# Principalization and resolution

*Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).*

The lessons [Order reduction for ideals](order-reduction-for-ideals.md) and [Order reduction for marked ideals](order-reduction-for-marked-ideals.md) prove order reduction in every dimension, compatibly with smooth morphisms, change of field and closed embeddings. This lesson derives the main theorems from it. Principalization turns an ideal sheaf into the ideal sheaf of a divisor with simple normal crossings by a sequence of smooth blow-ups that is compatible with smooth morphisms. Resolution of singularities follows by embedding a reduced scheme in a smooth one and stopping the principalization of its ideal at the step that blows up the strict transform of the scheme itself. At the end we derive the forms of these theorems used in other courses of this collection: a proper morphism from a smooth scheme that is an isomorphism over a dense open subset, smooth compactifications whose boundary is a divisor with simple normal crossings, and the elimination of indeterminacies of rational maps.

We use [Smooth blow-ups and transforms of ideals](smooth-blowups-and-transforms-of-ideals.md), [Blow-up sequences and the main theorems](blow-up-sequences-and-the-main-theorems.md), [Order reduction for ideals](order-reduction-for-ideals.md) and [Order reduction for marked ideals](order-reduction-for-marked-ideals.md), especially its Corollary 3.2: Theorems 4.1 and 4.2 of the second lesson hold in every dimension. Conventions: \(k\) has characteristic zero, smooth schemes are smooth of finite type over \(k\), and smooth schemes that are not equidimensional are treated piece by piece as in Remark 3.3 of the second lesson.

## 1. Separating the components of a divisor

The order reduction functors work with an *ordered* divisor, but a divisor with simple normal crossings has no preferred ordering of its components, and an arbitrary ordering would not be compatible with smooth morphisms. We therefore first make the components pairwise disjoint by blowing up their intersections, the deepest ones first. Afterwards all of them together form one smooth member, and the exceptional divisors of these blow-ups are ordered by the step that created them.

In this lesson an *snc divisor* on a smooth scheme \(X\) is a reduced effective divisor \(E\) whose irreducible components, listed in any order, form an ordered divisor with simple normal crossings in the sense of [Smooth blow-ups and transforms of ideals, Definition 3.2](smooth-blowups-and-transforms-of-ideals.md#3-blowing-up-a-smooth-centre). Its *depth* at \(x\in X\), written \(\delta_E(x)\), is the number of components of \(E\) through \(x\), and \(\operatorname{Sing}E=\{x:\delta_E(x)\ge2\}\). Write \(\mathcal I_E\) for the ideal sheaf of \(E\).

**Lemma 1.1.** Let \(E\) be an snc divisor on a smooth scheme \(X\).

1. \(\operatorname{ord}_x\mathcal I_E=\delta_E(x)\) for every \(x\in X\). Hence \(\{x:\delta_E(x)\ge d\}=\operatorname{cosupp}(\mathcal I_E,d)\) is closed, and \(\operatorname{Sing}E\) has codimension \(\ge2\).
2. Let \(d\ge1\) with \(\delta_E\le d\) everywhere, and let \(Z\) be the reduced closed subscheme with support \(\{\delta_E=d\}\). Then \(Z\) is the disjoint union of the nonempty intersections \(C_1\cap\cdots\cap C_d\) of \(d\) distinct components, and it is smooth of codimension \(d\). If \(D\) is a reduced divisor such that \(E+D\) is an snc divisor, then \(Z\) is a smooth centre having snc with \(E+D\).
3. Let \(h:Y\to X\) be a smooth morphism. Then \(h^{-1}E\) is an snc divisor on \(Y\), its components are the components of the \(h^{-1}(C)\), two components lying over the same \(C\) are disjoint, and \(\delta_{h^{-1}E}(y)=\delta_E(h(y))\). The same holds for \(E_{k'}\subset X_{k'}\) and a field extension \(k\subset k'\).

**Proof.** (1) Near \(x\) choose coordinates in which the components through \(x\) are \(V(z_1),\ldots,V(z_\delta)\), \(\delta=\delta_E(x)\); then \(\mathcal I_E=(z_1\cdots z_\delta)\) near \(x\). Every partial derivative of \(z_1\cdots z_\delta\) of order \(<\delta\) is divisible by some \(z_i\) with \(i\le\delta\), hence lies in \(\mathfrak m_x\), while \(\partial_1\cdots\partial_\delta(z_1\cdots z_\delta)=1\). By the Taylor criterion ([Smooth blow-ups and transforms of ideals, Proposition 2.3](smooth-blowups-and-transforms-of-ideals.md#2-order-of-vanishing-and-derivative-ideals)), \(\operatorname{ord}_x\mathcal I_E=\delta\). The set \(\operatorname{Sing}E\) lies in the union of the pairwise intersections of components, which have codimension \(2\).

(2) Let \(\delta_E(x)=d\). The components not through \(x\) are closed and miss a neighbourhood \(U\) of \(x\). In coordinates on \(U\) adapted to \(E+D\), with the components of \(E\) through \(x\) equal to \(V(z_1),\ldots,V(z_d)\), we get \(\{\delta_E=d\}\cap U=V(z_1,\ldots,z_d)\). This subscheme is smooth, hence reduced, and it is a coordinate subspace for coordinates adapted to \(E+D\), which is the snc condition. Two different sets of \(d\) components cannot meet, because a common point would have depth \(>d\). So the intersections are disjoint and \(Z\) is their union.

(3) Let \(y\in Y\) and \(x=h(y)\). Coordinates near \(x\) adapted to \(E\), completed by fibre coordinates as in the proof of [Smooth blow-ups and transforms of ideals, Proposition 2.6(4)](smooth-blowups-and-transforms-of-ideals.md#2-order-of-vanishing-and-derivative-ideals), form coordinates near \(y\). In them \(h^{-1}E=V(z_1\cdots z_\delta)\) near \(y\), so \(h^{-1}E\) is reduced near \(y\), with \(\delta_E(x)\) components through \(y\) crossing normally; they are the components through \(y\) of the \(h^{-1}(C_i)\). Each \(h^{-1}(C)\) is smooth, so its components are disjoint. For a field extension, use the base change of the coordinates. \(\square\)

**Construction 1.2 (the separating sequence).** Let \(X\) be smooth of dimension \(n\) and \(E\) an snc divisor on \(X\). The blow-up sequence \(\mathcal B^{\rm sep}(X,E)\) has one step for each \(d=n,n-1,\ldots,2\), in this order. At the step \(d\), let \(E_{(d)}\) be the strict transform of \(E\) on the current scheme, let \(Z_d\) be the reduced closed subscheme with support \(\{\delta_{E_{(d)}}\ge d\}\), and blow it up, with exceptional divisor \(F_d\) (empty if \(Z_d=\emptyset\)). Let \(\Pi_{\rm sep}:X'\to X\) be the composite, \(E'\) the strict transform of \(E\) on \(X'\), and \(G^d\) the strict transform of \(F_d\) on \(X'\). Put

\[
E^{\rm sep}=(E',G^2,G^3,\ldots,G^n),
\]

an ordered divisor with \(n\) members.

**Proposition 1.3.**

1. Before the step \(d\), the total transform of \(E\), consisting of the strict transforms of the components of \(E\) and of \(F_n,\ldots,F_{d+1}\), is an snc divisor, and \(\delta_{E_{(d)}}\le d\). The centre \(Z_d\) is empty or a smooth centre of codimension \(d\) having snc with this total transform. After the step, the strict transform of \(E\) has depth \(\le d-1\) everywhere.
2. Every centre lies over \(\operatorname{Sing}E\), so \(\Pi_{\rm sep}\) is an isomorphism over \(X\setminus\operatorname{Sing}E\).
3. The components of \(E'\) are pairwise disjoint, and \(E^{\rm sep}\) is an ordered snc divisor whose members together form the total transform of \(E\). If \(\mathcal I\) is an ideal sheaf on \(X\) that is nonzero on every component of \(X\), then \(\Pi_{\rm sep}^*\mathcal I\) is nonzero on every component of \(X'\).
4. Let \(h:Y\to X\) be a smooth morphism with \(\dim Y=n'\ge n\), and \(h':Y'\to X'\) the induced morphism. Then \(\mathcal B^{\rm sep}(Y,h^{-1}E)\) is \(h^*\mathcal B^{\rm sep}(X,E)\) with its empty blow-ups deleted, and \(E^{\rm sep}_Y\) is \(h'^{-1}E^{\rm sep}_X\) followed by \(n'-n\) empty members. For a field extension the same holds with \(n'=n\).

