Lemma 109.22.3 (0E76)—The Stacks project

Lemma 109.22.3. There exist an open substack $\mathcal{C}\! \mathit{urves}^{stable} \subset \mathcal{C}\! \mathit{urves}$ such that

given a family of curves $f : X \to S$ the following are equivalent
1. the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{stable}$,
2. $X \to S$ is a stable family of curves,
given $X$ a scheme proper over a field $k$ with $\dim (X) \leq 1$ the following are equivalent
1. the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{stable}$,
2. the singularities of $X$ 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,
3. the singularities of $X$ are at-worst-nodal, $\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, and $\omega _{X_ s}$ is ample.

Proof. By the discussion in Section 109.6 it suffices to look at families $f : X \to S$ of prestable curves. By Lemma 109.22.1 we obtain the desired openness of the locus in question. Formation of this open commutes with arbitrary base change, either because the (non)existence of rational tails or bridges is insensitive to ground field extensions by Algebraic Curves, Lemmas 53.22.6 and 53.23.6 or because ampleness is insenstive to base field extensions by Descent, Lemma 35.25.6. $\square$

The Stacks project

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