The Stacks project

Lemma 109.21.3. There exist an open substack $\mathcal{C}\! \mathit{urves}^{semistable} \subset \mathcal{C}\! \mathit{urves}$ such that

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

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

   2. $X \to S$ is a semistable family of curves,

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

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

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

   3. the singularities of $X$ are at-worst-nodal, $\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, and $\omega _{X_ s}^{\otimes m}$ is globally generated for $m \geq 2$.