**Proof.** (1) Induction over the steps. Initially the total transform is \(E\), and \(\delta_E\le n\) because the components through a point are distinct coordinate hyperplanes. Suppose the claim holds before the step \(d\). Lemma 1.1(2), applied to the snc divisor \(E_{(d)}\) and to the divisor \(D\) formed by the strict transforms of the \(F_{d'}\) with \(d'>d\), shows that \(Z_d\) is a smooth centre of codimension \(d\) having snc with the total transform, and by [Smooth blow-ups and transforms of ideals, Lemma 3.3](smooth-blowups-and-transforms-of-ideals.md#3-blowing-up-a-smooth-centre) the new total transform is snc. Let \(q\) be a point over \(z\in Z_d\). The components of \(E_{(d)}\) through \(z\) are exactly the \(d\) components \(C_1,\ldots,C_d\) whose intersection contains \(z\), because \(d\) is the maximal depth. Take coordinates with \(C_i=V(z_i)\) and \(Z_d=V(z_1,\ldots,z_d)\) near \(z\). On the chart \(U_l\) of [Smooth blow-ups and transforms of ideals, Proposition 3.1](smooth-blowups-and-transforms-of-ideals.md#3-blowing-up-a-smooth-centre) the strict transform of \(C_l\) is empty, as in the proof of Lemma 3.3 there. So at most \(d-1\) of the strict transforms of the \(C_i\) pass through \(q\), and strict transforms of components not through \(z\) do not pass through points over \(z\). Away from \(Z_d\) the blow-up is an isomorphism and the depth was already \(\le d-1\).

(2) A point of \(Z_d\) lies on the strict transforms of at least two distinct components \(C,C'\) of \(E\), so its image lies on \(C\cap C'\subset\operatorname{Sing}E\). Since all centres lie over the closed set \(\operatorname{Sing}E\), the composite is an isomorphism over its complement.

(3) After the step \(d=2\), every point lies on at most one component of \(E'\). By (1) the total transform is snc. Each \(F_d\) is smooth by Proposition 3.1 of the first lesson, and its components lie over the disjoint components of \(Z_d\), so they are disjoint. Under each later blow-up, whose centre has snc with the strict transform of \(F_d\), that strict transform stays smooth with disjoint components: in the charts of Lemma 3.3 of the first lesson it is a coordinate hyperplane or empty, and strict transforms of disjoint closed subsets are disjoint. So every member of \(E^{\rm sep}\) is a smooth divisor, and since all members together form the snc total transform, \(E^{\rm sep}\) is an ordered snc divisor. Finally, every centre has codimension \(\ge2\), so each component of \(X'\) is the strict transform of a component of \(X\) and meets the dense open subset \(\Pi_{\rm sep}^{-1}(X\setminus\operatorname{Sing}E)\), over which \(\Pi_{\rm sep}^*\mathcal I\) is \(\mathcal I\).

(4) By Lemma 1.1(3), \(h\) preserves depth. Blowing up commutes with flat base change ([Stacks, Tag 0805](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/divisors.html#divisors-lemma-flat-base-change-blowing-up)), and so does the strict transform ([Stacks, Tag 080F](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/divisors.html#divisors-lemma-strict-transform-flat)). By induction over the steps, the centre of the step \(d\) on \(Y\) is the inverse image of the centre of the step \(d\) on \(X\) (both are reduced, \(h^{-1}(Z_d)\) being smooth), the strict transform of \(h^{-1}E\) is the pull-back of the strict transform of \(E\), and the exceptional divisors pull back to the exceptional divisors. For \(d>n\) the depth on \(X\), and hence on \(Y\), is \(<d\), so the steps \(d=n',\ldots,n+1\) on \(Y\) have empty centres, and \(G^d_Y=\emptyset\) for these \(d\). For a field extension the argument is the same, with Lemma 1.1(3) for \(E_{k'}\). \(\square\)

**Remark 1.4.** Kollár makes the components disjoint in \(s-1\) steps, where \(s\) is the number of components of \(E\), blowing up the \(s\)-fold intersection first. That number is not preserved by smooth morphisms: an open subset can meet fewer components, and an étale morphism can split a component into several disjoint ones. Depth is a pointwise invariant, preserved by smooth morphisms by Lemma 1.1(3), which is what Proposition 1.3(4) needs. On a given \(X\) the two procedures differ only by empty blow-ups.

## 2. Empty members

Proposition 1.3(4) produces on \(Y\) an ordered divisor with more members than the pull-back of the one on \(X\), all of them empty. The next lemma shows that empty members do not affect order reduction.

**Lemma 2.1.** Let \((X,\mathcal I,E)\) be a triple with \(E=(E^1,\ldots,E^s)\), and let \(E^+\) be obtained from \(E\) by inserting empty members at arbitrary positions, the member \(E^i\) moving to the position \(\varphi(i)\) for an increasing map \(\varphi\). Then

\[
\mathcal B^{\rm ord}_m(X,\mathcal I,E^+)=\mathcal B^{\rm ord}_m(X,\mathcal I,E),\qquad \mathcal B^{\rm mord}_m(X,\mathcal I,m,E^+)=\mathcal B^{\rm mord}_m(X,\mathcal I,m,E),
\]

that is, the two sequences have the same centres. Along them, the transforms of \(E^+\) are obtained from the transforms of \(E\) by inserting empty members at the positions outside the image of \(\varphi\); the members created during the sequences are appended at the end of both. The same holds for the functors \(\mathcal B^{\rm dis}_{m,j}\) of [Order reduction for ideals, Lemma 2.1](order-reduction-for-ideals.md#2-moving-the-high-order-locus-off-a-divisor), with \(j\) replaced by \(\varphi(j)\).

**Proof.** By induction on \(\dim X\); in dimension \(0\) all sequences are empty. The constructions of the three previous lessons use the ordered divisor only in the following ways:

- through the components of all members together, in the monomial and non-monomial parts of [Order reduction for marked ideals, Definition 1.1](order-reduction-for-marked-ideals.md#1-monomial-and-non-monomial-parts);
- through one member \(E^j\) at a time, together with the restrictions of the other members to \(E^j\) or to a hypersurface of maximal contact, in Lemma 2.1 and in Step B of the proof of [Order reduction for ideals, Theorem 3.1](order-reduction-for-ideals.md#3-order-reduction-for-ideals-in-dimension-n);
- through the order of the indices: Step A of that proof runs through the old members in increasing order, and Step 3 of [Order reduction for marked ideals, Theorem 2.1](order-reduction-for-marked-ideals.md#2-the-construction) chooses the lexicographically smallest index tuple;
- by appending new members at the end.

Each of these is unchanged by inserting empty members. An empty member has no components. For an empty \(E^j\), \(\mathcal B^{\rm dis}_{m,j}\) is the empty sequence: its first centre is empty, and its second part lives on the empty scheme. The restrictions of \(E^+\) to a hypersurface are the restrictions of \(E\) with empty members inserted, so the induction hypothesis applies in dimension \(\dim X-1\). An increasing map preserves the lexicographic order of index tuples of equal length. A member appended at the end of \(E\) corresponds to one appended at the end of \(E^+\). Tuning, cosupports and the auxiliary ideals \(N(\mathcal I)\) and \(\mathcal K_t\) do not involve the ordering. Following the constructions step by step, including the gluing, which is local, the centres agree. \(\square\)

## 3. Principalization

**Theorem 3.1 (principalization; Theorem 4.3 of the second lesson).** For a triple \((X,\mathcal I,E)\) in which \(E\) is an snc divisor, define \(\mathcal B^{\rm pri}(X,\mathcal I,E)\) as \(\mathcal B^{\rm sep}(X,E)\) followed by

\[
\mathcal B^{\rm mord}_1\bigl(X',\Pi_{\rm sep}^*\mathcal I,1,E^{\rm sep}\bigr),
\]

and write \(\Pi:X_r\to X\) for the composite. Then:

1. all centres are smooth and have snc with \(E\) and its total transforms;
2. \(\Pi^*\mathcal I=\mathcal O_{X_r}(-D)\) for an effective divisor \(D\) whose support is a union of components of the total transform of \(E\), hence an snc divisor;
3. every centre lies over \(\operatorname{cosupp}(\mathcal I,1)\cup\operatorname{Sing}E\), so \(\Pi\) is an isomorphism over \(X\setminus\bigl(\operatorname{cosupp}(\mathcal I,1)\cup\operatorname{Sing}E\bigr)\);
4. \(\mathcal B^{\rm pri}\) commutes with smooth morphisms and change of field;
5. \(\mathcal B^{\rm pri}\) commutes with closed embeddings when \(E=\emptyset\).

