The Stacks project

Lemma 109.20.3. There is a decomposition into open and closed substacks

\[ \mathcal{C}\! \mathit{urves}^{prestable} = \coprod \nolimits _{g \geq 0} \mathcal{C}\! \mathit{urves}^{prestable}_ g \]

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

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

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

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

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

Proof. Since we have seen that $\mathcal{C}\! \mathit{urves}^{prestable}$ is contained in $\mathcal{C}\! \mathit{urves}^{h0, 1}$, this follows from Lemmas 109.20.2 and 109.9.4. $\square$

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