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

and it suffices to check the following: given any $2$-commutative diagram

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

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

the image of the set $T$ is exactly the open subset

Thus it suffices to show that $T$ is quasi-compact. By Lemma 109.4.1 we see that

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

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

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

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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