Theorem 109.25.3. 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 109.22.6. Deligne-Mumford is Lemma 109.22.7. Openness of $\mathcal{M}_ g$ is Lemma 109.22.8. We know that $\overline{\mathcal{M}}_ g \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated by Lemma 109.25.1 and we know that $\overline{\mathcal{M}}_ g$ is quasi-compact by Lemma 109.25.2. 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 106.11.3 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 101.39.1). (Observe that we don't need to worry about $2$-arrows too much, see Morphisms of Stacks, Lemma 101.39.3). 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 109.24.3. $\square$
Comments (0)