The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Theorem 101.25.3. 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 101.22.6. Deligne-Mumford is Lemma 101.22.7. Openness of $\mathcal{M}_ g$ is Lemma 101.22.8. We know that $\overline{\mathcal{M}}_ g \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated by Lemma 101.25.1 and we know that $\overline{\mathcal{M}}_ g$ is quasi-compact by Lemma 101.25.2. 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 98.11.2 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 93.38.1). (Observe that we don't need to worry about $2$-arrows too much, see Morphisms of Stacks, Lemma 93.38.3). 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 101.24.3. $\square$


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