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The Stacks project

54.15 Embedded resolution

Given a curve on a surface there is a blowing up which turns the curve into a strict normal crossings divisor. In this section we will use that a one dimensional locally Noetherian scheme is normal if and only if it is regular (Algebra, Lemma 10.119.7). We will also use that any point on a locally Noetherian scheme specializes to a closed point (Properties, Lemma 28.5.9).

Lemma 54.15.1. Let Y be a one dimensional integral Noetherian scheme. The following are equivalent

  1. there exists an alteration X \to Y with X regular,

  2. there exists a resolution of singularities of Y,

  3. there exists a finite sequence Y_ n \to Y_{n - 1} \to \ldots \to Y_1 \to Y of blowups in closed points with Y_ n regular, and

  4. the normalization Y^\nu \to Y is finite.

Proof. The implications (3) \Rightarrow (2) \Rightarrow (1) are immediate. The implication (1) \Rightarrow (4) follows from Lemma 54.13.1. Observe that a normal one dimensional scheme is regular hence the implication (4) \Rightarrow (2) is clear as well. Thus it remains to show that the equivalent conditions (1), (2), and (4) imply (3).

Let f : X \to Y be a resolution of singularities. Since the dimension of Y is one we see that f is finite by Varieties, Lemma 33.17.2. We will construct factorizations

X \to \ldots \to Y_2 \to Y_1 \to Y

where Y_ i \to Y_{i - 1} is a blowing up of a closed point and not an isomorphism as long as Y_{i - 1} is not regular. Each of these morphisms will be finite (by the same reason as above) and we will get a corresponding system

f_*\mathcal{O}_ X \supset \ldots \supset f_{2, *}\mathcal{O}_{Y_2} \supset f_{1, *}\mathcal{O}_{Y_1} \supset \mathcal{O}_ Y

where f_ i : Y_ i \to Y is the structure morphism. Since Y is Noetherian, this increasing sequence of coherent submodules must stabilize (Cohomology of Schemes, Lemma 30.10.1) which proves that for some n the scheme Y_ n is regular as desired. To construct Y_ i given Y_{i - 1} we pick a singular closed point y_{i - 1} \in Y_{i - 1} and we let Y_ i \to Y_{i - 1} be the corresponding blowup. Since X is regular of dimension 1 (and hence the local rings at closed points are discrete valuation rings and in particular PIDs), the ideal sheaf \mathfrak m_{y_{i - 1}} \cdot \mathcal{O}_ X is invertible. By the universal property of blowing up (Divisors, Lemma 31.32.5) this gives us a factorization X \to Y_ i. Finally, Y_ i \to Y_{i - 1} is not an isomorphism as \mathfrak m_{y_{i - 1}} is not an invertible ideal. \square

Lemma 54.15.2. Let X be a Noetherian scheme. Let Y \subset X be an integral closed subscheme of dimension 1 satisfying the equivalent conditions of Lemma 54.15.1. Then there exists a finite sequence

X_ n \to X_{n - 1} \to \ldots \to X_1 \to X

of blowups in closed points such that the strict transform of Y in X_ n is a regular curve.

Proof. Let Y_ n \to Y_{n - 1} \to \ldots \to Y_1 \to Y be the sequence of blowups given to us by Lemma 54.15.1. Let X_ n \to X_{n - 1} \to \ldots \to X_1 \to X be the corresponding sequence of blowups of X. This works because the strict transform is the blowup by Divisors, Lemma 31.33.2. \square

Let X be a locally Noetherian scheme. Let Y, Z \subset X be closed subschemes. Let p \in Y \cap Z be a closed point. Assume that Y is integral of dimension 1 and that the generic point of Y is not contained in Z. In this situation we can consider the invariant

54.15.2.1
\begin{equation} \label{resolve-equation-multiplicity} m_ p(Y \cap Z) = \text{length}_{\mathcal{O}_{X, p}}(\mathcal{O}_{Y \cap Z, p}) \end{equation}

This is an integer \geq 1. Namely, if I, J \subset \mathcal{O}_{X, p} are the ideals corresponding to Y, Z, then we see that \mathcal{O}_{Y \cap Z, p} = \mathcal{O}_{X, p}/I + J has support equal to \{ \mathfrak m_ p\} because we assumed that Y \cap Z does not contain the unique point of Y specializing to p. Hence the length is finite by Algebra, Lemma 10.62.3.

Lemma 54.15.3. In the situation above let X' \to X be the blowing up of X in p. Let Y', Z' \subset X' be the strict transforms of Y, Z. If \mathcal{O}_{Y, p} is regular, then

  1. Y' \to Y is an isomorphism,

  2. Y' meets the exceptional fibre E \subset X' in one point q and m_ q(Y \cap E) = 1,

  3. if q \in Z' too, then m_ q(Y \cap Z') < m_ p(Y \cap Z).

