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The Stacks project

Lemma 109.23.5. Let $g \geq 2$. There is a morphism of algebraic stacks over $\mathbf{Z}$

\[ stabilization : \mathcal{C}\! \mathit{urves}^{prestable}_ g \longrightarrow \overline{\mathcal{M}}_ g \]

which sends a prestable family of curves $X \to S$ of genus $g$ to the stable family $Y \to S$ associated to it in Lemma 109.23.4.

Proof. To see this is true, it suffices to check that the construction of Lemma 109.23.4 is compatible with base change (and isomorphisms but that's immediate), see the (abuse of) language for algebraic stacks introduced in Properties of Stacks, Section 100.2. To see this it suffices to check properties (1) and (2) of Lemma 109.23.4 are stable under base change. This is immediately clear for (1). For (2) this follows either from the fact that the contractions of Algebraic Curves, Lemmas 53.22.6 and 53.23.6 are stable under ground field extensions, or because the conditions characterizing the morphisms on fibres in Algebraic Curves, Lemma 53.24.2 are preserved under ground field extensions. $\square$

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