\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*}

The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0E9C. Beware of the difference between the letter 'O' and the digit '0'.