The Stacks project

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

  1. $C$ has semistable reduction (Semistable Reduction, Definition 55.14.6), or

  2. 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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