text{id} \times f)^*\mathcal{M}^{\otimes n}) & = \chi (Y_ t, \mathcal{N}_ t^{\otimes n} \otimes _{\mathcal{O}_{Y_ t}} f_ t^*\mathcal{L}_ t^{\otimes n}) \\ & = n\deg (\mathcal{N}_ t) + n\deg (f_ t^*\mathcal{L}_ t) + \chi (Y_ t, \mathcal{O}_{Y_ t}) \\ & = Q(n) + n\deg (\mathcal{L}_ t) \\ & = Q(n) + P(n) - P(0) \end{align*}

by Riemann-Roch for proper curves, more precisely by Varieties, Definition 33.43.1 and Lemma 33.43.7 and the fact that $f_ t$ is an isomorphism. Setting $P'(t) = Q(t) + P(t) - P(0)$ we find

\[ \mathit{Isom}_ U(Y, X) = \mathit{Isom}_ U(Y, X) \cap \mathit{Mor}^{P', \mathcal{M}}_ U(Y, X) \]

The intersection is an intersection of open subspaces of $\mathit{Mor}_ U(Y, X)$, see Moduli Stacks, Lemma 106.10.3 and Remark 106.10.4. Now $\mathit{Mor}^{P', \mathcal{M}}_ U(Y, X)$ is a Noetherian algebraic space as it is of finite presentation over $U$ by Moduli Stacks, Lemma 106.10.5. Thus the intersection is a Noetherian algebraic space too and the proof is finished. $\square$

Remark 107.5.2. The boundedness argument in the proof of Lemma 107.5.1 does not work for moduli of surfaces and in fact, the result is wrong, for example because K3 surfaces over fields can have infinite discrete automorphism groups. The “reason” the argument does not work is that on a projective surface $S$ over a field, given ample invertible sheaves $\mathcal{N}$ and $\mathcal{L}$ with Hilbert polynomials $Q$ and $P$, there is no a priori bound on the Hilbert polynomial of $\mathcal{N} \otimes _{\mathcal{O}_ S} \mathcal{L}$. In terms of intersection theory, if $H_1$, $H_2$ are ample effective Cartier divisors on $S$, then there is no (upper) bound on the intersection number $H_1 \cdot H_2$ in terms of $H_1 \cdot H_1$ and $H_2 \cdot H_2$.

Lemma 107.5.3. The morphism $\mathcal{C}\! \mathit{urves}\to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is quasi-separated and locally of finite presentation.

Proof. To check $\mathcal{C}\! \mathit{urves}\to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is quasi-separated we have to show that its diagonal is quasi-compact and quasi-separated. This is immediate from Lemma 107.5.1. To prove that $\mathcal{C}\! \mathit{urves}\to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite presentation, it suffices to show that $\mathcal{C}\! \mathit{urves}$ is limit preserving, see Limits of Stacks, Proposition 100.3.8. This is Quot, Lemma 97.15.6. $\square$

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