**Proof.** By Proposition 1.3(3), \((X',\Pi_{\rm sep}^*\mathcal I,1,E^{\rm sep})\) is a marked triple, so the second part is defined. By [Order reduction for marked ideals, Corollary 3.2](order-reduction-for-marked-ideals.md#3-closed-embeddings-and-the-end-of-the-induction) it is a blow-up sequence of order \(\ge1\) at whose end the marked transform \(\mathcal I_r\) has maximal order \(<1\), that is, \(\mathcal I_r=\mathcal O_{X_r}\).

(1) For the first part this is Proposition 1.3(1). In the second part each centre has snc with the current transform of \(E^{\rm sep}\), whose members are smooth divisors that together form the total transform of \(E\), by Proposition 1.3(3) and [Smooth blow-ups and transforms of ideals, Lemma 3.3](smooth-blowups-and-transforms-of-ideals.md#3-blowing-up-a-smooth-centre). A smooth member has disjoint components, so near a point it is a single smooth hypersurface or absent, and snc with the ordered divisor is snc with the total transform.

(2) Let the second part consist of the blow-ups \(\pi_i:X_{i+1}\to X_i\), \(s\le i<r\), with \(X_s=X'\) and exceptional divisors \(F_{i+1}\subset X_{i+1}\). With marking \(1\) the transform is \(\mathcal I_{i+1}=\mathcal O(F_{i+1})\cdot\pi_i^*\mathcal I_i\) ([Smooth blow-ups and transforms of ideals, Definition 4.2](smooth-blowups-and-transforms-of-ideals.md#4-transforms-of-ideals-and-of-marked-ideals)), so \(\pi_i^*\mathcal I_i=\mathcal O(-F_{i+1})\cdot\mathcal I_{i+1}\). Pull-backs of ideals compose. Pulling back to \(X_r\) and using \(\mathcal I_r=\mathcal O_{X_r}\) gives

\[
\Pi^*\mathcal I=\mathcal O_{X_r}\Bigl(-\sum_{i=s}^{r-1}\Pi_{r,i+1}^*F_{i+1}\Bigr).
\]

The support of \(\Pi_{r,i+1}^*F_{i+1}\) is the strict transform of \(F_{i+1}\) together with the exceptional divisors of later steps that lie over it. All of these are components of the total transform of \(E\); in particular the support is nowhere dense, so \(\Pi_{r,i+1}^*F_{i+1}\) is an effective Cartier divisor.

(3) The centres of the first part lie over \(\operatorname{Sing}E\) by Proposition 1.3(2). For the second part, \(\operatorname{cosupp}(\Pi_{\rm sep}^*\mathcal I,1)=\Pi_{\rm sep}^{-1}\operatorname{cosupp}(\mathcal I,1)\), because a local homomorphism maps the maximal ideal into the maximal ideal. Each centre \(Z_i\) has \(\operatorname{ord}_{Z_i}\mathcal I_i\ge1\), so \(Z_i\subset\operatorname{cosupp}(\mathcal I_i,1)\) and \(F_{i+1}\) lies over \(\operatorname{cosupp}(\mathcal I_i,1)\). Outside \(\pi_i^{-1}\operatorname{cosupp}(\mathcal I_i,1)\) the ideal \(\mathcal I_{i+1}=\mathcal O(F_{i+1})\cdot\pi_i^*\mathcal I_i\) is the unit ideal. By induction every centre lies over \(\operatorname{cosupp}(\mathcal I,1)\cup\operatorname{Sing}E\).

(4) Let \(h:Y\to X\) be smooth, with induced \(h':Y'\to X'\). By Proposition 1.3(4) the first part on \(Y\) is the pull-back of the first part on \(X\) with empty blow-ups deleted, and \(E^{\rm sep}_Y\) is \(h'^{-1}E^{\rm sep}_X\) followed by empty members. The ideal on \(Y'\) is \(h'^*\Pi_{\rm sep}^*\mathcal I\). By Lemma 2.1,

\[
\mathcal B^{\rm mord}_1\bigl(Y',h'^*\Pi_{\rm sep}^*\mathcal I,1,E^{\rm sep}_Y\bigr)=\mathcal B^{\rm mord}_1\bigl(Y',h'^*\Pi_{\rm sep}^*\mathcal I,1,h'^{-1}E^{\rm sep}_X\bigr),
\]

and by Corollary 3.2 of the previous lesson this is \(h'^*\mathcal B^{\rm mord}_1(X',\Pi_{\rm sep}^*\mathcal I,1,E^{\rm sep}_X)\) with its empty blow-ups deleted. Concatenating the two parts gives the claim. Change of field is the same.

(5) If \(E=\emptyset\), the first part is empty and \(E^{\rm sep}\) consists of \(n\) empty members, so \(\mathcal B^{\rm pri}(X,\mathcal I,\emptyset)=\mathcal B^{\rm mord}_1(X,\mathcal I,1,\emptyset)\) by Lemma 2.1. Statement (2) of Theorem 4.2 of the second lesson, proved in every dimension by Corollary 3.2 of the previous lesson, is the required compatibility with closed embeddings. \(\square\)

**Remark 3.2 (the singular locus of \(E\)).** Kollár's statement of principalization asserts that \(\Pi\) is an isomorphism over \(X\setminus\operatorname{cosupp}\mathcal I\) also when \(E\ne\emptyset\). For the construction above this fails when components of \(E\) meet outside the cosupport: for \(\mathcal I=\mathcal O\) and \(E=V(xy)\) on \(\mathbf A^2\), \(\mathcal B^{\rm pri}\) is the blow-up of the origin (Exercise 6.2). A rule that omitted the separating steps whenever \(\mathcal I=\mathcal O\) would not commute with open immersions: if \(\operatorname{cosupp}\mathcal I\ne\emptyset\) and an open \(U\subset X\) meets \(\operatorname{Sing}E\) but not \(\operatorname{cosupp}\mathcal I\), the restriction to \(U\) of the sequence of \(X\) blows up \(\operatorname{Sing}E\cap U\). All applications below use \(E=\emptyset\), where (3) is the classical statement.

**Corollary 3.3 (principalization).** Let \(X\) be a smooth scheme and \(\mathcal I\) an ideal sheaf nonzero on every component of \(X\).

1. There is a projective morphism \(\Pi:X'\to X\), a composite of blow-ups of smooth centres, such that \(X'\) is smooth, \(\Pi^*\mathcal I=\mathcal O_{X'}(-D)\) for an effective divisor \(D\) with snc support, and \(\Pi\) is an isomorphism over \(X\setminus\operatorname{cosupp}(\mathcal I,1)\).
2. The open set \(X\setminus\operatorname{cosupp}(\mathcal I,1)\) is dense in \(X\), and its inverse image is dense in \(X'\); in particular \(\Pi\) is birational.
3. If \(T\subset X\) is a closed subset containing no component of \(X\), applying (1) to the reduced ideal of \(T\) gives \(\Pi\), an isomorphism over \(X\setminus T\), with \(\Pi^{-1}(T)\) the support of an snc divisor.

These sequences are those of \(\mathcal B^{\rm pri}(\,\cdot\,,\emptyset)\) and are therefore compatible with smooth morphisms and change of field.

**Proof.** Take \(\Pi\) from \(\mathcal B^{\rm pri}(X,\mathcal I,\emptyset)\) and apply Theorem 3.1. A blow-up of a closed subscheme is projective ([Stacks, Tag 02NS](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/divisors.html#divisors-lemma-blowing-up-projective)), and a composite of projective morphisms to the quasi-compact and quasi-separated \(X\) is projective ([Stacks, Tag 0C4P](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-composition-projective)). The closed set \(\operatorname{cosupp}(\mathcal I,1)\) contains no component of \(X\), so its complement is dense. No centre contains a component of the scheme being blown up ([Blow-up sequences and the main theorems, Lemma 2.3](blow-up-sequences-and-the-main-theorems.md#2-triples-and-the-blow-ups-they-allow)), so each component of \(X'\) is the strict transform of a component of \(X\) and meets the inverse image of this dense open set. For (3), \(\Pi^{-1}(T)=V(\Pi^*\mathcal I_T)=\operatorname{Supp}D\). \(\square\)

**Corollary 3.4 (elimination of indeterminacy).** Let \(X\) be a smooth scheme, \(U\subset X\) a dense open subset, \(Y\) a proper scheme over \(k\), and \(f:U\to Y\) a morphism. There are a composite of blow-ups of smooth centres \(\Pi:X'\to X\) that is an isomorphism over \(U\), and a morphism \(f':X'\to Y\) with \(f'=f\circ\Pi\) on \(\Pi^{-1}(U)\).

