The Stacks project

Lemma 108.16.3. 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.

Proof. If we set

\[ \mathcal{C}\! \mathit{urves}^{smooth, h0} = \mathcal{C}\! \mathit{urves}^{smooth} \cap \mathcal{C}\! \mathit{urves}^{h0, 1} \]

then we see that (1) holds by Lemmas 108.9.1 and 108.16.1. In fact, this also gives the equivalence of (2)(a) and (2)(b). To finish the proof we have to show that (2)(b) is equivalent to each of (2)(c), (2)(d), and (2)(e).

A smooth scheme over a field is geometrically normal (Varieties, Lemma 33.25.4), smoothness is preserved under base change (Morphisms, Lemma 29.34.5), and being smooth is fpqc local on the target (Descent, Lemma 35.23.27). Keeping this in mind, the equivalence of (2)(b), (2)(c), 2(d), and (2)(e) follows from Varieties, Lemma 33.10.7. $\square$

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