Lemma 55.14.7. Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$. The following are equivalent

there exists a proper smooth model for $C$,

there exists a minimal model for $C$ which is smooth over $R$,

any minimal model is smooth over $R$.

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