The Stacks project

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

    2. $f_*\mathcal{O}_ X = \mathcal{O}_ S$, this holds after arbitrary base change, and the fibres of $f$ have dimension $1$,

  2. given a scheme $X$ 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 $\mathcal{C}\! \mathit{urves}^{h0, 1}$,

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

Proof. Given a family of curves $X \to S$ the set of $s \in S$ where $\kappa (s) = H^0(X_ s, \mathcal{O}_{X_ s})$ is open in $S$ by Derived Categories of Spaces, Lemma 73.26.2. Also, the set of points in $S$ where the fibre has dimension $1$ is open by More on Morphisms of Spaces, Lemma 74.31.5. Moreover, if $f : X \to S$ is a family of curves all of whose fibres have dimension $1$ (and in particular $f$ is surjective), then condition (1)(b) is equivalent to $\kappa (s) = H^0(X_ s, \mathcal{O}_{X_ s})$ for every $s \in S$, see Derived Categories of Spaces, Lemma 73.26.7. Thus we see that the lemma follows from the general discussion in Section 107.6. $\square$


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