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

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

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

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

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

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

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

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

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

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