**Proof.** By [Stacks, Tag 0C4V](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/divisors.html#divisors-lemma-extend-rational-map-after-modification) there are a proper morphism \(p:X''\to X\) that is an isomorphism over \(U\) and a morphism \(f'':X''\to Y\) that agrees with \(f\circ p\) on \(p^{-1}(U)\). By [Stacks, Tag 081T](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/flat.html#flat-lemma-dominate-modification-by-blowup) there is a blow-up \(b:B_{\mathcal J}X\to X\) of a finite type ideal \(\mathcal J\) with \(V(\mathcal J)\cap U=\emptyset\) that factors through \(p\). Since \(U\) is dense, \(\mathcal J\) is nonzero on every component of \(X\). Let \(\Pi\) be as in Corollary 3.3 for \(\mathcal J\). Then the scheme-theoretic inverse image \(\Pi^{-1}V(\mathcal J)\) is an effective Cartier divisor, so \(\Pi\) factors through \(b\) by the universal property of blowing up ([Stacks, Tag 0806](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/divisors.html#divisors-lemma-universal-property-blowing-up)). Let \(f'\) be the composite \(X'\to B_{\mathcal J}X\to X''\to Y\). Over \(U\) all the maps to \(X\) are isomorphisms, so \(f'=f\circ\Pi\) on \(\Pi^{-1}(U)\). \(\square\)

## 4. Resolution of singularities

*The smooth locus.* For a scheme \(X\) of finite type over \(k\), let \(X^{\rm sm}\) be the open set of points at which \(X\to\operatorname{Spec}k\) is smooth, and \(\operatorname{Sing}X=X\setminus X^{\rm sm}\). We use three facts.

- (a) \(x\in X^{\rm sm}\) if and only if \(\mathcal O_{X,x}\) is regular. One direction is [Stacks, Tag 056S](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/varieties.html#varieties-lemma-smooth-regular). Conversely, geometric regularity at \(x\) only requires regularity of the local rings after finite purely inseparable extensions of \(k\) ([Stacks, Tag 038U](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/varieties.html#varieties-lemma-geometrically-regular-at-point)); in characteristic zero the only such extension is \(k\), and geometric regularity at \(x\) is smoothness at \(x\) ([Stacks, Tag 038X](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/varieties.html#varieties-lemma-geometrically-regular-smooth)).
- (b) If \(X\) is reduced, \(X^{\rm sm}\) is dense in \(X\): \(X\) is geometrically reduced because \(k\) is perfect ([Stacks, Tag 020I](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/varieties.html#varieties-lemma-perfect-reduced)), and a geometrically reduced scheme locally of finite type over a field has a dense smooth open subset ([Stacks, Tag 056V](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/varieties.html#varieties-lemma-geometrically-reduced-dense-smooth-open)).
- (c) For a smooth morphism \(h:Y\to X\), \(Y^{\rm sm}=h^{-1}(X^{\rm sm})\), and \(Y\) is reduced if \(X\) is ([Stacks, Tag 034E](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-reduced-local-smooth)). For a field extension \(k\subset k'\), \((X_{k'})^{\rm sm}\) is the inverse image of \(X^{\rm sm}\). Indeed, if \(h(y)\in X^{\rm sm}\), then \(Y\to\operatorname{Spec}k\) is smooth at \(y\) as a composite of smooth morphisms ([Stacks, Tag 01VA](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-composition-smooth)). If \(y\in Y^{\rm sm}\), then \(\mathcal O_{X,h(y)}\to\mathcal O_{Y,y}\) is a flat local homomorphism with regular target, so its source is regular ([Stacks, Tag 00OF](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-flat-under-regular)), and (a) applies. For the field extension use the flat morphism \(X_{k'}\to X\) in the same way, with [Stacks, Tag 01VB](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-base-change-smooth) for the base change.

**Lemma 4.1 (principalizing a smooth subscheme).** Let \(P\) be a smooth scheme and \(S\subset P\) a smooth closed subscheme containing no component of \(P\). Then \(\mathcal B^{\rm pri}(P,\mathcal I_S,\emptyset)\) consists of a single blow-up, with centre \(S\).

**Proof.** As in the proof of Theorem 3.1(5), \(\mathcal B^{\rm pri}(P,\mathcal I_S,\emptyset)=\mathcal B^{\rm mord}_1(P,\mathcal I_S,1,\emptyset)\). This sequence and the one-step sequence with centre \(S\) both commute with open immersions, after deleting empty blow-ups, and both are formed on a disjoint union by running the pieces simultaneously ([Blow-up sequences and the main theorems, Remark 3.3](blow-up-sequences-and-the-main-theorems.md#3-blow-up-sequence-functors-and-functoriality)). A blow-up sequence is determined by its pull-back along a surjective local isomorphism (proof of Theorem 5.1 there), so it suffices to compare the two sequences on the disjoint union of the members of an open cover of \(P\). On \(P\setminus S\) both are empty, the first because \(\mathcal I_S\) is the unit ideal there.

By [Smooth blow-ups and transforms of ideals, Lemma 1.2](smooth-blowups-and-transforms-of-ideals.md#1-coordinates-and-derivations) and the remark after it, the points of \(S\) have open neighbourhoods \(U\) with coordinates such that \(S\cap U=V(z_1,\ldots,z_c)\), where \(c\ge1\) because \(S\) contains no component of \(P\). Put \(Y=V(z_2,\ldots,z_c)\subset U\), so that \(Y=U\) if \(c=1\). Then \(S\cap U=V(z_1|_Y)\) is a smooth hypersurface of \(Y\), and its ideal \(\mathcal J=\mathcal O_Y(-S)\) is nonzero on every component of \(Y\). By statement (2) of Theorem 4.2 of the second lesson, \(\mathcal B^{\rm mord}_1(U,\mathcal I_S,1,\emptyset)\) is the push-forward of \(\mathcal B^{\rm mord}_1(Y,\mathcal J,1,\emptyset)\), and this equals \(\mathcal B^{\rm ord}_1(Y,\mathcal J,\emptyset)\) by [Order reduction for marked ideals, Theorem 2.1](order-reduction-for-marked-ideals.md#2-the-construction), since \(\operatorname{maxord}\mathcal J=1\).

Now follow the construction in the proof of [Order reduction for ideals, Theorem 3.1](order-reduction-for-ideals.md#3-order-reduction-for-ideals-in-dimension-n). The hypersurface \(H=S\cap U\) is an MC-hypersurface for \(\mathcal J\), because \(MC(\mathcal J)=D^0(\mathcal J)=\mathcal J\) is generated by \(z_1|_Y\). Step A is empty because \(E=\emptyset\). The tuned ideal is \(W(\mathcal J)=\mathcal J\), of order \(w=1\) ([Tuning of ideals, Definition 2.3](tuning-of-ideals.md#2-tuning)). Step B is \(\mathcal B^{\rm dis}_{1,0}(Y,\mathcal J,(H))\). In it, Step 1 of [Order reduction for ideals, Lemma 2.1](order-reduction-for-ideals.md#2-moving-the-high-order-locus-off-a-divisor) blows up the union of the components of \(H\) contained in \(\operatorname{cosupp}(\mathcal J,1)=H\), that is, all of \(H\). Afterwards \(W_0=\mathcal O_Y(H)\cdot\mathcal J=\mathcal O_Y\), and the remaining part of \(H\) is empty, so Step 2 is empty. Thus \(\mathcal B^{\rm ord}_1(Y,\mathcal J,\emptyset)\) is the single blow-up of \(S\cap U\), and its push-forward is the blow-up of \(U\) with centre \(S\cap U\). \(\square\)

**Definition 4.2.** An *embedding* of a reduced scheme \(X\) of finite type over \(k\) is a closed immersion \(X\subset P\) into a smooth scheme \(P\) such that \(X\) contains no component of \(P\). For an embedding, let \(Z_i\subset P_i\) be the centres of \(\mathcal B^{\rm pri}(P,\mathcal I_X,\emptyset)\), \(\Pi_i:P_i\to P\) the composites, \(X_i\subset P_i\) the strict transforms of \(X\), and \(P^0=P\setminus\operatorname{Sing}X\), an open subset of \(P\).

**Proposition 4.3.** Let \(X\subset P\) be an embedding with \(X\ne\emptyset\).

1. There is exactly one index \(j\) such that \(Z_j\) meets \(\Pi_j^{-1}(P^0)\). The morphism \(\Pi_j\) is an isomorphism over \(P^0\), and it identifies \(Z_j\cap\Pi_j^{-1}(P^0)\) with \(X^{\rm sm}\). Every \(Z_i\) with \(i<j\) lies over \(\operatorname{Sing}X\).
2. \(X_j\) is the union of the connected components of \(Z_j\) that meet \(\Pi_j^{-1}(P^0)\). In particular \(X_j\) is smooth, and \(X_j\cap\Pi_j^{-1}(X^{\rm sm})\) is dense in \(X_j\).
3. \(\rho=\Pi_j|_{X_j}:X_j\to X\) is projective and is an isomorphism over \(X^{\rm sm}\).
4. \(\rho^{-1}(\operatorname{Sing}X)\) is the support of the restriction to \(X_j\) of the total transform \(E_j\) of the empty divisor along the first \(j\) steps, and this restriction is an snc divisor on \(X_j\).

