Lemma 107.22.5. There is a decomposition into open and closed substacks

where each $\overline{\mathcal{M}}_ g$ 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 $\overline{\mathcal{M}}_ g$,

$X \to S$ is a stable family of curves and $R^1f_*\mathcal{O}_ X$ is a locally free $\mathcal{O}_ S$-module of rank $g$,

given $X$ a scheme 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 $\overline{\mathcal{M}}_ g$,

the singularities of $X$ are at-worst-nodal, $\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, the genus of $X$ is $g$, and $X$ has no rational tails or bridges.

the singularities of $X$ are at-worst-nodal, $\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, the genus of $X$ is $g$, and $\omega _{X_ s}$ is ample.

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