The Stacks project

Lemma 99.15.12. The $1$-morphism (

\[ \mathcal{C}\! \mathit{urves}\longrightarrow \mathcal{S}\! \mathit{paces}'_{fp, flat, proper} \]

is representable by open and closed immersions.

Proof. Since ( is a fully faithful embedding of categories it suffices to show the following: given an object $X \to S$ of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ there exists an open and closed subscheme $U \subset S$ such that a morphism $S' \to S$ factors through $U$ if and only if the base change $X' \to S'$ of $X \to S$ has relative dimension $\leq 1$. This follows immediately from More on Morphisms of Spaces, Lemma 76.31.5. $\square$

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