The Stacks project

Situation 99.15.1. We define a category $\mathcal{C}\! \mathit{urves}$ as follows:

  1. Objects are families of curves. More precisely, an object is a morphism $f : X \to S$ where the base $S$ is a scheme, the total space $X$ is an algebraic space, and $f$ is flat, proper, of finite presentation, and has relative dimension $\leq 1$ (Morphisms of Spaces, Definition 67.33.2).

  2. A morphism $(X' \to S') \to (X \to S)$ between objects is given by a pair $(f, g)$ where $f : X' \to X$ is a morphism of algebraic spaces and $g : S' \to S$ is a morphism of schemes which fit into a commutative diagram

    \[ \xymatrix{ X' \ar[d] \ar[r]_ f & X \ar[d] \\ S' \ar[r]^ g & S } \]

    inducing an isomorphism $X' \to S' \times _ S X$, in other words, the diagram is cartesian.

The forgetful functor

\[ p : \mathcal{C}\! \mathit{urves}\longrightarrow \mathit{Sch}_{fppf},\quad (X \to S) \longmapsto S \]

is how we view $\mathcal{C}\! \mathit{urves}$ as a category over $\mathit{Sch}_{fppf}$ (see Section 99.2 for notation).

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