Lemma 54.17.2. Let $S$ be a Noetherian scheme. Let $X$ and $Y$ be proper integral schemes over $S$ which are regular of dimension $2$. Then $X$ and $Y$ are $S$-birational if and only if there exists a diagram of $S$-morphisms

$X = X_0 \leftarrow X_1 \leftarrow \ldots \leftarrow X_ n = Y_ m \to \ldots \to Y_1 \to Y_0 = Y$

where each morphism is a blowup in a closed point.

Proof. Let $U \subset X$ be open and let $f : U \to Y$ be the given $S$-rational map (which is invertible as an $S$-rational map). By Lemma 54.4.3 we can factor $f$ as $X_ n \to \ldots \to X_1 \to X_0 = X$ and $f_ n : X_ n \to Y$. Since $X_ n$ is proper over $S$ and $Y$ separated over $S$ the morphism $f_ n$ is proper. Clearly $f_ n$ is birational. Hence $f_ n$ is a composition of contractions by Lemma 54.17.1. We omit the proof of the converse. $\square$

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