We call the step \(j\) the *distinguished step*, and write \(R_P(X)=X_j\) and \(\rho_P=\rho\). For \(X=\emptyset\) put \(R_P(X)=\emptyset\).

**Proof.** (1) The subscheme \(X^{\rm sm}=X\cap P^0\) is smooth and closed in \(P^0\), nonempty by fact (b), and it contains no component of \(P^0\): such a component would be dense in a component of \(P\), which would then lie in the closed set \(X\). By Theorem 3.1(4) for the open immersion \(P^0\subset P\), the restriction of \(\mathcal B^{\rm pri}(P,\mathcal I_X,\emptyset)\) to \(P^0\) is \(\mathcal B^{\rm pri}(P^0,\mathcal I_{X^{\rm sm}},\emptyset)\) after deleting empty blow-ups, and by Lemma 4.1 this is the single blow-up of \(X^{\rm sm}\). So exactly one step has a centre meeting the inverse image of \(P^0\). The earlier steps restrict to empty blow-ups over \(P^0\), so \(\Pi_j\) is an isomorphism over \(P^0\), and the centre \(Z_j\) restricted to \(\Pi_j^{-1}(P^0)\cong P^0\) is \(X^{\rm sm}\). All centres lie over \(\operatorname{cosupp}(\mathcal I_X,1)=X\) by Theorem 3.1(3), and those with \(i<j\) do not meet \(\Pi_i^{-1}(P^0)\), so they lie over \(X\setminus P^0=\operatorname{Sing}X\).

(2) Put \(\Omega_i=X_i\cap\Pi_i^{-1}(P^0)\) for \(i\le j\). Then \(\Pi_i\) maps \(\Omega_i\) isomorphically onto \(X^{\rm sm}\), and \(\Omega_i\cap Z_i=\emptyset\) for \(i<j\). The strict transform \(X_{i+1}\) is the closure of \(\pi_i^{-1}(X_i\setminus Z_i)\) ([Stacks, Tag 080D](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/divisors.html#divisors-definition-strict-transform)), and \(\pi_i\) is an isomorphism over \(P_i\setminus Z_i\). Since \(\Omega_0=X^{\rm sm}\) is dense in \(X_0=X\) by fact (b), induction shows that \(\Omega_i\) is dense in \(X_i\) for \(i\le j\): if \(\Omega_i\) is dense in \(X_i\), it is dense in the open subset \(X_i\setminus Z_i\), so \(\Omega_{i+1}=\pi_i^{-1}(\Omega_i)\) is dense in \(\pi_i^{-1}(X_i\setminus Z_i)\), and therefore in its closure \(X_{i+1}\). By (1), \(\Omega_j=Z_j\cap\Pi_j^{-1}(P^0)\), both being the copy of \(X^{\rm sm}\) in \(\Pi_j^{-1}(P^0)\cong P^0\). This is an open subset of the smooth scheme \(Z_j\), whose connected components are irreducible, so its closure is the union \(Z'\) of the connected components of \(Z_j\) that meet it. Thus \(X_j\) and \(Z'\) have the same underlying set. Both are reduced: \(Z'\) because it is smooth, and \(X_j\) because it is obtained from the reduced scheme \(X\) by successive blow-ups \(B_{Z_i\cap X_i}X_i\) ([Stacks, Tag 080E](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/divisors.html#divisors-lemma-strict-transform)), whose Rees algebras are subrings of polynomial rings over reduced rings. Hence \(X_j=Z'\). Finally \(X_j\cap\Pi_j^{-1}(X^{\rm sm})=\Omega_j\), which is dense.

(3) \(\Pi_j\) is projective (Tags 02NS and 0C4P), hence so is its base change \(P_j\times_PX\to X\) ([Stacks, Tag 02V6](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-base-change-projective)). The closed immersion \(X_j\to P_j\times_PX\) is projective, and so is the composite \(\rho\) (Tag 0C4P). By (2), \(\rho^{-1}(X^{\rm sm})=\Omega_j\), which \(\Pi_j\) maps isomorphically onto \(X^{\rm sm}\).

(4) Let \(\operatorname{Ex}_j\subset P_j\) be the union of the inverse images of \(Z_0,\ldots,Z_{j-1}\); it is the union of the members of \(E_j\). By induction on \(i\), \(\Pi_i\) maps \(P_i\setminus\operatorname{Ex}_i\) isomorphically onto an open subset of \(P\), and \(X_i\setminus\operatorname{Ex}_i\) onto an open subset of \(X\), because the strict transform agrees with the inverse image away from the centre. Since \(X_j\) is smooth, the points of \(X_j\setminus\operatorname{Ex}_j\) map to points of \(X^{\rm sm}\). Since every \(Z_i\) with \(i<j\) lies over \(\operatorname{Sing}X\), the points of \(X_j\cap\operatorname{Ex}_j\) map to \(\operatorname{Sing}X\). Hence \(\rho^{-1}(\operatorname{Sing}X)=X_j\cap\operatorname{Ex}_j\). By Theorem 3.1(1), \(Z_j\) has snc with \(E_j\), and no member of \(E_j\) contains a component of \(X_j\), because every component of \(X_j\) meets \(\Omega_j\), which is disjoint from \(\operatorname{Ex}_j\). At a point of \(X_j\) take coordinates adapted to \(E_j\) and \(Z_j\): \(Z_j=V(z_{a_1},\ldots,z_{a_t})\), and every member through the point is \(V(z_c)\) with \(c\notin\{a_1,\ldots,a_t\}\). Their restrictions to \(X_j\) are distinct coordinate hyperplanes of \(X_j\). So the restriction of \(E_j\) to \(X_j\) is an snc divisor with support \(X_j\cap\operatorname{Ex}_j\). \(\square\)

**Lemma 4.4.** Let \(X\subset P\) be an embedding.

1. If \(P'\subset P\) is open and \(X'=X\cap P'\), then \(X'\subset P'\) is an embedding and \(R_{P'}(X')=\rho_P^{-1}(X')\) over \(X'\).
2. If \(\iota:P\to P''\) is a closed immersion into a smooth scheme such that \(X\subset P''\) is also an embedding, then \(R_{P''}(X)=R_P(X)\) over \(X\).
3. An isomorphism \(P\to P_2\) that maps \(X\) isomorphically onto \(X_2\subset P_2\) induces an isomorphism \(R_P(X)\to R_{P_2}(X_2)\) over \(X\to X_2\).

**Proof.** (1) \(X'\) contains no component of \(P'\), since a component of \(P'\) is dense in a component of \(P\). If \(X'=\emptyset\), both sides are empty. Otherwise, by Theorem 3.1(4), the restriction of \(\mathcal B^{\rm pri}(P,\mathcal I_X,\emptyset)\) to \(P'\) is \(\mathcal B^{\rm pri}(P',\mathcal I_{X'},\emptyset)\) after deleting empty blow-ups. Strict transforms commute with restriction to open subsets, and \(P'\setminus\operatorname{Sing}X'=P'\cap P^0\) by fact (c). The distinguished step for \(P\) restricts to a nonempty step, because its centre contains the copy of \(X'^{\rm sm}\ne\emptyset\); it is the distinguished step for \(P'\).

(2) The ideal \(\mathcal I^P_X\) of \(X\) in \(P\) is nonzero on every component of \(P\), and \(\mathcal O_{P''}/\mathcal I^{P''}_X=\iota_*(\mathcal O_P/\mathcal I^P_X)\). By Theorem 3.1(5), \(\mathcal B^{\rm pri}(P'',\mathcal I^{P''}_X,\emptyset)=\iota_*\mathcal B^{\rm pri}(P,\mathcal I^P_X,\emptyset)\). The two sequences have the same centres \(Z_i\subset P_i\subset P''_i\), and the strict transforms of \(X\) agree, since both are obtained by the successive blow-ups \(B_{Z_i\cap X_i}X_i\) (Tag 080E). A centre meets the inverse image of \(P''\setminus\operatorname{Sing}X\) if and only if it meets the inverse image of \(P\setminus\operatorname{Sing}X\), because it lies in \(P_i\). So the distinguished steps agree.

