Lemma 107.9.1. There exist an open substack $\mathcal{C}\! \mathit{urves}^{h0, 1} \subset \mathcal{C}\! \mathit{urves}$ 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 $\mathcal{C}\! \mathit{urves}^{h0, 1}$,

$f_*\mathcal{O}_ X = \mathcal{O}_ S$, this holds after arbitrary base change, and the fibres of $f$ 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 $\mathcal{C}\! \mathit{urves}^{h0, 1}$,

$H^0(X, \mathcal{O}_ X) = k$ and $\dim (X) = 1$.

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