resolution of singularities steven dale cutkoskyfaculty.missouri.edu/~cutkoskys/booksamp.pdf ·...

Resolution of Singularities

Steven Dale Cutkosky

Department of Mathematics, University of Missouri, Columbia,Missouri 65211

To Hema, Ashok and Maya

Contents

Preface vii

Chapter 1. Introduction 1

§1.1. Notation 2

Chapter 2. Non-singularity and Resolution of Singularities 3

§2.1. Newton’s method for determining the branches of a planecurve 3

§2.2. Smoothness and non-singularity 7

§2.3. Resolution of singularities 9

§2.4. Normalization 10

§2.5. Local uniformization and generalized resolution problems 11

Chapter 3. Curve Singularities 17

§3.1. Blowing up a point on A2 17

§3.2. Completion 22

§3.3. Blowing up a point on a non-singular surface 25

§3.4. Resolution of curves embedded in a non-singular surface I 26

§3.5. Resolution of curves embedded in a non-singular surface II 29

Chapter 4. Resolution Type Theorems 37

§4.1. Blow-ups of ideals 37

§4.2. Resolution type theorems and corollaries 40

Chapter 5. Surface Singularities 45

§5.1. Resolution of surface singularities 45

v

vi Contents

§5.2. Embedded resolution of singularities 56

Chapter 6. Resolution of Singularities in Characteristic Zero 61§6.1. The operator 4 and other preliminaries 62§6.2. Hypersurfaces of maximal contact and induction in resolution 66§6.8. Resolution of singularities in characteristic zero 99

Chapter 7. Resolution of Surfaces in Positive Characteristic 131

Chapter 8. Local Uniformization and Resolution of Surfaces 153

Chapter 9. Ramification of Valuations and Simultaneous Resolution 161

Appendix. Smoothness and Non-singularity II 163§A.1. Proofs of the basic theorems 163§A.2. Non-singularity and uniformizing parameters 169§A.3. Higher derivations 171§A.4. Upper semi-continuity of νq(I) 174

Bibliography 179

Preface

The notion of singularity is basic to mathematics. In elementary algebrasingularity appears as a multiple root of a polynomial. In geometry a pointin a space is non-singular if it has a tangent space whose dimension is thesame as that of the space. Both notions of singularity can be detectedthrough the vanishing of derivitives.

Over an algebraically closed field, a variety is non-singular at a pointif there exists a tangent space at the point which has the same dimensionas the variety. More generally, a variety is non-singular at a point if itslocal ring is a regular local ring. A fundamental problem is to remove asingularity by simple algebraic mappings. That is, can a given variety bedesingularized by a proper, birational morphism from a non-singular variety?This is always possible in all dimensions, over fields of characteristic zero.We give a complete proof of this in Chapter 6.

We also treat positive characteristic, developing the basic tools neededfor this study, and giving a proof of resolution of surface singularities inpositive characteristic in Chapter 7.

In Section 2.5 we discuss important open problems, such as resolutionof singularities in positive characteristic and local monomialization of mor-phisms.

Chapter 8 gives a classification of valuations in algebraic function fieldsof surfaces, and a modernization of Zariski’s original proof of local uni-formization for surfaces in characteristic zero.

This book has evolved out of lectures given at the University of Mis-souri and at the Chennai Mathematics Institute, in Chennai, (also knownas Madras), India. It can be used as part of a one year introductory sequence

vii

viii Preface

in algebraic geometry, and would provide an exciting direction after the ba-sic notions of schemes and sheaves have been covered. A core course onresolution is covered in Chapters 2 through 6. The major ideas of resolutionhave been introduced by the end of Section 6.2, and after reading this far,a student will find the resolution theorems of Section 6.8 quite believable,and have a good feel for what goes into their proofs.

Chapters 7 and 8 cover additional topics. These two chapters are inde-pendent, and can be chosen as possible followups to the basic material inthe first 5 chapters. Chapter 7 gives a proof of resolution of singularitiesfor surfaces in positive characteristic, and Chapter 8 gives a proof of localuniformization and resolution of singularities for algebraic surfaces. Thischapter provides an introduction to valuation theory in algebraic geometry,and to the problem of local uniformization.

The appendix proves foundational results on the singular locus that weneed. On a first reading, I recommend that the reader simply look upthe statements as needed in reading the main body of the book. Versionsof almost all of these statements are much easier over algebraically closedfields of characteristic zero, and most of the results can be found in this casein standard textbooks in algebraic geometry.

I assume that the reader has some familiarity with algebraic geometryand commutative algebra, such as can be obtained from an introductorycourse on these subjects. This material is covered in books such as Atiyahand MacDonald [13] or the basic sections of Eisenbud’s book [37], andthe first two chapters of Hartshorne’s book on algebraic geometry [47], orEisenbud and Harris’s book on schemes [38].