(3) An isomorphism is a smooth morphism; apply Theorem 3.1(4). \(\square\)

**Lemma 4.5 (Kollár).** Let \(U\) be an affine scheme of finite type over \(k\), and \(i_1:U\to\mathbf A^a\), \(i_2:U\to\mathbf A^b\) closed immersions. There is an automorphism \(\sigma\) of \(\mathbf A^{a+b}\) with \(\sigma\circ(i_1,0)=(0,i_2)\).

**Proof.** Let \(x\) and \(y\) be the coordinates of \(\mathbf A^a\) and \(\mathbf A^b\). Since \(i_1^*:k[x]\to\mathcal O(U)\) is surjective, there is a morphism \(g:\mathbf A^a\to\mathbf A^b\) with \(g\circ i_1=i_2\), and similarly \(f:\mathbf A^b\to\mathbf A^a\) with \(f\circ i_2=i_1\). The automorphism \(\sigma_1(x,y)=(x,y+g(x))\) maps \((i_1,0)\) to \((i_1,i_2)\), and \(\sigma_2(x,y)=(x+f(y),y)\) maps \((0,i_2)\) to \((i_1,i_2)\). Take \(\sigma=\sigma_2^{-1}\circ\sigma_1\). \(\square\)

**Proposition 4.6.** For two embeddings \(X\subset P_1\) and \(X\subset P_2\) there is a unique isomorphism \(R_{P_1}(X)\to R_{P_2}(X)\) over \(X\).

**Proof.** *Uniqueness.* Let \(\varphi,\psi\) be two such isomorphisms. On \(\rho_{P_1}^{-1}(X^{\rm sm})\) both equal \(\rho_{P_2}^{-1}\circ\rho_{P_1}\), by Proposition 4.3(3). This open subset is dense in the reduced scheme \(R_{P_1}(X)\) (Proposition 4.3(2)), so its scheme-theoretic closure is \(R_{P_1}(X)\); and \(R_{P_2}(X)\to X\) is separated, being projective. So \(\varphi=\psi\) by [Stacks, Tag 01RH](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-equality-of-morphisms). By uniqueness, isomorphisms constructed over the members of an open cover of \(X\) agree on overlaps and glue, so it suffices to construct one over a neighbourhood of each point \(x\in X\). Over \(X^{\rm sm}\) both sides are identified with \(X^{\rm sm}\). Let \(x\in\operatorname{Sing}X\).

*Common affine neighbourhoods.* Choose an affine open \(P_2'\ni x\) of \(P_2\). The set \(X\setminus P_2'\) is closed in \(P_1\), so there is an affine open \(P_1'\ni x\) of \(P_1\) with \(U=X\cap P_1'\subset P_2'\). The affine scheme \(U\) is open in the affine scheme \(X\cap P_2'\), so there is \(g\in\mathcal O(P_2')\) with \(x\in X\cap D(g)\subset U\). Then \(U'=X\cap D(g)\) is the principal open subset of \(U\) where the restriction of \(g\) is invertible; lifting this restriction to \(g'\in\mathcal O(P_1')\) gives \(U'=X\cap D(g')\). So \(Q_1=D(g')\subset P_1\) and \(Q_2=D(g)\subset P_2\) are affine opens with \(X\cap Q_1=X\cap Q_2=U'\ni x\), and \(U'\) is not smooth.

*Comparison.* Choose closed immersions \(Q_1\subset\mathbf A^a\) and \(Q_2\subset\mathbf A^b\), and let \(i_1:U'\to\mathbf A^a\) and \(i_2:U'\to\mathbf A^b\) be the composite closed immersions. Since \(U'\) is not smooth, its image is not the whole affine space in any of the closed immersions into \(\mathbf A^a\), \(\mathbf A^b\) and \(\mathbf A^{a+b}\) used below, so all of them are embeddings, these spaces being irreducible; \(U'\subset Q_1\) and \(U'\subset Q_2\) are embeddings by Lemma 4.4(1). By Lemma 4.4(1), (2), (2), then (3) with Lemma 4.5, then (2), (2), (1),

\[
R_{P_1}(X)|_{U'}=R_{Q_1}(U')=R_{\mathbf A^a}(U')=R_{\mathbf A^{a+b}}\bigl(U',(i_1,0)\bigr)\cong R_{\mathbf A^{a+b}}\bigl(U',(0,i_2)\bigr)=R_{\mathbf A^b}(U')=R_{Q_2}(U')=R_{P_2}(X)|_{U'},
\]

where \(R_{\mathbf A^{a+b}}(U',\iota)\) denotes the construction for the embedding \(\iota\). Every identification is over \(U'\); for the middle one this holds because \(\sigma\circ(i_1,0)=(0,i_2)\). \(\square\)

**Lemma 4.7.** Let \(h:Y\to X\) be a smooth morphism of schemes of finite type over \(k\), \(y\in Y\), and \(U\subset X\) an affine open containing \(h(y)\), with a closed immersion \(U\subset A\) into a smooth affine scheme \(A\). There are an open \(V\ni y\) with \(h(V)\subset U\), a smooth morphism \(h_A:A_V\to A\), and a closed immersion \(V\subset A_V\) such that \(V=U\times_AA_V\).

**Proof.** Write \(U=\operatorname{Spec}R\). By [Stacks, Tag 054L](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-smooth-etale-over-affine-space) and [Stacks, Tag 02GT](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-etale-locally-standard-etale) there are an affine open neighbourhood \(V\) of \(y\) with \(h(V)\subset U\), a principal open subset \(\operatorname{Spec}C\) of \(\mathbf A^d_U\), \(C=R[x_1,\ldots,x_d]_q\), and an isomorphism \(V\cong\operatorname{Spec}\bigl(C[t]_g/(f)\bigr)\) over \(U\), where \(f\in C[t]\) is monic and \(\partial f/\partial t\) is invertible in \(C[t]_g/(f)\). Write \(A=\operatorname{Spec}B\) with \(B\to R\) surjective. Lift \(q\) to \(Q\in B[x]\), so that \(B[x]_Q\to C\) is surjective, and lift \(f\) to a monic \(F\in B[x]_Q[t]\) and \(g\) to \(G\in B[x]_Q[t]\). Put \(A_V=\operatorname{Spec}\bigl(B[x]_Q[t]_{G\cdot\partial F/\partial t}/(F)\bigr)\). It is standard étale over the open subset \(\operatorname{Spec}B[x]_Q\) of \(\mathbf A^d_A\), hence étale over it ([Stacks, Tag 00UC](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-standard-etale)), so \(A_V\to A\) is smooth. Its base change to \(U\) is \(\operatorname{Spec}\bigl(C[t]_{g\cdot\partial f/\partial t}/(f)\bigr)=V\), because \(\partial f/\partial t\) is already invertible on \(V\). \(\square\)

**Theorem 4.8 (resolution of singularities; Theorem 4.4 of the second lesson).** Let \(X\) be a reduced scheme of finite type over \(k\). There is a proper morphism \(\rho_X:R(X)\to X\) from a smooth scheme, unique up to unique isomorphism over \(X\), such that \(R(X)|_U\cong R_P(U)\) over \(U\) for every open \(U\subset X\) and every embedding \(U\subset P\). It has the following properties.

1. \(\rho_X\) is an isomorphism over \(X^{\rm sm}\), and \(\rho_X^{-1}(X^{\rm sm})\) is dense in \(R(X)\).
2. \(\rho_X^{-1}(\operatorname{Sing}X)\) is the support of an snc divisor.
3. For a smooth morphism \(h:Y\to X\), \(R(Y)=R(X)\times_XY\) canonically. For a field extension \(k\subset k'\), \(R(X_{k'})=R(X)_{k'}\).
4. If \(X\) has an embedding \(X\subset P\), then \(R(X)=R_P(X)\), the strict transform of \(X\) at the distinguished step of the smooth blow-up sequence \(\mathcal B^{\rm pri}(P,\mathcal I_X,\emptyset)\), and \(\rho_X\) is projective. Every quasi-projective \(X\) has an embedding into an open subset of a projective space.

