54.4 Dominating by quadratic transformations
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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