Lemma 109.22.8 (0E87)—The Stacks project

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Table of contents
Part 8: Topics in Moduli Theory
Chapter 109: Moduli of Curves
Section 109.22: Stable curves
Lemma 109.22.8 (cite)

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Lemma 109.22.8. Let $g \geq 2$ . The inclusion

$|\mathcal{M}_ g| \subset |\overline{\mathcal{M}}_ g|$

is that of an open dense subset.

Proof. Since $\overline{\mathcal{M}}_ g \subset \mathcal{C}\! \mathit{urves}^{lci+}$ is open and since $\mathcal{C}\! \mathit{urves}^{smooth} \cap \overline{\mathcal{M}}_ g = \mathcal{M}_ g$ this follows immediately from Lemma 109.17.1. $\square$

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