The Stacks project

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

  1. there exists a proper smooth model for $C$,

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

  3. any minimal model is smooth over $R$.

Proof. If $X$ is a smooth proper model, then the special fibre is connected (Lemma 55.9.4) and smooth, hence irreducible. This immediately implies that it is minimal. Thus (1) implies (2). To finish the proof we have to show that (2) implies (3). This is clear if the genus of $C$ is $> 0$, since then the minimal model is unique (Lemma 55.10.1). On the other hand, if the minimal model is not unique, then the morphism $X \to \mathop{\mathrm{Spec}}(R)$ is smooth for any minimal model as its special fibre will be isomorphic to $\mathbf{P}^1_ k$ by Lemma 55.12.4. $\square$

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