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Final theorem on \bar{M}_g

At least for now. This finally accomplishes the original goal of the
Stacks project.
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aisejohan committed Aug 14, 2017
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  1. +62 −0 moduli-curves.tex
@@ -3356,6 +3356,68 @@ \section{Properties of the stack of stable curves}
as desired.
\end{proof}

\noindent
Here is the main theorem of this section.

\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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