Lemma 54.17.1. Let $f : X \to Y$ be a proper birational morphism between integral Noetherian schemes regular of dimension $2$. Then $f$ is a sequence of blowups in closed points.
54.17 Factorization birational maps
Proper birational morphisms between nonsingular surfaces are given by sequences of quadratic transforms.
Proof. Let $V \subset Y$ be the maximal open over which $f$ is an isomorphism. Then $V$ contains all codimension $1$ points of $V$ (Varieties, Lemma 33.17.3). Let $y \in Y$ be a closed point not contained in $V$. Then we want to show that $f$ factors through the blowup $b : Y' \to Y$ of $Y$ at $y$. Namely, if this is true, then at least one (and in fact exactly one) component of the fibre $f^{-1}(y)$ will map isomorphically onto the exceptional curve in $Y'$ and the number of curves in fibres of $X \to Y'$ will be strictly less that the number of curves in fibres of $X \to Y$, so we conclude by induction. Some details omitted.
By Lemma 54.4.3 we know that there exists a sequence of blowing ups
\[ X' = X_ n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X \]
in closed points lying over the fibre $f^{-1}(y)$ and a morphism $X' \to Y'$ such that
\[ \xymatrix{ X' \ar[d]_{f'} \ar[r] & X \ar[d]^ f \\ Y' \ar[r] & Y } \]
is commutative. We want to show that the morphism $X' \to Y'$ factors through $X$ and hence we can use induction on $n$ to reduce to the case where $X' \to X$ is the blowup of $X$ in a closed point $x \in X$ mapping to $y$.
Let $E \subset X'$ be the exceptional fibre of the blowing up $X' \to X$. If $E$ maps to a point in $Y'$, then we obtain the desired factorization by Lemma 54.16.1. We will prove that if this is not the case we obtain a contradiction. Namely, if $f'(E)$ is not a point, then $E' = f'(E)$ must be the exceptional curve in $Y'$. Picture
\[ \xymatrix{ E \ar[r] \ar[d]_ g & X' \ar[d]_{f'} \ar[r] & X \ar[d]^ f \\ E' \ar[r] & Y' \ar[r] & Y } \]
Arguing as before $f'$ is an isomorphism in an open neighbourhood of the generic point of $E'$. Hence $g : E \to E'$ is a finite birational morphism. Then the inverse of $g$ (a rational map) is everywhere defined by Morphisms, Lemma 29.42.5 and $g$ is an isomorphism. Consider the map
\[ g^*\mathcal{C}_{E'/Y'} \longrightarrow \mathcal{C}_{E/X'} \]
of Morphisms, Lemma 29.31.3. Since the source and target are invertible modules of degree $1$ on $E = E' = \mathbf{P}^1_\kappa $ and since the map is nonzero (as $f'$ is an isomorphism in the generic point of $E$) we conclude it is an isomorphism. By Morphisms, Lemma 29.32.18 we conclude that $\Omega _{X'/Y'}|_ E = 0$. This means that $f'$ is unramified at every point of $E$ (Morphisms, Lemma 29.35.14). Hence $f'$ is quasi-finite at every point of $E$ (Morphisms, Lemma 29.35.10). Hence the maximal open $V' \subset Y'$ over which $f'$ is an isomorphism contains $E'$ by Varieties, Lemma 33.17.3. This in turn implies that the inverse image of $y$ in $X'$ is $E'$. Hence the inverse image of $y$ in $X$ is $x$. Hence $x \in X$ is in the maximal open over which $f$ is an isomorphism by Varieties, Lemma 33.17.3. This is a contradiction as we assumed that $y$ is not in this open. $\square$
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 where each morphism is a blowup in a closed point.
\[ X = X_0 \leftarrow X_1 \leftarrow \ldots \leftarrow X_ n = Y_ m \to \ldots \to Y_1 \to Y_0 = Y \]
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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