Definition 54.5.1. Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. Let $x \in X$ be a closed point. The normalized blowup of $X$ at $x$ is the composition $X'' \to X' \to X$ where $X' \to X$ is the blowup of $X$ in $x$ and $X'' \to X'$ is the normalization of $X'$.
54.5 Dominating by normalized blowups
In this section we prove that a modification of a surface can be dominated by a sequence of normalized blowups in points.
Here the normalization $X'' \to X'$ is defined as the scheme $X'$ has an open covering by opens which have finitely many irreducible components by Divisors, Lemma 31.32.10. See Morphisms, Definition 29.54.1 for the definition of the normalization.
In general the normalized blowing up need not be proper even when $X$ is Noetherian. Recall that a scheme is Nagata if it has an open covering by affines which are spectra of Nagata rings (Properties, Definition 28.13.1).
Lemma 54.5.2. In Definition 54.5.1 if $X$ is Nagata, then the normalized blowing up of $X$ at $x$ is normal, Nagata, and proper over $X$.
Proof. The blowup morphism $X' \to X$ is proper (as $X$ is locally Noetherian we may apply Divisors, Lemma 31.32.13). Thus $X'$ is Nagata (Morphisms, Lemma 29.18.1). Therefore the normalization $X'' \to X'$ is finite (Morphisms, Lemma 29.54.11) and we conclude that $X'' \to X$ is proper as well (Morphisms, Lemmas 29.44.11 and 29.41.4). It follows that the normalized blowing up is a normal (Morphisms, Lemma 29.54.5) Nagata algebraic space. $\square$
In the following lemma we need to assume $X$ is Noetherian in order to make sure that it has finitely many irreducible components. Then the properness of $f : Y \to X$ assures that $Y$ has finitely many irreducible components too and it makes sense to require $f$ to be birational (Morphisms, Definition 29.50.1).
Lemma 54.5.3. Let $X$ be a scheme which is Noetherian, Nagata, and has dimension $2$. 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.
\[ \xymatrix{ X_ n \ar[r] \ar[d] & X_{n - 1} \ar[r] & \ldots \ar[r] & X_1 \ar[r] & X_0 \ar[d] \\ Y \ar[rrrr] & & & & X } \]
Proof. We will use the results of Morphisms, Sections 29.18, 29.52, and 29.54 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 Varieties, Lemma 33.17.3. Hence there is a finite set of closed points $T \subset X$ such that $f$ is an isomorphism over $X \setminus T$. For each $x \in T$ the fibre $Y_ x$ is a proper geometrically connected scheme of dimension $1$ over $\kappa (x)$, see More on Morphisms, Lemma 37.53.6. Thus
\[ BadCurves(f) = \{ C \subset Y\text{ closed} \mid \dim (C) = 1, f(C) = \text{a point}\} \]
is a finite set. We will prove the lemma by induction on the number of elements of $BadCurves(f)$. The base case is the case where $BadCurves(f)$ is empty, and in that case $f$ is an isomorphism.
Fix $x \in T$. Let $X' \to X$ be the normalized blowup of $X$ at $x$ and let $Y'$ be the normalization of $Y \times _ X X'$. Picture
\[ \xymatrix{ Y' \ar[r]_{f'} \ar[d] & X' \ar[d] \\ Y \ar[r]^ f & X } \]
Let $x' \in X'$ be a closed point lying over $x$ such that the fibre $Y'_{x'}$ has dimension $\geq 1$. Let $C' \subset Y'$ be an irreducible component of $Y'_{x'}$, i.e., $C' \in BadCurves(f')$. Since $Y' \to Y \times _ X X'$ is finite we see that $C'$ must map to an irreducible component $C \subset Y_ x$. If is clear that $C \in BadCurves(f)$. Since $Y' \to Y$ is birational and hence an isomorphism over points of codimension $1$ in $Y$, we see that we obtain an injective map
\[ BadCurves(f') \longrightarrow BadCurves(f) \]
Thus it suffices to show that after a finite number of these normalized blowups we get rid at of at least one of the bad curves, i.e., the displayed map is not surjective.
We will get rid of a bad curve using an argument due to Zariski. Pick $C \in BadCurves(f)$ lying over our $x$. Denote $\mathcal{O}_{Y, C}$ the local ring of $Y$ at the generic point of $C$. Choose an element $u \in \mathcal{O}_{X, C}$ whose image in the residue field $R(C)$ is transcendental over $\kappa (x)$ (we can do this because $R(C)$ has transcendence degree $1$ over $\kappa (x)$ by Varieties, Lemma 33.20.3). We can write $u = a/b$ with $a, b \in \mathcal{O}_{X, x}$ as $\mathcal{O}_{Y, C}$ and $\mathcal{O}_{X, x}$ have the same fraction fields. By our choice of $u$ it must be the case that $a, b \in \mathfrak m_ x$. Hence
\[ N_{u, a, b} = \min \{ \text{ord}_{\mathcal{O}_{Y, C}}(a), \text{ord}_{\mathcal{O}_{Y, C}}(b)\} > 0 \]
Thus we can do descending induction on this integer. Let $X' \to X$ be the normalized blowing up of $x$ and let $Y'$ be the normalization of $X' \times _ X Y$ as above. We will show that if $C$ is the image of some bad curve $C' \subset Y'$ lying over $x' \in X'$, then there exists a choice of $a', b' \mathcal{O}_{X', x'}$ such that $N_{u, a', b'} < N_{u, a, b}$. This will finish the proof. Namely, since $X' \to X$ factors through the blowing up, we see that there exists a nonzero element $d \in \mathfrak m_{x'}$ such that $a = a' d$ and $b = b' d$ (namely, take $d$ to be the local equation for the exceptional divisor of the blowup). Since $Y' \to Y$ is an isomorphism over an open containing the generic point of $C$ (seen above) we see that $\mathcal{O}_{Y', C'} = \mathcal{O}_{Y, C}$. Hence
\[ \text{ord}_{\mathcal{O}_{Y, C}}(a) = \text{ord}_{\mathcal{O}_{Y', C'}}(a' d) = \text{ord}_{\mathcal{O}_{Y', C'}}(a') + \text{ord}_{\mathcal{O}_{Y', C'}}(d) > \text{ord}_{\mathcal{O}_{Y', C'}}(a') \]
Similarly for $b$ and the proof is complete. $\square$
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.
\[ X_ n \to X_{n - 1} \to \ldots \to X_1 \to X_0 \to X \]
Proof. 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 54.5.3 to the morphism $p : X' \to X$. The composition $X_ n \to X' \to Y$ is the desired morphism. $\square$
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