**Proof.** *Construction.* Cover \(X\) by affine opens \(U_\alpha\) with closed immersions \(U_\alpha\subset\mathbf A^{N_\alpha}\), \(N_\alpha>\dim U_\alpha\); these are embeddings. Put \(R_\alpha=R_{\mathbf A^{N_\alpha}}(U_\alpha)\). The open subset \(U_{\alpha\beta}=U_\alpha\cap U_\beta\) is closed in the open subset \(\mathbf A^{N_\alpha}\setminus(U_\alpha\setminus U_{\alpha\beta})\), and similarly for \(\beta\). By Lemma 4.4(1) and Proposition 4.6 there is a unique isomorphism \(\varphi_{\alpha\beta}:R_\alpha|_{U_{\alpha\beta}}\to R_\beta|_{U_{\alpha\beta}}\) over \(U_{\alpha\beta}\). By the uniqueness in Proposition 4.6, \(\varphi_{\beta\gamma}\circ\varphi_{\alpha\beta}=\varphi_{\alpha\gamma}\) over \(U_\alpha\cap U_\beta\cap U_\gamma\), so the \(R_\alpha\) glue to a scheme \(R(X)\) over \(X\) ([Stacks, Section 01JA](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/schemes.html#schemes-section-glueing-schemes)). The same uniqueness argument applies to any two schemes over \(X\) with the stated local description, so \(R(X)\) has the stated characterization and is independent of the cover up to unique isomorphism. Properness is local on the base ([Stacks, Tag 01W2](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/morphisms.html#morphisms-lemma-proper-local-on-the-base)).

(1) This is local on \(X\) and follows from Proposition 4.3(2),(3).

(2) Locally on \(R(X)\), the set \(\rho_X^{-1}(\operatorname{Sing}X)\) is the support of an snc divisor, by Proposition 4.3(4). Let \(C\) be an irreducible component of \(\rho_X^{-1}(\operatorname{Sing}X)\). For each \(\alpha\) with \(C\cap R_\alpha\ne\emptyset\), the set \(C\cap R_\alpha\) is irreducible, and, as for any open subset of a noetherian space, it is an irreducible component of \(\rho_X^{-1}(\operatorname{Sing}X)\cap R_\alpha\). So it is a component of the local snc divisor, hence smooth, and distinct global components restrict to distinct local components. Thus every component is smooth, and at every point the components through it are distinct components of a local snc divisor, so they cross normally.

(3) Let \(h:Y\to X\) be smooth. By fact (c), \(Y\) is reduced and \(Y^{\rm sm}=h^{-1}(X^{\rm sm})\). The scheme \(R(X)\times_XY\) is smooth over the smooth scheme \(R(X)\), hence smooth over \(k\) and reduced. Its projection to \(Y\) is proper and an isomorphism over \(Y^{\rm sm}\), and the inverse image of \(Y^{\rm sm}\) is dense, being the inverse image of the dense open subset \(\rho_X^{-1}(X^{\rm sm})\) under the smooth, hence open, projection to \(R(X)\). As in Proposition 4.6, a \(Y\)-isomorphism \(R(Y)\to R(X)\times_XY\) is therefore unique if it exists, and it suffices to construct one over a neighbourhood of each \(y\in Y\).

Choose an affine open \(U\ni h(y)\) of \(X\) with an embedding \(U\subset A\) into a smooth affine scheme, and \(V\) and \(h_A:A_V\to A\) as in Lemma 4.7. Then \(V\subset A_V\) is an embedding: a component \(W\) of \(A_V\) contained in \(V\) would give \(h_A(W)\subset U\) with \(h_A(W)\) a nonempty open subset of \(A\), and then \(U\) would contain a component of \(A\). Since \(V=U\times_AA_V\), the ideal of \(V\) is \(h_A^*\mathcal I_U\), so Theorem 3.1(4) gives \(\mathcal B^{\rm pri}(A_V,\mathcal I_V,\emptyset)=h_A^*\mathcal B^{\rm pri}(A,\mathcal I_U,\emptyset)\) with empty blow-ups deleted. The strict transforms of \(V\) are the pull-backs of those of \(U\) (Tag 080F), and \(A_V\setminus\operatorname{Sing}V=h_A^{-1}(A\setminus\operatorname{Sing}U)\) by fact (c) for the smooth morphism \(V\to U\). The distinguished step for \(A\) pulls back to a step whose centre meets the inverse image of \(A_V\setminus\operatorname{Sing}V\), because \(h(V)\) is a nonempty open subset of \(U\) and therefore meets \(U^{\rm sm}\); so it is the distinguished step for \(A_V\). Hence \(R_{A_V}(V)=R_A(U)\times_UV\), that is, \(R(Y)|_V\cong(R(X)\times_XY)|_V\) over \(V\).

For a field extension, let \(U\subset\mathbf A^N\) be an embedding of an affine open subset. Then \(U_{k'}\subset\mathbf A^N_{k'}\) is an embedding of the reduced scheme \(U_{k'}\) (Tag 020I), \(\mathcal B^{\rm pri}\) commutes with the change of field, strict transforms commute with the flat base change, and \((U_{k'})^{\rm sm}=(U^{\rm sm})_{k'}\) by fact (c). So \(R_{\mathbf A^N_{k'}}(U_{k'})=R_{\mathbf A^N}(U)_{k'}\), and these identifications glue by uniqueness.

(4) By Lemma 4.4(1) and Proposition 4.6, \(R_P(X)\) has the local description that characterizes \(R(X)\), so \(R(X)=R_P(X)\); projectivity is Proposition 4.3(3). If \(X\) is quasi-projective, choose a locally closed immersion \(X\subset\mathbf P^N\subset\mathbf P^{N+1}\), let \(\overline X\) be the closure of \(X\), and put \(P=\mathbf P^{N+1}\setminus(\overline X\setminus X)\). Then \(X\) is closed in \(P\), and \(P\) is irreducible of dimension \(N+1>\dim X\), so \(X\subset P\) is an embedding. \(\square\)

**Remark 4.9.** Kollár proves that the resolution is an isomorphism over the smooth locus by transporting the smoothness of an exceptional divisor between étale neighbourhoods of smooth points, for embeddings of codimension at least two. The argument above uses Lemma 4.1 instead, which identifies the distinguished step directly from the compatibility of \(\mathcal B^{\rm pri}\) with open immersions; no condition on the codimension of the embedding is needed. For a hypersurface the distinguished step is a trivial blow-up. The induced blow-ups of \(X\) itself, with centres \(Z_i\cap X_i\), need not have smooth centres; only the ambient sequence is smooth.

## 5. Consequences used in the programme

**Corollary 5.1 (proper smooth models).** Every reduced scheme \(X\) of finite type over \(k\) admits a proper morphism from a smooth scheme that is an isomorphism over a dense open subset of \(X\): the morphism \(\rho_X:R(X)\to X\), an isomorphism over \(X^{\rm sm}\). If \(X\) is quasi-projective, \(\rho_X\) is projective.

**Proof.** Theorem 4.8(1) and (4), with fact (b) for the density of \(X^{\rm sm}\). \(\square\)

This is the form in which resolution is used to compare the étale and singular cohomology of finite-type complex schemes.

**Corollary 5.2 (smooth compactification).** Let \(X\) be a smooth separated scheme of finite type over \(k\). There is an open immersion \(X\subset X'\) into a smooth proper \(k\)-scheme such that \(X\) is dense in \(X'\) and \(X'\setminus X\) is the support of an snc divisor. If \(X\) is quasi-projective, \(X'\) can be chosen projective.

**Proof.** By Nagata's theorem ([Stacks, Tag 0F41](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/flat.html#flat-theorem-nagata); a compactification there is a quasi-compact open immersion into a proper scheme, [Stacks, Section 0ATT](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/flat.html#flat-section-compactify)) there is an open immersion \(X\subset\overline X\) with \(\overline X\) proper over \(k\). Replace \(\overline X\) by the closure of \(X\) with its reduced structure: it is proper, and \(X\) is still an open subscheme because \(X\) is reduced. If \(X\) is quasi-projective, take instead for \(\overline X\) the closure of \(X\) in a projective space, with its reduced structure. Since \(X\) is open in \(\overline X\) and smooth, \(X\subset\overline X^{\rm sm}\). By Theorem 4.8, \(R(\overline X)\) is smooth and proper over \(k\), it contains \(\rho^{-1}(X)\cong X\) as an open subscheme, and \(X\) is dense in it, because \(X\) is dense in \(\overline X^{\rm sm}\) and \(\rho^{-1}(\overline X^{\rm sm})\) is dense in \(R(\overline X)\). The closed subset \(T=R(\overline X)\setminus X\) contains no component of \(R(\overline X)\), so Corollary 3.3(3) gives a composite of smooth blow-ups \(\Pi:X'\to R(\overline X)\) that is an isomorphism over \(X\), with \(\Pi^{-1}(T)\) the support of an snc divisor. Then \(X'\) is smooth and proper, it contains \(X\) as a dense open subscheme by Corollary 3.3(2), and \(X'\setminus X=\Pi^{-1}(T)\). In the quasi-projective case all morphisms are projective, so \(X'\) is projective over \(k\) (Tag 0C4P). \(\square\)

**Example 5.3 (the node).** Let \(X=V(xy)\subset\mathbf A^2\). Then \(\operatorname{Sing}X\) is the origin. By Proposition 4.3, the steps of \(\mathcal B^{\rm pri}(\mathbf A^2,(xy),\emptyset)\) before the distinguished one have centres over the origin, and \(R(X)\) is the strict transform of \(X\) at the distinguished step, a smooth curve. The earlier centres are points or curves lying over the origin, and blowing them up does not change the strict transforms of the two axes up to isomorphism. So \(R(X)\) is the disjoint union of two lines mapping isomorphically to the two axes, which is the normalization of \(X\), and \(\rho^{-1}(\operatorname{Sing}X)\) consists of two points. Exercise 6.4 identifies the steps.

