The Stacks project

Lemma 109.22.7. The stacks $\overline{\mathcal{M}}$ and $\overline{\mathcal{M}}_ g$ are open substacks of $\mathcal{C}\! \mathit{urves}^{DM}$. In particular, $\overline{\mathcal{M}}$ and $\overline{\mathcal{M}}_ g$ are DM (Morphisms of Stacks, Definition 101.4.2) as well as Deligne-Mumford stacks (Algebraic Stacks, Definition 94.12.2).

Proof. Proof of the first assertion. Let $X$ be a scheme proper over a field $k$ whose singularities are at-worst-nodal, $\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, the genus of $X$ is $\geq 2$, and $X$ has no rational tails or bridges. We have to show that the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \overline{\mathcal{M}} \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{DM}$. We may first replace $k$ by the algebraic closure (since we already know the relevant stacks are open substacks of the algebraic stack $\mathcal{C}\! \mathit{urves}$). By Lemmas 109.22.3, 109.7.3, and 109.7.4 it suffices to show that $\text{Der}_ k(\mathcal{O}_ X, \mathcal{O}_ X) = 0$. This is proven in Algebraic Curves, Lemma 53.25.3.

Since $\mathcal{C}\! \mathit{urves}^{DM}$ is the maximal open substack of $\mathcal{C}\! \mathit{urves}$ which is DM, we see this is true also for the open substack $\overline{\mathcal{M}}$ of $\mathcal{C}\! \mathit{urves}^{DM}$. Finally, a DM algebraic stack is Deligne-Mumford by Morphisms of Stacks, Theorem 101.21.6. $\square$

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