Lemma 109.21.4. There is a decomposition into open and closed substacks
where each \mathcal{C}\! \mathit{urves}^{semistable}_ 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}^{semistable}_ g,
X \to S is a semistable 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 \mathcal{C}\! \mathit{urves}^{semistable}_ 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 tail,
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}^{\otimes m} is globally generated for m \geq 2.
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