A regular proper model of a curve is obtained by successive blowups from a minimal model

Lemma 55.8.5. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$. If $X$ is a regular proper model for $C$, then there exists a sequence of morphisms

$X = X_ m \to X_{m - 1} \to \ldots \to X_1 \to X_0$

of proper regular models of $C$, such that each morphism is a contraction of an exceptional curve of the first kind, and such that $X_0$ is a minimal model.

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. $\square$

Comment #2114 by Matthew Emerton on

Suggested slogan: A regular proper model of a curve is obtained by successive blow-ups from a minimal model

Comment #2120 by Ariyan on

Typos: "squence" --> "sequence"

Comment #2135 by on

@#2114: Thanks for the slogans

@#2120: Thanks for the typo

See here.

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