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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