The Stacks project

Lemma 54.4.3. Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is regular 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

\[ X_ n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X \]

and an $S$-morphism $f_ n : X_ n \to Y$ such that $X_{i + 1} \to X_ i$ blowing up of $X_ i$ at a closed point not lying over $U$ and $f_ n$ and $f$ agree.

Proof. We may assume $U$ contains every point of codimension $1$, see Morphisms, Lemma 29.42.5. Hence the complement $T \subset X$ of $U$ is a finite set of closed points whose local rings are regular of dimension $2$. Applying Divisors, Lemma 31.36.2 we find a proper morphism $p : X' \to X$ which is an isomorphism over $U$ and a morphism $f' : X' \to Y$ agreeing with $f$ over $U$. Apply Lemma 54.4.2 to the morphism $p : X' \to X$. The composition $X_ n \to X' \to Y$ is the desired morphism. $\square$


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