Lemma 109.24.1 (0E8D)—The Stacks project

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Table of contents
Part 8: Topics in Moduli Theory
Chapter 109: Moduli of Curves
Section 109.24: Stable reduction theorem
Lemma 109.24.1 (cite)

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Lemma 109.24.1. Let $R$ be a discrete valuation ring with fraction field $K$ . Let $C$ be a smooth projective curve over $K$ with $K = H^0(C, \mathcal{O}_ C)$ having genus $g \geq 2$ . The following are equivalent

$C$ has semistable reduction (Semistable Reduction, Definition 55.14.6), or
there is a stable family of curves over $R$ with generic fibre $C$ .

Proof. Since a stable family of curves is also prestable, it is immediate that (2) implies (1). Conversely, given a prestable family of curves over $R$ with generic fibre $C$ , we can contract it to a stable family of curves by Lemma 109.23.4. Since the generic fibre already is stable, it does not get changed by this procedure and the proof is complete. $\square$

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