Lemma 109.22.1 (0E74)—The Stacks project

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Part 8: Topics in Moduli Theory
Chapter 109: Moduli of Curves
Section 109.22: Stable curves
Lemma 109.22.1 (cite)

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Lemma 109.22.1. Let $f : X \to S$ be a prestable family of curves of genus $g \geq 2$. Let $s \in S$ be a point of the base scheme. The following are equivalent

$X_ s$ does not have a rational tail and does not have a rational bridge (Algebraic Curves, Examples 53.22.1 and 53.23.1), and
$\omega _{X/S}$ is ample on $f^{-1}(U)$ for some $s \in U \subset S$ open.

Proof. Assume (2). Then $\omega _{X_ s}$ is ample on $X_ s$. By Algebraic Curves, Lemmas 53.22.2 and 53.23.2 we conclude that (1) holds (we also use the characterization of ample invertible sheaves in Varieties, Lemma 33.44.15).

Assume (1). Then $\omega _{X_ s}$ is ample on $X_ s$ by Algebraic Curves, Lemmas 53.23.6. We conclude by Descent on Spaces, Lemma 74.13.2. $\square$

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