I thank Professors Seshadri and Ed Dunne for their encouragement towrite this book, and Laura Ghezzi, Tai Ha, Krishna Hanamanthu, OlgaKashcheyeva and Emanoil Theodorescu for their helpful comments on pre-liminary versions of the manuscript.

For financial support during the preparation of this book I thank theNational Science Foundation, the National Board of Higher Mathematics ofIndia, the Mathematical Sciences Research Insititute and the University ofMissouri.

Steven Dale Cutkosky

Chapter 1

Introduction

An algebraic variety X is defined locally by the vanishing of a system ofpolynomial equations fi ∈ K[x1, . . . , xn],

f1 = · · · = fm = 0.

If K is algebraically closed, points of X in this chart are α = (α1, . . . , αn) ∈An

K which satisfy this system. The tangent space Tα(X) at a point α ∈ Xis the linear subspace of An

K defined by the system of linear equations

L1 = · · · = Lm = 0,

where Li is defined by

Li =n∑

j=1

∂fi

∂xj(α)(xj − αj).

We have that dim Tα(X) ≥ dim X, and X is non-singular at the point αif dim Tα(X) = dim X. The locus of points in X which are singular is aproper closed subset of X.

The fundamental problem of resolution of singularities is to performsimple algebraic transformations of X so that the transform Y of X is non-singular everywhere. To be precise, we seek a resolution of singularitiesof X; that is, a proper birational morphism Φ : Y → X such that Y isnon-singular.

The problem of resolution when K has characteristic zero has been stud-ied for some time. In fact we will see (Chapters 2 and 3) that the method ofNewton for determining the analytical branches of a plane curve singularityextends to give a proof of resolution for algebraic curves. The first algebraicproof of resolution of surface singularities is due to Zariski [86]. We give

1

2 1. Introduction

a modern treatment of this proof in Chapter 8. A study of this proof issurely the best introduction to the methods and ideas underlying the recentemphasis on valuation-theoretic methods in resolution problems.

The existence of a resolution of singularities has been completely solvedby Hironaka [52], in all dimensions, when K has characteristic zero. We givea simplified proof of this theorem, based on the proof of canonical resolutionby Encinas and Villamayor ([40], [41]), in Chapter 6.

When K has positive characteristic, resolution is known for curves, sur-faces and 3-folds (with char(K) > 5). The first proof in positive characteris-tic of resolution of surfaces and of resolution for 3-folds is due to Abhyankar[1], [4].

We give several proofs, in Chapters 2, 3 and 4, of resolution of curvesin arbitrary characteristic. In Chapter 5 we give a proof of resolution ofsurfaces in arbitrary characteristic.

1.1. Notation

The notation of Hartshorne [47] will be followed, with the following differ-ences and additions.

By a variety over a field K (or a K-variety), we will mean an open subsetof an equidimensional reduced subscheme of the projective space Pn

K . Thusan integral variety is a “quasi-projective variety” in the classical sense. Acurve is a one-dimensional variety. A surface is a two-dimensional variety,and a 3-fold is a three-dimensional variety. A subvariety Y of a variety Xis a closed subscheme of X which is a variety.

An affine ring is a reduced ring which is of finite type over a field K.If X is a variety, and I is an ideal sheaf on X, we denote V (I) =

spec(OX/I) ⊂ X. If Y is a subscheme of a variety X, we denote the idealof Y in X by IY . If W1,W2 are subschemes of a variety of X, we will denotethe scheme-theoretic intersection of W1 and W2 by W1 · W2. This is thesubscheme

W1 ·W2 = V (IW1 + IW2) ⊂W.

A hypersurface is a codimension one subvariety of a non-singular variety.

Chapter 2

Non-singularity andResolution ofSingularities

2.1. Newton’s method for determining the branches of aplane curve

Newton’s algorithm for solving f(x, y) = 0 by a fractional power seriesy = y(x

1m ) can be thought of as a generalization of the implicit function

theorem to general analytic functions. We begin with this algorithm becauseof its simplicity and elegance, and because this method contains some ofthe most important ideas in resolution. We will see (in Section 2.5) thatthe algorithm immediately gives a local solution to resolution of analyticplane curve singularities, and that it can be interpreted to give a globalsolution to resolution of plane curve singularities (in Section 3.5). All ofthe proofs of resolution in this book can be viewed as generalizations ofNewton’s algorithm, with the exception of the proof that curve singularitiescan be resolved by normalization (Theorems 2.14 and 4.3).

Suppose K is an algebraically closed field of characteristic 0, K[[x, y]] isa ring of power series in two variables and f ∈ K[[x, y]] is a non-unit, suchthat x - f . Write f =

∑i,j aijx

iyj with aij ∈ K. Let

mult(f) = mini+ j | aij 6= 0,

and

mult(f(0, y)) = minj | a0j 6= 0.

3

4 2. Non-singularity and Resolution of Singularities

Set r0 = mult(f(0, y)) ≥ mult(f). Set

δ0 = min

i

r0 − j: j < r0 and aij 6= 0

.

