The Stacks project

Lemma 55.8.3. Let $X$ be a model of a smooth curve $C$ over $K$. Then there exists a resolution of singularities of $X$ and any resolution is a model of $C$.

Proof. We check condition (4) of Lipman's theorem (Resolution of Surfaces, Theorem 54.14.5) hold. This is clear from Lemma 55.8.2 except for the statement that $X^\nu $ has finitely many singular points. To see this we can use that $R$ is J-2 by More on Algebra, Proposition 15.48.7 and hence the nonsingular locus is open in $X^\nu $. Since $X^\nu $ is normal of dimension $\leq 2$, the singular points are closed, hence closedness of the singular locus means there are finitely many of them (as $X$ is quasi-compact). Observe that any resolution of $X$ is a modification of $X$ (Resolution of Surfaces, Definition 54.14.1). This will be an isomorphism over the normal locus of $X$ by Varieties, Lemma 33.17.3. Since the set of normal points includes $C = X_ K$ we conclude any resolution is a model of $C$. $\square$

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