Lemma 107.10.1. There exist an open substack $\mathcal{C}\! \mathit{urves}^{geomred} \subset \mathcal{C}\! \mathit{urves}$ such that

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

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

the fibres of the morphism $X \to S$ are geometrically reduced (More on Morphisms of Spaces, Definition 74.29.2),

given a scheme $X$ proper over a field $k$ with $\dim (X) \leq 1$ the following are equivalent

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

$X$ is geometrically reduced over $k$.

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