δ0 = ∞ if and only if f = uyr0 , where u is a unit in K[[x, y]]. Suppose thatδ0 <∞. Then we can write

f =∑

i+δ0j≥δ0r0

aijxiyj

with a0r0 6= 0, and the weighted leading form

Lδ0(x, y) =∑

i+δ0j=δ0r0

aijxiyj = a0r0y

r0 + terms of lower degree in y

has at least two non-zero terms. We can thus choose 0 6= c1 ∈ K so that

Lδ0(1, c1) =∑

i+δ0j=δ0r0

aijcj1 = 0.

Write δ0 = p0

q0, where q0, p0 are relatively prime positive integers. We make

a transformationx = xq0

1 , y = xp01 (y1 + c1).

Thenf = xr0p0

1 f1(x1, y1),where

(2.1) f1(x1, y1) =∑

i+δ0j=δ0r0

aij(c1 + y1)j + x1H(x1, y1).

By our choice of c1, f1(0, 0) = 0. Set r1 = mult(f1(0, y1)). We see thatr1 ≤ r0. We have an expansion

f1 =∑

aij(1)xi1y

j1.

Set

δ1 = min

i

r1 − j: j < r1 and aij(1) 6= 0

,

and write δ1 = p1

q1with p1, q1 relatively prime. We can then choose c2 ∈ K

for f1, in the same way that we chose c1 for f , and iterate this process,obtaining a sequence of transformations

(2.2)x = xq0

1 , y = xp01 (y1 + c1),

x1 = xq12 , y1 = xp1

2 (y2 + c2),...

Either this sequence of transformations terminates after a finite number nof steps with δn = ∞, or we can construct an infinite sequence of trans-formations with δn < ∞ for all n. This allows us to write y as a series inascending fractional powers of x.

2.1. Newton’s method for determining the branches of a plane curve 5

As our first approximation, we can use our first transformation to solvefor y in terms of x and y1:

y = c1xδ0 + y1x

δ0 .

Now the second transformation gives us

y = c1xδ0 + c2x

δ0+δ1q0 + y2x

δ0+δ1q0 .

We can iterate this procedure to get the formal fractional series

(2.3) y = c1xδ0 + c2x

δ0+δ1q0 + c3x

δ0+δ1q0

+δ2

q0q1 + · · · .

Theorem 2.1. There exists an i0 such that δi ∈ N for i ≥ i0.

Proof. ri = mult(fi(0, yi)) are monotonically decreasing, and positive forall i, so it suffices to show that ri = ri+1 implies δi ∈ N. Without loss ofgenerality, we may assume that i = 0 and r0 = r1. f1(x1, y1) is given by theexpression (2.1). Set

g(t) = f1(0, t) =∑

i+δ0j=δ0r0

aij(c1 + t)j .

g(t) has degree r0. Since r1 = r0, we also have mult(g(t)) = r0. Thusg(t) = a0r0t

r0 , and ∑i+δ0j=δ0r0

aijtj = a0r0(t− c1)r0 .

In particular, since K has characteristic 0, the binomial theorem shows that

(2.4) ai,r0−1 6= 0,

where i is a natural number with i+ δ0(r0 − 1) = δ0r0. Thus δ0 ∈ N.

We can thus find a natural number m, which we can take to be thesmallest possible, and a series

p(t) =∑

biti

such that (2.3) becomes

(2.5) y = p(x1m ).

For n ∈ N, set

pn(t) =n∑

i=1

biti.

Using induction, we can show that

mult(f(tm, pn(t)) →∞


as n→∞, and thus f(tm, p(t)) = 0. Thus

(2.6) y =∑

bixim

is a branch of the curve f = 0. This expansion is called a Puiseux series(when r0 = mult(f)), in honor of Puiseux, who introduced this theory intoalgebraic geometry.

Remark 2.2. Our proof of Theorem 2.1 is not valid in positive character-istic, since we cannot conclude (2.4). Theorem 2.1 is in fact false over fieldsof positive characteristic. See Exercise 2.4 at the end of this section.

Suppose that f ∈ K[[x, y]] is irreducible, and that we have found asolution y = p(x

1m ) to f(x, y) = 0. We may suppose that m is the smallest

natural number for which it is possible to write such a series. y − p(x1m )

divides f in R1 = K[[x1m , y]]. Let ω be a primitive m-th root of unity in K.

Since f is invariant under the K-algebra automorphism φ of R1 determinedby x

1m → ωx

1m and y → y, it follows that y − p(ωjx

1m )) | f in R1 for all j,

and thus y = p(ωjx1m ) is a solution to f(x, y) = 0 for all j. These solutions

are distinct for 0 ≤ j ≤ m− 1, by our choice of m. The series

g =m−1∏j=0

(y − p(ωjx

1m )

)is invariant under φ, so g ∈ K[[x, y]] and g | f in K[[x, y]], the ring ofinvariants of R1 under the action of the group Zm generated by φ. Since fis irreducible,

f = u

m−1∏j=0

(y − p(ωjx

1m )

) ,

where u is a unit in K[[x, y]].

