Lemma 107.24.1. 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 $C$.

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