Lemma 107.20.2. There exist an open substack $\mathcal{C}\! \mathit{urves}^{prestable} \subset \mathcal{C}\! \mathit{urves}$ such that

given a family of curves $f : X \to S$ the following are equivalent

the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{prestable}$,

$X \to S$ is a prestable family of curves,

given $X$ a scheme proper over a field $k$ with $\dim (X) \leq 1$ the following are equivalent

the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{prestable}$,

the singularities of $X$ are at-worst-nodal, $\dim (X) = 1$, and $k = H^0(X, \mathcal{O}_ X)$.

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