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@@ -3356,6 +3356,68 @@ \section{Properties of the stack of stable curves} |
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as desired. |
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\end{proof} |
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\noindent |
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Here is the main theorem of this section. |
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\begin{theorem} |
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\label{theorem-stable-smooth-proper} |
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Let $g \geq 2$. The algebraic stack $\overline{\mathcal{M}}_g$ is a |
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Deligne-Mumford stack, proper and smooth over $\Spec(\mathbf{Z})$. |
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Moreover, the locus $\mathcal{M}_g$ parametrizing smooth curves |
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is a dense open substack. |
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\end{theorem} |
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\begin{proof} |
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Most of the properties mentioned in the statement have already been shown. |
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Smoothness is Lemma \ref{lemma-stable-curves-smooth}. |
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Deligne-Mumford is Lemma \ref{lemma-stable-curves-deligne-mumford}. |
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Openness of $\mathcal{M}_g$ is Lemma \ref{lemma-smooth-dense-in-stable}. |
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We know that $\overline{\mathcal{M}}_g \to \Spec(\mathbf{Z})$ |
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is separated by Lemma \ref{lemma-stable-separated} and we know that |
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$\overline{\mathcal{M}}_g$ is quasi-compact by |
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Lemma \ref{lemma-stable-quasi-compact}. |
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Thus, to show that $\overline{\mathcal{M}}_g \to \Spec(\mathbf{Z})$ |
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is proper and finish the proof, we may apply |
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More on Morphisms of Stacks, Lemma |
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\ref{stacks-more-morphisms-lemma-refined-valuative-criterion-proper} |
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to the morphisms $\mathcal{M}_g \to \overline{\mathcal{M}}_g$ and |
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$\overline{\mathcal{M}}_g \to \Spec(\mathbf{Z})$. |
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Thus it suffices to check the following: given any $2$-commutative diagram |
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$$ |
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\xymatrix{ |
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\Spec(K) \ar[r] \ar[d]_j & |
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\mathcal{M}_g \ar[r] & |
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\overline{\mathcal{M}}_g \ar[d] \\ |
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\Spec(A) \ar[rr] & & \Spec(\mathbf{Z}) |
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} |
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$$ |
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where $A$ is a discrete valuation ring with field of fractions $K$, there |
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exist an extension $K'/K$ of fields, a valuation ring $A' \subset K'$ |
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dominating $A$ such that the category of dotted arrows for the |
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induced diagram |
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$$ |
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\xymatrix{ |
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\Spec(K') \ar[r] \ar[d]_{j'} & \overline{\mathcal{M}}_g \ar[d] \\ |
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\Spec(A') \ar[r] \ar@{..>}[ru] & \Spec(\mathbf{Z}) |
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} |
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$$ |
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is nonempty (Morphisms of Stacks, Definition |
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\ref{stacks-morphisms-definition-fill-in-diagram}). |
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(Observe that we don't need to worry about |
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$2$-arrows too much, see Morphisms of Stacks, Lemma |
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\ref{stacks-morphisms-lemma-cat-dotted-arrows-independent}). |
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Unwinding what this means using that |
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$\mathcal{M}_g$, resp.\ $\overline{\mathcal{M}}_g$ are the algebraic |
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stacks parametrizing smooth, resp.\ stable families of genus $g$ curves, |
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we find that what we have to prove is exactly the result contained |
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in the stable reduction theorem, i.e., Theorem |
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\ref{theorem-stable-reduction}. |
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\end{proof} |
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