The Stacks Project


Tag 0E9C

Chapter 99: Moduli of Curves > Section 99.25: Properties of the stack of stable curves

Theorem 99.25.3. Let $g \geq 2$. The algebraic stack $\overline{\mathcal{M}}_g$ is a Deligne-Mumford stack, proper and smooth over $\mathop{\rm Spec}(\mathbf{Z})$. 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 99.22.6. Deligne-Mumford is Lemma 99.22.7. Openness of $\mathcal{M}_g$ is Lemma 99.22.8. We know that $\overline{\mathcal{M}}_g \to \mathop{\rm Spec}(\mathbf{Z})$ is separated by Lemma 99.25.1 and we know that $\overline{\mathcal{M}}_g$ is quasi-compact by Lemma 99.25.2. Thus, to show that $\overline{\mathcal{M}}_g \to \mathop{\rm Spec}(\mathbf{Z})$ is proper and finish the proof, we may apply More on Morphisms of Stacks, Lemma 96.11.2 to the morphisms $\mathcal{M}_g \to \overline{\mathcal{M}}_g$ and $\overline{\mathcal{M}}_g \to \mathop{\rm Spec}(\mathbf{Z})$. Thus it suffices to check the following: given any $2$-commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(K) \ar[r] \ar[d]_j & \mathcal{M}_g \ar[r] & \overline{\mathcal{M}}_g \ar[d] \\ \mathop{\rm Spec}(A) \ar[rr] & & \mathop{\rm Spec}(\mathbf{Z}) } $$ 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 $$ \xymatrix{ \mathop{\rm Spec}(K') \ar[r] \ar[d]_{j'} & \overline{\mathcal{M}}_g \ar[d] \\ \mathop{\rm Spec}(A') \ar[r] \ar@{..>}[ru] & \mathop{\rm Spec}(\mathbf{Z}) } $$ is nonempty (Morphisms of Stacks, Definition 91.38.1). (Observe that we don't need to worry about $2$-arrows too much, see Morphisms of Stacks, Lemma 91.38.3). 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 99.24.3. $\square$

    The code snippet corresponding to this tag is a part of the file moduli-curves.tex and is located in lines 3362–3368 (see updates for more information).

    \begin{theorem}
    \label{theorem-stable-smooth-proper}
    Let $g \geq 2$. The algebraic stack $\overline{\mathcal{M}}_g$ is a
    Deligne-Mumford stack, proper and smooth over $\Spec(\mathbf{Z})$.
    Moreover, the locus $\mathcal{M}_g$ parametrizing smooth curves
    is a dense open substack.
    \end{theorem}
    
    \begin{proof}
    Most of the properties mentioned in the statement have already been shown.
    Smoothness is Lemma \ref{lemma-stable-curves-smooth}.
    Deligne-Mumford is Lemma \ref{lemma-stable-curves-deligne-mumford}.
    Openness of $\mathcal{M}_g$ is Lemma \ref{lemma-smooth-dense-in-stable}.
    We know that $\overline{\mathcal{M}}_g \to \Spec(\mathbf{Z})$
    is separated by Lemma \ref{lemma-stable-separated} and we know that
    $\overline{\mathcal{M}}_g$ is quasi-compact by
    Lemma \ref{lemma-stable-quasi-compact}.
    Thus, to show that $\overline{\mathcal{M}}_g \to \Spec(\mathbf{Z})$
    is proper and finish the proof, we may apply
    More on Morphisms of Stacks, Lemma
    \ref{stacks-more-morphisms-lemma-refined-valuative-criterion-proper}
    to the morphisms $\mathcal{M}_g \to \overline{\mathcal{M}}_g$ and
    $\overline{\mathcal{M}}_g \to \Spec(\mathbf{Z})$.
    Thus it suffices to check the following: given any $2$-commutative diagram
    $$
    \xymatrix{
    \Spec(K) \ar[r] \ar[d]_j &
    \mathcal{M}_g \ar[r] &
    \overline{\mathcal{M}}_g \ar[d] \\
    \Spec(A) \ar[rr] & & \Spec(\mathbf{Z})
    }
    $$
    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
    $$
    \xymatrix{
    \Spec(K') \ar[r] \ar[d]_{j'} & \overline{\mathcal{M}}_g \ar[d] \\
    \Spec(A') \ar[r] \ar@{..>}[ru] & \Spec(\mathbf{Z})
    }
    $$
    is nonempty (Morphisms of Stacks, Definition
    \ref{stacks-morphisms-definition-fill-in-diagram}).
    (Observe that we don't need to worry about
    $2$-arrows too much, see Morphisms of Stacks, Lemma
    \ref{stacks-morphisms-lemma-cat-dotted-arrows-independent}).
    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
    \ref{theorem-stable-reduction}.
    \end{proof}

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