The Stacks project

Lemma 107.7.3. There exist an open substack $\mathcal{C}\! \mathit{urves}^{DM} \subset \mathcal{C}\! \mathit{urves}$ with the following properties

  1. $\mathcal{C}\! \mathit{urves}^{DM} \subset \mathcal{C}\! \mathit{urves}$ is the maximal open substack which is DM,

  2. given a family of curves $X \to S$ the following are equivalent

    1. the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{DM}$,

    2. the group algebraic space $\mathit{Aut}_ S(X)$ is unramified over $S$,

  3. given $X$ a proper scheme over a field $k$ of dimension $\leq 1$ the following are equivalent

    1. the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{DM}$,

    2. $\mathit{Aut}(X)$ is geometrically reduced over $k$ and has dimension $0$,

    3. $\mathit{Aut}(X) \to \mathop{\mathrm{Spec}}(k)$ is unramified.

Proof. The existence of an open substack with property (1) is Morphisms of Stacks, Lemma 99.22.1. The points of this open substack are characterized by (3)(c) by Morphisms of Stacks, Lemma 99.22.2. The equivalence of (3)(b) and (3)(c) is the statement that an algebraic space $G$ which is locally of finite type, geometrically reduced, and of dimension $0$ over a field $k$, is unramified over $k$. First, $G$ is a scheme by Spaces over Fields, Lemma 70.9.1. Then we can take an affine open in $G$ and observe that it will be proper over $k$ and apply Varieties, Lemma 33.9.3. Minor details omitted.

Part (2) is true because (3) holds. Namely, the morphism $\mathit{Aut}_ S(X) \to S$ is locally of finite type. Thus we can check whether $\mathit{Aut}_ S(X) \to S$ is unramified at all points of $\mathit{Aut}_ S(X)$ by checking on fibres at points of the scheme $S$, see Morphisms of Spaces, Lemma 65.38.10. But after base change to a point of $S$ we fall back into the equivalence of (3)(a) and (3)(c). $\square$


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