Remark 2.3. Some letters of Newton developing this idea are translated(from Latin) in [18].

After we have defined non-singularity, we will return to this algorithmin (2.7) of Section 2.5, to see that we have actually constructed a resolutionof singularities of a plane curve singularity.

Exercise 2.4.

1. Construct a Puiseux series solution to

f(x, y) = y4 − 2x3y2 − 4x5y + x6 − x7 = 0

over the complex numbers.

2.2. Smoothness and non-singularity 7

2. Apply the algorithm of this section to the equation

f(x, y) = yp + yp+1 + x = 0

over an algebraically closed field k of characteristic p. What is theresulting fractional series?

2.2. Smoothness and non-singularity

Definition 2.5. Suppose that X is a scheme. X is non-singular at P ∈ Xif OX,P is a regular local ring.

Recall that a local ring R, with maximal ideal m, is regular if the di-mension of m/m2 as an R/m vector space is equal to the Krull dimensionof R.

For varieties over a field, there is a related notion of smoothness.Suppose that K[x1, . . . , xn] is a polynomial ring over a field K,

f1, . . . , fm ∈ K[x1, . . . , xn].

We define the Jacobian matrix

J(f ;x) = J(f1, . . . , fm;x1, . . . , xn) =

∂f1

∂x1· · · ∂f1

∂xn...

∂fm

∂x1· · · ∂fm

∂xn

.

Let I = (f1, . . . , fm) be the ideal generated by f1, . . . , fm, and define R =K[x1, . . . , xn]/I. Suppose that P ∈ spec(R) has ideal m in K[x1, . . . , xn]and ideal n in R. Let K(P ) = Rn/nn. We will say that J(f ;x) has rankl at P if the image of the l-th Fitting ideal Il(J(f ;x)) of l × l minors ofJ(f, x) in K(P ) is K(P ) and the image of the s-th Fitting ideal Is(J(f ;x))in K(P ) is (0) for s > l.

Definition 2.6. Suppose that X is a variety of dimension s over a field K,and P ∈ X. Suppose that U = spec(R) is an affine neighborhood of P suchthat R ∼= K[x1, . . . , xn]/I with I = (f1, . . . , fm). Then X is smooth over Kif J(f ;x) has rank n− s at P .

This definition depends only on P and X and not on any of the choicesof U , x or f . We verify this is the appendix.

Theorem 2.7. Let K be a field. The set of points in a K-variety X whichare smooth over K is an open set of X.

Theorem 2.8. Suppose that X is a variety over a field K and P ∈ X.

1. Suppose that X is smooth over K at P . Then P is a non-singularpoint of X.


2. Suppose that P is a non-singular point of X and K(P ) is separablygenerated over K. Then X is smooth over K at P .

In the case when K is algebraically closed, Theorems 2.7 and 2.8 areproven in Theorems I.5.3 and I.5.1 of [47]. Recall that an algebraicallyclosed field is perfect. We will give the proofs of Theorems 2.7 and 2.8 forgeneral fields in the appendix.

Corollary 2.9. Suppose that X is a variety over a perfect field K andP ∈ X. Then X is non-singular at P if and only if X is smooth at P overK.

Proof. This is immediate since an algebraic function field over a perfect fieldK is always separably generated over K (Theorem 13, Section 13, ChapterII, [92]).

In the case when X is an affine variety over an algebraically closedfield K, the notion of smoothness is geometrically intuitive. Suppose thatX = V (I) = V (f1, . . . , fm) ⊂ An

K is an s-dimensional affine variety, whereI = (f1, . . . , fm) is a reduced and equidimensional ideal. We interpret theclosed points of X as the set of solutions to f1 = · · · = fm = 0 in Kn. Weidentify a closed point p = (a1, . . . , an) ∈ V (I) ⊂ An

K with the maximalideal m = (x1 − a1, . . . , xn − an) of K[x1, . . . , xn]. For 1 ≤ i ≤ m,

fi ≡ fi(p) + Li mod m2,

where

Li =n∑

j=1

∂fi

∂xi(p)(xj − aj).

But fi(p) = 0 for all i since p is a point of X. The tangent space to X inAn

K at the point p is

Tp(X) = V (L1, . . . , Lm) ⊂ AnK .

We see that dimTp(X) = n − rank(J(f ;x)(p)). Thus dimTp(X) ≥ s (byRemark A.9), and X is non-singular at p if and only if dimTp(X) = s.

Theorem 2.10. Suppose that X is a variety over a field K. Then the setof non-singular points of X is an open dense set of X.

This theorem is proven when K is algebraically closed in Corollary I.5.3[47]. We will prove Theorem 2.10 when K is perfect in the appendix. Thegeneral case is proven in the corollary to Theorem 11 [85].