Proof. Since \mathcal{O}_{X, p} \to \mathcal{O}_{Y, p} is surjective and \mathcal{O}_{Y, p} is a discrete valuation ring, we can pick an element x_1 \in \mathfrak m_ p mapping to a uniformizer in \mathcal{O}_{Y, p}. Choose an affine open U = \mathop{\mathrm{Spec}}(A) containing p such that x_1 \in A. Let \mathfrak m \subset A be the maximal ideal corresponding to p. Let I, J \subset A be the ideals defining Y, Z in \mathop{\mathrm{Spec}}(A). After shrinking U we may assume that \mathfrak m = I + (x_1), in other words, that V(x_1) \cap U \cap Y = \{ p\} scheme theoretically. We conclude that p is an effective Cartier divisor on Y and since Y' is the blowing up of Y in p (Divisors, Lemma 31.33.2) we see that Y' \to Y is an isomorphism by Divisors, Lemma 31.32.7. The relationship \mathfrak m = I + (x_1) implies that \mathfrak m^ n \subset I + (x_1^ n) hence we can define a map

\psi : A[\textstyle {\frac{\mathfrak m}{x_1}}] \longrightarrow A/I

by sending y/x_1^ n \in A[\frac{\mathfrak m}{x_1}] to the class of a in A/I where a is chosen such that y \equiv ax_1^ n \bmod I. Then \psi corresponds to the morphism of Y \cap U into X' over U given by Y' \cong Y. Since the image of x_1 in A[\frac{\mathfrak m}{x_1}] cuts out the exceptional divisor we conclude that m_ q(Y', E) = 1. Finally, since J \subset \mathfrak m implies that the ideal J' \subset A[\frac{\mathfrak m}{x_1}] certainly contains the elements f/x_1 for f \in J. Thus if we choose f \in J whose image \overline{f} in A/I has minimal valuation equal to m_ p(Y \cap Z), then we see that \psi (f/x_1) = \overline{f}/x_1 in A/I has valuation one less proving the last part of the lemma. \square

Lemma 54.15.4. Let X be a Noetherian scheme. Let Y_ i \subset X, i = 1, \ldots , n be an integral closed subschemes of dimension 1 each satisfying the equivalent conditions of Lemma 54.15.1. Then there exists a finite sequence

X_ n \to X_{n - 1} \to \ldots \to X_1 \to X

of blowups in closed points such that the strict transform Y'_ i \subset X_ n of Y_ i in X_ n are pairwise disjoint regular curves.

Proof. It follows from Lemma 54.15.2 that we may assume Y_ i is a regular curve for i = 1, \ldots , n. For every i \not= j and p \in Y_ i \cap Y_ j we have the invariant m_ p(Y_ i \cap Y_ j) (54.15.2.1). If the maximum of these numbers is > 1, then we can decrease it (Lemma 54.15.3) by blowing up in all the points p where the maximum is attained. If the maximum is 1 then we can separate the curves using the same lemma by blowing up in all these points p. \square

When our curve is contained on a regular surface we often want to turn it into a divisor with normal crossings.

Lemma 54.15.5. Let X be a regular scheme of dimension 2. Let Z \subset X be a proper closed subscheme. There exists a sequence

X_ n \to \ldots \to X_1 \to X

of blowing ups in closed points such that the inverse image Z_ n of Z in X_ n is an effective Cartier divisor.

Proof. Let D \subset Z be the largest effective Cartier divisor contained in Z. Then \mathcal{I}_ Z \subset \mathcal{I}_ D and the quotient is supported in closed points by Divisors, Lemma 31.15.8. Thus we can write \mathcal{I}_ Z = \mathcal{I}_{Z'} \mathcal{I}_ D where Z' \subset X is a closed subscheme which set theoretically consists of finitely many closed points. Applying Lemma 54.4.1 we find a sequence of blowups as in the statement of our lemma such that \mathcal{I}_{Z'}\mathcal{O}_{X_ n} is invertible. This proves the lemma. \square

Lemma 54.15.6. Let X be a regular scheme of dimension 2. Let Z \subset X be a proper closed subscheme such that every irreducible component Y \subset Z of dimension 1 satisfies the equivalent conditions of Lemma 54.15.1. Then there exists a sequence

X_ n \to \ldots \to X_1 \to X

of blowups in closed points such that the inverse image Z_ n of Z in X_ n is an effective Cartier divisor supported on a strict normal crossings divisor.

Proof. Let X' \to X be a blowup in a closed point p. Then the inverse image Z' \subset X' of Z is supported on the strict transform of Z and the exceptional divisor. The exceptional divisor is a regular curve (Lemma 54.3.1) and the strict transform Y' of each irreducible component Y is either equal to Y or the blowup of Y at p. Thus in this process we do not produce additional singular components of dimension 1. Thus it follows from Lemmas 54.15.5 and 54.15.4 that we may assume Z is an effective Cartier divisor and that all irreducible components Y of Z are regular. (Of course we cannot assume the irreducible components are pairwise disjoint because in each blowup of a point of Z we add a new irreducible component to Z, namely the exceptional divisor.)

Assume Z is an effective Cartier divisor whose irreducible components Y_ i are regular. For every i \not= j and p \in Y_ i \cap Y_ j we have the invariant m_ p(Y_ i \cap Y_ j) (54.15.2.1). If the maximum of these numbers is > 1, then we can decrease it (Lemma 54.15.3) by blowing up in all the points p where the maximum is attained (note that the “new” invariants m_{q_ i}(Y'_ i \cap E) are always 1). If the maximum is 1 then, if p \in Y_1 \cap \ldots \cap Y_ r for some r > 2 and not any of the others (for example), then after blowing up p we see that Y'_1, \ldots , Y'_ r do not meet in points above p and m_{q_ i}(Y'_ i, E) = 1 where Y'_ i \cap E = \{ q_ i\} . Thus continuing to blowup points where more than 3 of the components of Z meet, we reach the situation where for every closed point p \in X there is either (a) no curves Y_ i passing through p, (b) exactly one curve Y_ i passing through p and \mathcal{O}_{Y_ i, p} is regular, or (c) exactly two curves Y_ i, Y_ j passing through p, the local rings \mathcal{O}_{Y_ i, p}, \mathcal{O}_{Y_ j, p} are regular and m_ p(Y_ i \cap Y_ j) = 1. This means that \sum Y_ i is a strict normal crossings divisor on the regular surface X, see Étale Morphisms, Lemma 41.21.2. \square


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