The Stacks project

Lemma 107.3.1. Let $T \to B$ be a morphism of algebraic spaces. The category

\[ \mathop{Mor}\nolimits _ B(T, B\text{-}\mathcal{C}\! \mathit{urves}) = \mathop{Mor}\nolimits (T, \mathcal{C}\! \mathit{urves}) \]

is the category of families of curves over $T$.

Proof. A family of curves over $T$ is a morphism $f : X \to T$ of algebraic spaces, which is flat, proper, of finite presentation, and has relative dimension $\leq 1$ (Morphisms of Spaces, Definition 65.33.2). This is exactly the same as the definition in Quot, Situation 97.15.1 except that $T$ the base is allowed to be an algebraic space. Our default base category for algebraic stacks/spaces is the category of schemes, hence the lemma does not follow immediately from the definitions. Having said this, we encourage the reader to skip the proof.

By the product description of $B\text{-}\mathcal{C}\! \mathit{urves}$ given above, it suffices to prove the lemma in the absolute case. Choose a scheme $U$ and a surjective étale morphism $p : U \to T$. Let $R = U \times _ T U$ with projections $s, t : R \to U$.

Let $v : T \to \mathcal{C}\! \mathit{urves}$ be a morphism. Then $v \circ p$ corresponds to a family of curves $X_ U \to U$. The canonical $2$-morphism $v \circ p \circ t \to v \circ p \circ s$ is an isomorphism $\varphi : X_ U \times _{U, s} R \to X_ U \times _{U, t} R$. This isomorphism satisfies the cocycle condition on $R \times _{s, t} R$. By Bootstrap, Lemma 78.11.3 we obtain a morphism of algebraic spaces $X \to T$ whose pullback to $U$ is equal to $X_ U$ compatible with $\varphi $. Since $\{ U \to T\} $ is an étale covering, we see that $X \to T$ is flat, proper, of finite presentation by Descent on Spaces, Lemmas 72.10.13, 72.10.19, and 72.10.12. Also $X \to T$ has relative dimension $\leq 1$ because this is an étale local property. Hence $X \to T$ is a family of curves over $T$.

Conversely, let $X \to T$ be a family of curves. Then the base change $X_ U$ determines a morphism $w : U \to \mathcal{C}\! \mathit{urves}$ and the canonical isomorphism $X_ U \times _{U, s} R \to X_ U \times _{U, t} R$ determines a $2$-arrow $w \circ s \to w \circ t$ satisfying the cocycle condition. Thus a morphism $v : T = [U/R] \to \mathcal{C}\! \mathit{urves}$ by the universal property of the quotient $[U/R]$, see Groupoids in Spaces, Lemma 76.22.2. (Actually, it is much easier in this case to go back to before we introduced our abuse of language and direct construct the functor $\mathit{Sch}/T \to \mathcal{C}\! \mathit{urves}$ which “is” the morphism $T \to \mathcal{C}\! \mathit{urves}$.)

We omit the verification that the constructions given above extend to morphisms between objects and are mutually quasi-inverse. $\square$

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