Exercise 2.11. Consider the curve y2−x3 = 0 in A2K , over an algebraically

closed field K of characteristic 0 or p > 3.

2.3. Resolution of singularities 9

1. Show that at the point p1 = (1, 1), the tangent space Tp1(X) is theline

−3(x− 1) + 2(y − 1) = 0.2. Show that at the point p2 = (0, 0), the tangent space Tp2(X) is the

entire plane A2K .

3. Show that the curve is singular only at the origin p2.

2.3. Resolution of singularities

Suppose that X is a K-variety, where K is a field.

Definition 2.12. A resolution of singularities of X is a proper birationalmorphism φ : Y → X such that Y is a non-singular variety.

A birational morphism is a morphism φ : Y → X of varieties suchthat there is a dense open subset U of X such that φ−1(U) → U is anisomorphism. IfX and Y are integral andK(X) andK(Y ) are the respectivefunction fields of X and Y , φ is birational if and only if φ∗ : K(X) → K(Y )is an isomorphism.

A morphism of varieties φ : Y → X is proper if for every valuationring V with morphism α : spec(V ) → X, there is a unique morphism β :spec(V ) → Y such that φ β = α. If X and Y are integral, and K(X) isthe function field of X, then we only need consider valuation rings V suchthat K ⊂ V ⊂ K(X) in the definition of properness.

The geometric idea of properness is that every mapping of a “formal”curve germ into X lifts uniquely to a morphism to Y .

One consequence of properness is that every proper map is surjective.The properness assumption rules out the possibility of “resolving” by takingthe birational resolution map to be the inclusion of the open set of non-singular points into the given variety, or the mapping of the non-singularpoints of a partial resolution to the variety.

Birational proper morphisms of non-singular varieties (over a field ofcharacteristic zero) can be factored by alternating sequences of blow-ups andblow-downs of non-singular subvarieties, as shown by Abramovich, Karu,Matsuki and Wlodarczyk [11].

We can extend our definition of a resolution of singularities to arbitraryschemes. A reasonable category to consider is excellent (or quasi-excellent)schemes (defined in IV.7.8 [45] and on page 260 of [66]). The definition ofexcellence is extremely technical, but the idea is to give minimal conditionsensuring that the the singular locus is preserved by natural base extensionssuch as completion. There are examples of non-excellent schemes whichadmit a resolution of singularities. Rotthaus [74] gives an example of a


regular local ring R of dimension 3 containing a field which is not excellent.In this case, spec(R) is a resolution of singularities of spec(R).

2.4. Normalization

Suppose that R is an affine domain with quotient field L. Let S be theintegral closure of R in L. Since R is affine, S is finite over R, so thatS is affine (cf. Theorem 9, Chapter V [92]). We say that spec(S) is thenormalization of spec(R).

Suppose that V is a valuation ring of L such that R ⊂ V . V is integrallyclosed in L (although V need not be Noetherian). Thus S ⊂ V . Themorphism spec(V ) → spec(R) lifts to a morphism spec(V ) → spec(S), sothat spec(S) → spec(R) is proper.

If X is an integral variety, we can cover X by open affines spec(Ri)with normalization spec(Si). The spec(Si) patch to a variety Y called thenormalization of X. Y → X is proper, by our local proof.

For a general (reduced but not necessarily integral) variety X, the nor-malization of X is the disjoint union of the normalizations of the irreduciblecomponents of X.

A ring R is normal if Rp is an integrally closed domain for all p ∈spec(R). If R is an affine ring and spec(S) is the normalization of spec(R)we constructed above, then S is a normal ring.

Theorem 2.13 (Serre). A Noetherian ring A is normal if and only if Ais R1 (Ap is regular if ht(p) ≤ 1) and S2 (depthAp ≥ min(ht(p), 2) for allp ∈ spec(A)).

A proof of this theorem is given in Theorem 23.8 [66].

Theorem 2.14. Suppose that X is a 1-dimensional variety over a field K.Then the normalization of X is a resolution of singularities.

Proof. Let X be the normalization of X. All local rings OX,p of pointsp ∈ X are local rings of dimension 1 which are integrally closed, so Theorem2.13 implies they are regular.

