Lemma 89.5.4. Let $S$ be a scheme. Let $X$ be a Noetherian Nagata algebraic space over $S$ with $\dim (X) = 2$. Let $f : Y \to X$ be a proper birational morphism. Then there exists a commutative diagram
where $X_0 \to X$ is the normalization and where $X_{i + 1} \to X_ i$ is the normalized blowing up of $X_ i$ at a closed point.
Proof. Although one can prove this lemma directly for algebraic spaces, we will continue the approach used above to reduce it to the case of schemes.
We will use that Noetherian algebraic spaces are quasi-separated and hence points have well defined residue fields (for example by Decent Spaces, Lemma 68.11.4). We will use the results of Morphisms of Spaces, Sections 67.26, 67.35, and 67.49 without further mention. We may replace $Y$ by its normalization. Let $X_0 \to X$ be the normalization. The morphism $Y \to X$ factors through $X_0$. Thus we may assume that both $X$ and $Y$ are normal.
Assume $X$ and $Y$ are normal. The morphism $f : Y \to X$ is an isomorphism over an open which contains every point of codimension $0$ and $1$ in $Y$ and every point of $Y$ over which the fibre is finite, see Spaces over Fields, Lemma 72.3.3. Hence we see that there is a finite set of closed points $T \subset |X|$ such that $f$ is an isomorphism over $X \setminus T$. By More on Morphisms of Spaces, Lemma 76.39.5 there exists an $X \setminus T$-admissible blowup $Y' \to X$ which dominates $Y$. After replacing $Y$ by the normalization of $Y'$ we see that we may assume that $Y \to X$ is representable.
Say $T = \{ x_1, \ldots , x_ r\} $. Pick elementary étale neighbourhoods $(U_ i, u_ i) \to (X, x_ i)$ as in Section 89.3. For each $i$ the morphism $Y_ i = Y \times _ X U_ i \to U_ i$ is a proper birational morphism which is an isomorphism over $U_ i \setminus \{ u_ i\} $. Thus we may apply Resolution of Surfaces, Lemma 54.5.3 to find a sequence
of normalized blowing ups in closed points lying over $u_ i$ such that $X_{i, m_ i}$ dominates $Y_ i$. By Lemma 89.5.3 we find a sequence of normalized blowing ups
as in the statement of the lemma whose base change to our $U_ i$ produces the given sequences. It follows that $X_ m$ dominates $Y$ by the equivalence of categories of Lemma 89.3.1. $\square$
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