Lemma 107.8.2. There exist an open substack $\mathcal{C}\! \mathit{urves}^{CM, 1} \subset \mathcal{C}\! \mathit{urves}$ such that

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

the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{CM, 1}$,

the morphism $X \to S$ is Cohen-Macaulay and has relative dimension $1$ (Morphisms of Spaces, Definition 65.33.2),

given a scheme $X$ 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 $\mathcal{C}\! \mathit{urves}^{CM, 1}$,

$X$ is Cohen-Macaulay and $X$ is equidimensional of dimension $1$.

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