Lemma 109.20.2 (0E6U)—The Stacks project

Table of contents
Part 8: Topics in Moduli Theory
Chapter 109: Moduli of Curves
Section 109.20: Prestable curves
Lemma 109.20.2 (cite)

Lemma 109.20.2. There exist an open substack $\mathcal{C}\! \mathit{urves}^{prestable} \subset \mathcal{C}\! \mathit{urves}$ such that

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 $\mathcal{C}\! \mathit{urves}^{prestable}$,
2. $X \to S$ is a prestable family of curves,
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 $\mathcal{C}\! \mathit{urves}^{prestable}$,
2. the singularities of $X$ are at-worst-nodal, $\dim (X) = 1$, and $k = H^0(X, \mathcal{O}_ X)$.

Proof. Given a family of curves $X \to S$ we see that it is prestable if and only if the classifying morphism factors both through $\mathcal{C}\! \mathit{urves}^{nodal}$ and $\mathcal{C}\! \mathit{urves}^{h0, 1}$. An alternative is to use $\mathcal{C}\! \mathit{urves}^{grc, 1}$ (since a nodal curve is geometrically reduced hence has $H^0$ equal to the ground field if and only if it is connected). In a formula

\[ \mathcal{C}\! \mathit{urves}^{prestable} = \mathcal{C}\! \mathit{urves}^{nodal} \cap \mathcal{C}\! \mathit{urves}^{h0, 1} = \mathcal{C}\! \mathit{urves}^{nodal} \cap \mathcal{C}\! \mathit{urves}^{grc, 1} \]

Thus the lemma follows from Lemmas 109.9.1 and 109.18.1. $\square$

The Stacks project

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