As an example, consider the curve singularity y2 − x3 = 0 in A2, withaffine ring R = K[x, y]/(y2 − x3). In the quotient field L of R we have therelation ( y

x)2 − x = 0. Thus yx is integral over R. Since R[ y

x ] = K[ yx ] is a

regular ring, it must be the integral closure of R in L. Thus spec(R[ yx ]) →

spec(R) is a resolution of singularities.Normalization is in general not enough to resolve singularities in dimen-

sion larger than 1. As an example, consider the surface singularity X defined

2.5. Local uniformization and generalized resolution problems 11

by z2−xy = 0 in A3K , where K is a field. The singular locus of X is defined

by the ideal J = (z2 − xy, y, x, 2z).√J = (x, y, z), so the singular locus of

X is the origin in A2. Thus R = K[x, y, z]/(z2 − xy) is R1. Since z2 − xyis a regular element in K[x, y, z], R is Cohen-Macaulay, and hence R is S2

(Theorem 17.3 [66]). We conclude that X is normal, but singular.Kawasaki [58] has proven that under extremely mild assumptions a

Noetherian scheme admits a Macaulayfication (all local rings have maxi-mal depth). In general, Cohen-Macaulay schemes are far from being non-singular, but they do share many good homological properties with non-singular schemes.

A scheme which does not admit a resolution of singularities is givenby the example of Nagata (Example 3 in the appendix to [68]) of a one-dimensional Noetherian domain R whose integral closure R is not a finite R-module. We will show that such a ring cannot have a resolution of singulari-ties. Suppose that there exists a resolution of singularities φ : Y → spec(R).That is, Y is a regular scheme and φ is proper and birational. Let L bethe quotient field of R. Y is a normal scheme by Theorem 2.13. Thus Yhas an open cover by affine open sets U1, . . . , Us such that Ui

∼= spec(Ti),with Ti = R[gi1, . . . , git] where gij ∈ L, and each Ti is integrally closed. Theintegrally closed subring

T = Γ(Y,OY ) =s⋂

i=1

Ti

of L is a finite R-module since φ is proper (Theorem III, 3.2.1 [44] or Corol-lary II.5.20 [47] if φ is assumed to be projective).

Exercise 2.15.

1. Let K = Zp(t), where p is a prime and t is an indeterminate. LetR = K[x, y]/(xp + yp − t), X = spec(R). Prove that X is non-singular, but there are no points of X which are smooth over K.

2. Let K = Zp(t), where p > 2 is a prime and t is an indeterminate.Let R = K[x, y]/(x2 + yp − t), X = spec(R). Prove that X isnon-singular, and X is smooth over K at every point except at theprime (yp − t, x).

2.5. Local uniformization and generalized resolutionproblems

Suppose that f is irreducible in K[[x, y]]. Then the Puiseux series (2.5) ofSection 2.1 determines an inclusion

(2.7) R = K[[x, y]]/(f(x, y)) → K[[t]]


with x = tm, y = p(t) the series of (2.5). Since the m chosen in (2.5) ofSection 2.1 is the smallest possible, we can verify that t is in the quotientfield L of R. Since the quotient field of K[[t]] is generated by 1, t, . . . , tm−1

over L, we conclude that R and K[[t]] have the same quotient field. (2.7)is an explicit realization of the normalization of the curve singularity germf(x, y) = 0, and thus spec(K[[t]]) → spec(R) is a resolution of singularities.The quotient field L of K[[t]] has a valuation ν defined for non-zero h ∈ Lby

ν(h(t)) = n ∈ Z

if h(t) = tnu, where u is a unit in K[[t]]. This valuation can be understoodin R by the Puiseux series (2.6).

More generally, we can consider an algebraic function field L over a fieldK. Suppose that V is a valuation ring of L containing K. The problem oflocal uniformization is to find a regular local ring R, essentially of finite typeover K and with quotient field L such that the valuation ring V dominates R(R ⊂ V and the intersection of the maximal ideal of V with R is the maximalideal of R). The exact relationship of local uniformization to resolution ofsingularities is explained in Section 8.3

If L has dimension 1 over K, then the valuation rings V of L whichcontain K are precisely the local rings of points on the unique non-singularprojective curve C with function field L (cf. Section I.6 [47]). The Newtonmethod of Section 2.1 can be viewed as a solution to the local uniformizationproblem for complex analytic curves.

If L has dimension ≥ 2 over K, there are many valuations rings V of Lwhich are non-Noetherian (see the exercise of Section 8.1).

Zariski proved local uniformization for two-dimensional function fieldsover an algebraically closed field of characteristic zero in [86]. He was ableto patch together local solutions to prove the existence of a resolution ofsingularities for algebraic surfaces over an algebraically closed field of char-acteristic zero. He later was able to prove local uniformization for algebraicfunction fields of characteristic zero in [87]. This method leads to an ex-tremely difficult patching problem in higher dimensions, which Zariski wasable to solve in dimension 3 in [90]. Zariski’s proof of local uniformizationcan be considered as an extension of the Newton method to general val-uations. In Chapter 8, we present Zariski’s proof of resolution of surfacesingularities through local uniformization.

Local uniformization has been proven for two-dimensional function fieldsin positive characteristic by Abhyankar. Abhyankar has proven resolution ofsingularities in positive characteristic for surfaces and for three-dimensionalvarieties, [1],[3],[4].


There has recently been a resurgence of interest in local uniformizationin positive characteristic, and significant progress is being made. Some of therecent papers on this area are: Hauser [50], Heinzer, Rotthaus and Wiegand[51], Kuhlmann [59], [60], Piltant [71], Spivakovsky [77], Teissier [78].

