Lemma 54.5.4. Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is Noetherian, Nagata, and has dimension $2$. Let $Y$ be a proper scheme over $S$. Given an $S$-rational map $f : U \to Y$ from $X$ to $Y$ there exists a sequence

and an $S$-morphism $f_ n : X_ n \to Y$ such that $X_0 \to X$ is the normalization, $X_{i + 1} \to X_ i$ is the normalized blowing up of $X_ i$ at a closed point, and $f_ n$ and $f$ agree.

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