Lemma 109.25.1. Let $g \geq 2$. The stack $\overline{\mathcal{M}}_ g$ is separated.
109.25 Properties of the stack of stable curves
In this section we prove the basic structure result for $\overline{\mathcal{M}}_ g$ for $g \geq 2$.
Proof. The statement means that the morphism $\overline{\mathcal{M}}_ g \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated. We will prove this using the refined Noetherian valuative criterion as stated in More on Morphisms of Stacks, Lemma 106.11.4
Since $\overline{\mathcal{M}}_ g$ is an open substack of $\mathcal{C}\! \mathit{urves}$, we see $\overline{\mathcal{M}}_ g \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is quasi-separated and locally of finite presentation by Lemma 109.5.3. In particular the stack $\overline{\mathcal{M}}_ g$ is locally Noetherian (Morphisms of Stacks, Lemma 101.17.5). By Lemma 109.22.8 the open immersion $\mathcal{M}_ g \to \overline{\mathcal{M}}_ g$ has dense image. Also, $\mathcal{M}_ g \to \overline{\mathcal{M}}_ g$ is quasi-compact (Morphisms of Stacks, Lemma 101.8.2), hence of finite type. Thus all the preliminary assumptions of More on Morphisms of Stacks, Lemma 106.11.4 are satisfied for the morphisms
\[ \mathcal{M}_ g \to \overline{\mathcal{M}}_ g \quad \text{and}\quad \overline{\mathcal{M}}_ g \to \mathop{\mathrm{Spec}}(\mathbf{Z}) \]
and it suffices to check the following: given any $2$-commutative diagram
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & \mathcal{M}_ g \ar[r] & \overline{\mathcal{M}}_ g \ar[d] \\ \mathop{\mathrm{Spec}}(R) \ar[rr] \ar@{..>}[rru] & & \mathop{\mathrm{Spec}}(\mathbf{Z}) } \]
where $R$ is a discrete valuation ring with field of fractions $K$ the category of dotted arrows is either empty or a setoid with exactly one isomorphism class. (Observe that we don't need to worry about $2$-arrows too much, see Morphisms of Stacks, Lemma 101.39.3). Unwinding what this means using that $\mathcal{M}_ g$, resp. $\overline{\mathcal{M}}_ g$ are the algebraic stacks parametrizing smooth, resp. stable families of genus $g$ curves, we find that what we have to prove is exactly the uniqueness result stated and proved in Lemma 109.24.2. $\square$
Lemma 109.25.2. Let $g \geq 2$. The stack $\overline{\mathcal{M}}_ g$ is quasi-compact.
Proof. We will use the notation from Section 109.4. Consider the subset
\[ T \subset |\textit{PolarizedCurves}| \]
of points $\xi $ such that there exists a field $k$ and a pair $(X, \mathcal{L})$ over $k$ representing $\xi $ with the following two properties
$X$ is a stable genus $g$ curve, and
$\mathcal{L} = \omega _ X^{\otimes 3}$.
Clearly, under the continuous map
\[ |\textit{PolarizedCurves}| \longrightarrow |\mathcal{C}\! \mathit{urves}| \]
the image of the set $T$ is exactly the open subset
\[ |\overline{\mathcal{M}}_ g| \subset |\mathcal{C}\! \mathit{urves}| \]
Thus it suffices to show that $T$ is quasi-compact. By Lemma 109.4.1 we see that
\[ |\textit{PolarizedCurves}| \subset |\mathcal{P}\! \mathit{olarized}| \]
is an open and closed immersion. Thus it suffices to prove quasi-compactness of $T$ as a subset of $|\mathcal{P}\! \mathit{olarized}|$. For this we use the criterion of Moduli Stacks, Lemma 108.11.3. First, we observe that for $(X, \mathcal{L})$ as above the Hilbert polynomial $P$ is the function $P(t) = (6g - 6)t + (1 - g)$ by Riemann-Roch, see Algebraic Curves, Lemma 53.5.2. Next, we observe that $H^1(X, \mathcal{L}) = 0$ and $\mathcal{L}$ is very ample by Algebraic Curves, Lemma 53.24.3. This means exactly that with $n = P(3) - 1$ there is a closed immersion
\[ i : X \longrightarrow \mathbf{P}^ n_ k \]
such that $\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^1_ k}(1)$ as desired. $\square$
Here is the main theorem of this section.
Theorem 109.25.3. Let $g \geq 2$. The algebraic stack $\overline{\mathcal{M}}_ g$ is a Deligne-Mumford stack, proper and smooth over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Moreover, the locus $\mathcal{M}_ g$ parametrizing smooth curves is a dense open substack.
Proof. Most of the properties mentioned in the statement have already been shown. Smoothness is Lemma 109.22.6. Deligne-Mumford is Lemma 109.22.7. Openness of $\mathcal{M}_ g$ is Lemma 109.22.8. We know that $\overline{\mathcal{M}}_ g \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated by Lemma 109.25.1 and we know that $\overline{\mathcal{M}}_ g$ is quasi-compact by Lemma 109.25.2. Thus, to show that $\overline{\mathcal{M}}_ g \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is proper and finish the proof, we may apply More on Morphisms of Stacks, Lemma 106.11.3 to the morphisms $\mathcal{M}_ g \to \overline{\mathcal{M}}_ g$ and $\overline{\mathcal{M}}_ g \to \mathop{\mathrm{Spec}}(\mathbf{Z})$. Thus it suffices to check the following: given any $2$-commutative diagram
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d]_ j & \mathcal{M}_ g \ar[r] & \overline{\mathcal{M}}_ g \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[rr] & & \mathop{\mathrm{Spec}}(\mathbf{Z}) } \]
where $A$ is a discrete valuation ring with field of fractions $K$, there exist an extension $K'/K$ of fields, a valuation ring $A' \subset K'$ dominating $A$ such that the category of dotted arrows for the induced diagram
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K') \ar[r] \ar[d]_{j'} & \overline{\mathcal{M}}_ g \ar[d] \\ \mathop{\mathrm{Spec}}(A') \ar[r] \ar@{..>}[ru] & \mathop{\mathrm{Spec}}(\mathbf{Z}) } \]
is nonempty (Morphisms of Stacks, Definition 101.39.1). (Observe that we don't need to worry about $2$-arrows too much, see Morphisms of Stacks, Lemma 101.39.3). Unwinding what this means using that $\mathcal{M}_ g$, resp. $\overline{\mathcal{M}}_ g$ are the algebraic stacks parametrizing smooth, resp. stable families of genus $g$ curves, we find that what we have to prove is exactly the result contained in the stable reduction theorem, i.e., Theorem 109.24.3. $\square$
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