Some related resolution type problems are resolution of vector fields,resolution of differential forms and monomialization of morphisms.

Suppose that X is a variety which is smooth over a perfect field K, andD ∈ Hom(Ω1

X/K ,OX) is a vector field. If p ∈ X is a closed point, andx1, . . . , xn are regular parameters at p, then there is a local expression

D = a1(x)∂

∂x1+ · · ·+ an(x)

∂

∂xn,

where ai(x) ∈ OX,p. We can associate an ideal sheaf ID to D on X by

ID,p = (a1, . . . , an)

for p ∈ X.There are an effective divisor FD and an ideal sheaf JD such that V (JD)

has codimension ≥ 2 in X and ID = OX(−FD)JD. The goal of resolution ofvector fields is to find a proper morphism π : Y → X so that if D′ = π−1(D),then the ideal ID′ ⊂ OY is as simple as possible.

This was accomplished by Seidenberg for vector fields on non-singularsurfaces over an algebraically closed field of characteristic zero in [75]. Thebest statement that can be attained is that the order of JD′ at p (DefinitionA.17) is νp(JD′) ≤ 1 for all p ∈ Y . The basic invariant considered in theproof is the order νp(JD). While this is an upper semi-continuous functionon Y , it can go up under a monoidal transform. However, we have thatνq(JD′) ≤ νp(JD)+1 if π : Y → X is the blow-up of p, and q ∈ π−1(p). Thisshould be compared with the classical resolution problem, where a basicresult is that order cannot go up under a permissible monoidal transform(Lemma 6.4). Resolution of vector fields for smooth surfaces over a perfectfield has been proven by Cano [19]. Cano has also proven a local theoremfor resolution for vector fields over fields of characteristic zero [20], whichimplies local resolution (along a valuation) of a vector field. The statementis that we can achieve νq(JD′) ≤ 1 (at least locally along a valuation) after amorphism π : Y → X. There is an analogous problem for differential forms.A recent paper on this is [21].

We can also consider resolution problems for morphisms f : Y → X ofvarieties. The natural question to ask is if it is possible to perform monoidaltransforms (blow-ups of non-singular subvarieties) over X and Y to producea morphism which is a monomial mapping.


Definition 2.16. Suppose that Φ : X → Y is a dominant morphism of non-singular irreducible K-varieties (where K is a field of characteristic zero).Φ is monomial if for all p ∈ X there exists an etale neighborhood U of p,uniformizing parameters (x1, . . . , xn) on U , regular parameters (y1, . . . , ym)in OY,Φ(p), and a matrix (aij) of non-negative integers (which necessarilyhas rank m) such that

y1 = xa111 · · ·xa1n

n ,...

ym = xam11 · · ·xamn

n .

Definition 2.17. Suppose that Φ : X → Y is a dominant morphism ofintegral K-varieties. A morphism Ψ : X1 → Y1 is a monomializationof Φ [27] if there are sequences of blow-ups of non-singular subvarietiesα : X1 → X and β : Y1 → Y , and a morphism Ψ : X1 → Y1 such that thediagram

X1Ψ→ Y1

↓ ↓X

Φ→ Y

commutes, and Ψ is a monomial morphism.

If Φ : X → Y is a dominant morphism from a 3-dimensional variety toa surface (over an algebraically closed field of characteristic 0), then thereis a monomialization of Φ [27]. A generalized multiplicity is defined inthis paper, and it can go up, causing a very high complexity in the proof.An extension of this result to strongly prepared morphisms from n-folds tosurfaces is proven in [33]. It is not known if monomialization is true evenfor birational morphisms of varieties of dimension ≥ 3, although it is truelocally along a valuation, from the following Theorem 2.18. Theorem 2.18is proven when the quotient field of S is finite over the quotient field of Rin [25]. The proof for general field extensions is in [28].

Theorem 2.18 (Theorem 1.1 [26], Theorem 1.1 [28]). Suppose that R ⊂ Sare regular local rings, essentially of finite type over a field K of character-istic zero.

Let V be a valuation ring of K which dominates S. Then there existsequences of monoidal transforms R → R′ and S → S′ such that V domi-nates S′, S′ dominates R′ and there are regular parameters (x1, ...., xm) inR′, (y1, ..., yn) in S′, units δ1, . . . , δm ∈ S′ and an m × n matrix (aij) ofnon-negative integers such that rank(aij) = m is maximal and

(2.8)x1 = ya11

1 .....ya1nn δ1,

...xm = yam1

1 .....yamnn δm.


Thus (since char(K) = 0) there exists an etale extension S′ → S′′,where S′′ has regular parameters y1, . . . , yn such that x1, . . . , xm are puremonomials in y1, . . . , yn.

