Lemma 99.15.12. The 1-morphism (99.15.1.1)
\mathcal{C}\! \mathit{urves}\longrightarrow \mathcal{S}\! \mathit{paces}'_{fp, flat, proper}
is representable by open and closed immersions.
Proof. Since (99.15.1.1) 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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