Lemma 107.11.3. There is a decomposition into open and closed substacks

where each $\mathcal{C}\! \mathit{urves}^{grc, 1}_ g$ is characterized as follows:

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

the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{grc, 1}_ g$,

the geometric fibres of the morphism $f : X \to S$ are reduced, connected, of dimension $1$ and $R^1f_*\mathcal{O}_ X$ is a locally free $\mathcal{O}_ S$-module of rank $g$,

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 $\mathcal{C}\! \mathit{urves}^{grc, 1}_ g$,

$X$ is geometrically reduced, geometrically connected, has dimension $1$, and has genus $g$.

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