## 6. Exercises

**Exercise 6.1.** Let \(X\) be smooth and \(X\subset P\) an embedding. Show that \(\mathcal B^{\rm pri}(P,\mathcal I_X,\emptyset)\) is the blow-up of \(X\), and that \(R(X)=X\).

*Solution.* Here \(\operatorname{Sing}X=\emptyset\), and Lemma 4.1 with \(S=X\) shows that \(\mathcal B^{\rm pri}(P,\mathcal I_X,\emptyset)\) is the single blow-up of \(X\). Its centre meets \(P^0=P\), so the distinguished step is the first one, and \(R_P(X)=X_0=X\). For a smooth \(X\) without an embedding, Theorem 4.8(1) gives the same conclusion directly: \(\rho_X\) is an isomorphism over \(X^{\rm sm}=X\).

**Exercise 6.2.** Compute \(\mathcal B^{\rm pri}(\mathbf A^2,\mathcal O,V(xy))\).

*Solution.* Here \(n=2\), and \(\mathcal B^{\rm sep}\) has the single step \(d=2\): the locus of depth \(2\) of \(E=V(x)+V(y)\) is the origin, which is blown up. Then \(E^{\rm sep}=(E',G^2)\), where \(E'\) consists of the two disjoint strict transforms of the axes and \(G^2\) is the exceptional curve. The second part \(\mathcal B^{\rm mord}_1(X',\mathcal O,1,E^{\rm sep})\) is empty, because the unit ideal has maximal order \(0<1\). So \(\Pi\) is the blow-up of the origin, although \(\operatorname{cosupp}(\mathcal O,1)=\emptyset\). This is why Theorem 3.1(3) excludes \(\operatorname{Sing}E\).

**Exercise 6.3.** Let \(E\) be the union of the three coordinate planes in \(\mathbf A^3\). Describe \(\mathcal B^{\rm sep}(\mathbf A^3,E)\) and \(E^{\rm sep}\).

*Solution.* At the step \(d=3\) the locus of depth \(3\) is the origin; blowing it up gives \(F_3\cong\mathbf P^2\). Two of the planes have strict transforms meeting along the strict transform of their common axis, which meets \(F_3\) in one of the three coordinate points of \(\mathbf P^2\); the strict transforms of the three axes are disjoint. So the depth is now \(\le2\), and the step \(d=2\) blows up the three disjoint curves, with exceptional divisor \(F_2\) of three components. Afterwards the strict transforms of the planes are pairwise disjoint. Thus \(E^{\rm sep}=(E',G^2,G^3)\), where \(G^2=F_2\) and \(G^3\), the strict transform of \(F_3\), is \(\mathbf P^2\) blown up in its three coordinate points, because each axis meets \(F_3\) transversally in one point.

**Exercise 6.4.** For \(X=V(xy)\subset\mathbf A^2\), show that \(\mathcal B^{\rm pri}(\mathbf A^2,(xy),\emptyset)\) begins with the blow-up of the origin, followed by the blow-up of the two points \(p_1,p_2\) where the strict transforms \(L_1',L_2'\) of the axes meet the exceptional curve \(F\), followed by the blow-up of the strict transforms \(L_1''\sqcup L_2''\), and that this third step is the distinguished one.

*Solution.* By Lemma 2.1, \(\mathcal B^{\rm pri}(\mathbf A^2,(xy),\emptyset)=\mathcal B^{\rm mord}_1(\mathbf A^2,(xy),1,\emptyset)\). Step 1 of [Order reduction for marked ideals, Theorem 2.1](order-reduction-for-marked-ideals.md#2-the-construction) first applies \(\mathcal B^{\rm ord}_2\) to \((xy)\), of maximal order \(2\) at the origin. Its tuned ideal is \(W_4((xy))=(x^2y^2)+xy\,(x,y)^2+(x,y)^4=(x,y)^4\). On the MC-hypersurface \(H=V(x-y)\), with coordinate \(t\), the restriction is \((t^4)\) with marking \(4\); its order reduction blows up the origin of \(H\), a trivial blow-up after which the marked transform is \(t^{-4}(t^4)=\mathcal O\), as in [Order reduction for ideals, Example 4.1](order-reduction-for-ideals.md#4-an-example). Pushed forward, this is the blow-up of the origin of \(\mathbf A^2\). The non-monomial part becomes the unmarked transform \(\mathcal O(2F)\pi^*(xy)\), the ideal of \(L_1'\sqcup L_2'\), of maximal order \(1\) (Lemma 1.2(3) of the previous lesson). The next round applies \(\mathcal B^{\rm ord}_1\) to it with \(E=(F)\). Its Step A is \(\mathcal B^{\rm dis}_{1,1}\): no component of \(F\) lies in the cosupport \(L_1'\sqcup L_2'\), and the second part pushes forward \(\mathcal B^{\rm mord}_1(F,\mathcal I_{\{p_1,p_2\}},1,\emptyset)\), which is the blow-up of \(\{p_1,p_2\}\) by Lemmas 2.1 and 4.1. In Step B the curve \(L_1'\sqcup L_2'\) is an MC-hypersurface, because \(MC\) of an ideal of maximal order \(1\) is the ideal itself; its strict transform is \(L_1''\sqcup L_2''\), and \(\mathcal B^{\rm dis}_{1,0}\) blows up the components of it contained in the cosupport, that is, all of \(L_1''\sqcup L_2''\), a trivial blow-up after which the ideal is \(\mathcal O\). This centre contains the generic points of the strict transform of \(X\), while the first two lie over the origin, so it is the distinguished step, and \(R(X)=L_1''\sqcup L_2''\).

## References

- [Kollár] J. Kollár, *Resolution of singularities — Seattle lecture*, arXiv:math/0508332, the theorems on principalization and resolution, the embedding lemmas, the gluing of local resolutions, and the deduction of principalization from order reduction. This lesson replaces the count of components by depth (Remark 1.4), adds Lemma 2.1 on empty members, records that the separating blow-ups take place over \(\operatorname{Sing}E\) (Remark 3.2), and identifies the distinguished step through Lemma 4.1. <https://arxiv.org/abs/math/0508332>
- [Włodarczyk] J. Włodarczyk, *Simple Hironaka resolution in characteristic zero*, J. Amer. Math. Soc. 18 (2005); free preprint arXiv:math/0401401, for the passage from principalization to resolution. <https://arxiv.org/abs/math/0401401>
- [Villamayor] O. Villamayor, *Constructiveness of Hironaka's resolution*, Ann. Sci. École Norm. Sup. (4) 22 (1989), 1–32, an early constructive form of the algorithm. <https://www.numdam.org/item/ASENS_1989_4_22_1_1_0/>
- [Bierstone–Milman] E. Bierstone and P. Milman, *Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant*, arXiv:alg-geom/9508005, a canonical resolution compatible with smooth morphisms, built from a local invariant. <https://arxiv.org/abs/alg-geom/9508005>
- [Encinas–Hauser] S. Encinas and H. Hauser, *Strong resolution of singularities in characteristic zero*, arXiv:math/0211423. <https://arxiv.org/abs/math/0211423>
- [Hauser] H. Hauser, *The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)*, Bull. Amer. Math. Soc. 40 (2003), 323–403, free from the AMS. <https://www.ams.org/journals/bull/2003-40-03/S0273-0979-03-00982-0/>
- [Temkin] M. Temkin, *Desingularization of quasi-excellent schemes in characteristic zero*, arXiv:math/0703678, which extends resolution from schemes of finite type over a field to noetherian quasi-excellent schemes of characteristic zero. <https://arxiv.org/abs/math/0703678>
- [Abramovich–Temkin–Włodarczyk] D. Abramovich, M. Temkin and J. Włodarczyk, *Functorial embedded resolution via weighted blowings up*, arXiv:1906.07106, a later functorial algorithm in which each step is a weighted blow-up determined by a single invariant. <https://arxiv.org/abs/1906.07106>

The theorem of this lesson is due to Hironaka. The functorial construction followed in the course is the one developed by Villamayor, Bierstone–Milman, Encinas–Hauser and Włodarczyk, in the presentation of Kollár.