Lemma 55.10.2. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$ and genus $> 0$. Let $X$ be the minimal model for $C$ (Lemma 55.10.1). Let $Y$ be a regular proper model for $C$. Then there is a unique morphism of models $Y \to X$ which is a sequence of contractions of exceptional curves of the first kind.

Proof. The existence and properties of the morphism $X \to Y$ follows immediately from Lemma 55.8.5 and the uniqueness of the minimal model. The morphism $Y \to X$ is unique because $C \subset Y$ is scheme theoretically dense and $X$ is separated (see Morphisms, Lemma 29.7.10). $\square$

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