Lemma 107.11.1. There exist an open substack $\mathcal{C}\! \mathit{urves}^{grc, 1} \subset \mathcal{C}\! \mathit{urves}$ 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 $\mathcal{C}\! \mathit{urves}^{grc, 1}$,

the geometric fibres of the morphism $X \to S$ are reduced, connected, and have dimension $1$,

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 $\mathcal{C}\! \mathit{urves}^{grc, 1}$,

$X$ is geometrically reduced, geometrically connected, and has dimension $1$.

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