The Stacks project

Proof. By Resolution of Surfaces, Lemma 54.16.11 we see that $X$ is projective over $R$. Hence $X$ has an ample invertible sheaf by More on Morphisms, Lemma 37.50.1 (we will use this below). Let $E \subset X$ be an exceptional curve of the first kind. See Resolution of Surfaces, Section 54.16. By Resolution of Surfaces, Lemma 54.16.8 we can contract $E$ by a morphism $X \to X'$ such that $X'$ is regular and is projective over $R$. Clearly, the number of irreducible components of $X'_ k$ is exactly one less than the number of irreducible components of $X_ k$. Thus we can only perform a finite number of these contractions until we obtain a minimal model. $\square$