Lemma 108.16.3. There exist an open substack $\mathcal{C}\! \mathit{urves}^{smooth, h0} \subset \mathcal{C}\! \mathit{urves}$ 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 $\mathcal{C}\! \mathit{urves}^{smooth}$,

$f_*\mathcal{O}_ X = \mathcal{O}_ S$, this holds after any base change, and $f$ is smooth of relative dimension $1$,

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 $\mathcal{C}\! \mathit{urves}^{smooth, h0}$,

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

$X$ is smooth, $\dim (X) = 1$, and $X$ is geometrically connected,

$X$ is smooth, $\dim (X) = 1$, and $X$ is geometrically integral, and

$X_{\overline{k}}$ is a smooth curve.

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