Using the result above we can prove that blowups in points dominate any modification of a regular $2$ dimensional scheme.

Let $X$ be a scheme. Let $x \in X$ be a closed point. As usual, we view $i : x = \mathop{\mathrm{Spec}}(\kappa (x)) \to X$ as a closed subscheme. The blowing up $X' \to X$ of $X$ at $x$ is the blowing up of $X$ in the closed subscheme $x \subset X$. Observe that if $X$ is locally Noetherian, then $X' \to X$ is projective (in particular proper) by Divisors, Lemma 31.32.13.

Lemma 54.4.1. Let $X$ be a Noetherian scheme. Let $T \subset X$ be a finite set of closed points $x$ such that $\mathcal{O}_{X, x}$ is regular of dimension $2$ for $x \in T$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals such that $\mathcal{O}_ X/\mathcal{I}$ is supported on $T$. Then there exists a sequence

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

where $X_{i + 1} \to X_ i$ is the blowing up of $X_ i$ at a closed point lying above a point of $T$ such that $\mathcal{I}\mathcal{O}_{X_ n}$ is an invertible ideal sheaf.

Proof. Say $T = \{ x_1, \ldots , x_ r\}$. Denote $I_ i$ the stalk of $\mathcal{I}$ at $x_ i$. Set

$n_ i = \text{length}_{\mathcal{O}_{X, x_ i}}(\mathcal{O}_{X, x_ i}/I_ i)$

This is finite as $\mathcal{O}_ X/\mathcal{I}$ is supported on $T$ and hence $\mathcal{O}_{X, x_ i}/I_ i$ has support equal to $\{ \mathfrak m_{x_ i}\}$ (see Algebra, Lemma 10.62.3). We are going to use induction on $\sum n_ i$. If $n_ i = 0$ for all $i$, then $\mathcal{I} = \mathcal{O}_ X$ and we are done.

Suppose $n_ i > 0$. Let $X' \to X$ be the blowing up of $X$ in $x_ i$ (see discussion above the lemma). Since $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i}) \to X$ is flat we see that $X' \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i})$ is the blowup of the ring $\mathcal{O}_{X, x_ i}$ in the maximal ideal, see Divisors, Lemma 31.32.3. Hence the square in the commutative diagram

$\xymatrix{ \text{Proj}(\bigoplus \nolimits _{d \geq 0} \mathfrak m_{x_ i}^ d) \ar[r] \ar[d] & X' \ar[d] \\ \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i}) \ar[r] & X }$

is cartesian. Let $E \subset X'$ and $E' \subset \text{Proj}(\bigoplus \nolimits _{d \geq 0} \mathfrak m_{x_ i}^ d)$ be the exceptional divisors. Let $d \geq 1$ be the integer found in Lemma 54.3.5 for the ideal $\mathcal{I}_ i \subset \mathcal{O}_{X, x_ i}$. Since the horizontal arrows in the diagram are flat, since $E' \to E$ is surjective, and since $E'$ is the pullback of $E$, we see that

$\mathcal{I}\mathcal{O}_{X'} \subset \mathcal{O}_{X'}(-dE)$

(some details omitted). Set $\mathcal{I}' = \mathcal{I}\mathcal{O}_{X'}(dE) \subset \mathcal{O}_{X'}$. Then we see that $\mathcal{O}_{X'}/\mathcal{I}'$ is supported in finitely many closed points $T' \subset |X'|$ because this holds over $X \setminus \{ x_ i\}$ and for the pullback to $\text{Proj}(\bigoplus \nolimits _{d \geq 0} \mathfrak m_{x_ i}^ d)$. The final assertion of Lemma 54.3.5 tells us that the sum of the lengths of the stalks $\mathcal{O}_{X', x'}/\mathcal{I}'\mathcal{O}_{X', x'}$ for $x'$ lying over $x_ i$ is $< n_ i$. Hence the sum of the lengths has decreased.

By induction hypothesis, there exists a sequence

$X'_ n \to \ldots \to X'_1 \to X'$

of blowups at closed points lying over $T'$ such that $\mathcal{I}'\mathcal{O}_{X'_ n}$ is invertible. Since $\mathcal{I}'\mathcal{O}_{X'}(-dE) = \mathcal{I}\mathcal{O}_{X'}$, we see that $\mathcal{I}\mathcal{O}_{X'_ n} = \mathcal{I}'\mathcal{O}_{X'_ n}(-d(f')^{-1}E)$ where $f' : X'_ n \to X'$ is the composition. Note that $(f')^{-1}E$ is an effective Cartier divisor by Divisors, Lemma 31.32.11. Thus we are done by Divisors, Lemma 31.13.7. $\square$

Lemma 54.4.2. Let $X$ be a Noetherian scheme. Let $T \subset X$ be a finite set of closed points $x$ such that $\mathcal{O}_{X, x}$ is a regular local ring of dimension $2$. Let $f : Y \to X$ be a proper morphism of schemes which is an isomorphism over $U = X \setminus T$. Then there exists a sequence

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

where $X_{i + 1} \to X_ i$ is the blowing up of $X_ i$ at a closed point $x_ i$ lying above a point of $T$ and a factorization $X_ n \to Y \to X$ of the composition.

Proof. By More on Flatness, Lemma 38.31.4 there exists a $U$-admissible blowup $X' \to X$ which dominates $Y \to X$. Hence we may assume there exists an ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$ such that $\mathcal{O}_ X/\mathcal{I}$ is supported on $T$ and such that $Y$ is the blowing up of $X$ in $\mathcal{I}$. By Lemma 54.4.1 there exists a sequence

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

where $X_{i + 1} \to X_ i$ is the blowing up of $X_ i$ at a closed point $x_ i$ lying above a point of $T$ such that $\mathcal{I}\mathcal{O}_{X_ n}$ is an invertible ideal sheaf. By the universal property of blowing up (Divisors, Lemma 31.32.5) we find the desired factorization. $\square$

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 $X_{i + 1} \to X_ i$ is the 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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