The Stacks project

Lemma 109.16.3. There exist an open substack $\mathcal{C}\! \mathit{urves}^{smooth, h0} \subset \mathcal{C}\! \mathit{urves}$ such that

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

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

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

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

   1. the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{smooth, h0}$,

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

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

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

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