The Stacks project

Lemma 99.15.2. The category $\mathcal{C}\! \mathit{urves}$ is fibred in groupoids over $\mathit{Sch}_{fppf}$.

Proof. Using the embedding (, the description of the image, and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.1) this reduces to the following statement: Given a morphism

\[ \xymatrix{ X' \ar[r] \ar[d] & X \ar[d] \\ S' \ar[r] & S } \]

in $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (recall that this implies in particular the diagram is cartesian) if $X \to S$ has relative dimension $\leq 1$, then $X' \to S'$ has relative dimension $\leq 1$. This follows from Morphisms of Spaces, Lemma 67.34.3. $\square$

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