The Stacks project

Proposition 55.8.6. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$. A minimal model exists.

Proof. Choose a closed immersion $C \to \mathbf{P}^ n_ K$. Let $X$ be the scheme theoretic image of $C \to \mathbf{P}^ n_ R$. Then $X \to \mathop{\mathrm{Spec}}(R)$ is a projective model of $C$ by Lemma 55.8.1. By Lemma 55.8.3 there exists a resolution of singularities $X' \to X$ and $X'$ is a model for $C$. Then $X' \to \mathop{\mathrm{Spec}}(R)$ is proper as a composition of proper morphisms. Then we may apply Lemma 55.8.5 to obtain a minimal model. $\square$

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