The standard theorems on resolution of singularities allow one to easilyfind R′ and S′ such that (2.8) holds, but, in general, we will not have theessential condition rank(aij) = m. The difficulty of the problem is to achievethis condition.

This result gives very simple structure theorems for the ramificationof valuations in characteristic zero function fields [35]. We discuss some ofthese results in Chapter 9. A generalization of monomialization in character-istic p function fields of algebraic surfaces is obtained in [34] and especiallyin [35].

We point out that while it seems possible that Theorem 2.18 does hold inpositive characteristic, there are simple examples in positive characteristicwhere a monomialization does not exist. The simplest example is the mapof curves

y = xp + xp+1

in characteristic p.A quasi-complete variety over a field K is an integral finite type K-

scheme which satisfies the existence part of the valuative criterion for proper-ness (Hironaka, Chapter 0, Section 6 of [52] and Chapter 8 of [26]).

The construction of a monomialization by quasi-complete varieties fol-lows from Theorem 2.18. Theorem 2.19 is proven for generically finite mor-phisms in [26] and for arbitrary morphisms in Theorem 1.2 [28].

Theorem 2.19 (Theorem 1.2 [26],Theorem 1.2 [28]). Let K be a field ofcharacteristic zero, Φ : X → Y a dominant morphism of proper K-varieties.Then there exist birational morphisms of non-singular quasi-complete K-varieties α : X1 → X and β : Y1 → Y , and a monomial morphism Ψ :X1 → Y1 such that the diagram

X1Ψ→ Y1

↓ ↓X Φ

→ Y

commutes and α and β are locally products of blow-ups of non-singular sub-varieties. That is, for every z ∈ X1, there exist affine neighborhoods V1 of zand V of x = α(z) such that α : V1 → V is a finite product of monoidal trans-forms, and there exist affine neighborhoods W1 of Ψ(z), W of y = β(Ψ(z))such that β : W1 →W is a finite product of monoidal transforms.


A monoidal transform of a non-singular K-scheme S is the map T → Sinduced by an open subset T of proj(

⊕In), where I is the ideal sheaf of a

non-singular subvariety of S.The proof of Theorem 2.19 follows from Theorem 2.18, by patching a

finite number of local solutions. The resulting schemes may not be separated.It is an extremely interesting question to determine if a monomialization

exists for all morphisms of varieties (over a field of characteristic zero). Thatis, the conclusions of Theorem 2.19 hold, but with the stronger conditionsthat α and β are products of monoidal transforms on proper varieties X1

and Y1.

Chapter 3

Curve Singularities

3.1. Blowing up a point on A2

Suppose that K is an algebraically closed field.Let

U1 = spec(K[s, t]) = A2K , U2 = spec(K[u, v]) = A2

K .

Define a K-algebra isomorphism

λ : K[s, t]s = K[s, t,1s] → K[u, v]v = K[u, v,

1v]

by λ(s) = 1v , λ(t) = uv. We define a K-variety B0 by patching U1 to U2 on

the open sets

U2 − V (v) = spec(K[u, v]v) and U1 − V (s) = spec(K[s, t]s)

by the isomorphism λ. The K-algebra homomorphisms

K[x, y] → K[s, t]

defined by x 7→ st, y 7→ t, and

K[x, y] → K[u, v]

defined by x 7→ u, y 7→ uv, are compatible with the isomorphism λ, so weget a morphism

π : B(p) = B0 → U0 = spec(K[x, y]) = A2K ,

where p denotes the origin of U0. Upon localization of the above maps, weget isomorphisms

K[x, y]y ∼= K[s, t]t and K[x, y]x ∼= K[u, v]u.

17

18 3. Curve Singularities

Thus π : U1 − V (t) ∼= U0 − V (y), π : U2 − V (u) ∼= U0 − V (x), and π is anisomorphism over U0 − V (x, y) = U0 − p. Moreover,

π−1(p) ∩ U1 = V (st, t) = V (t) ⊂ spec(K[s, t]) = spec(K[s]),

π−1(p) ∩ U2 = V (u, uv) = V (u) ⊂ spec(K[u, v]) = spec(K[1s]),

by the identification s = 1v . Thus π−1(p) ∼= P1. Set E = π−1(p). We have

“blown up p” into a codimension 1 subvariety of B(p), isomorphic to P1.Set R = K[x, y], m = (x, y). Then

U1 = spec(K[s, t]) = spec(K[x

y, y]) = spec(R[

x

y]),

U2 = spec(K[x,y

x]) = spec(R[

y

x]).

We see that

B(p) = spec(R[x

y]) ∪ spec(R[

y

x]) = proj(

⊕n≥0

mn).

Suppose that q ∈ π−1(p) is a closed point. If q ∈ U1, then its associatedideal mq is a maximal ideal of R1 = K[x

y , y] which contains (x, y)R1 = yR1.Thus mq = (y, x

y − α) for some α ∈ K. If we set y1 = y, x1 = xy − α, we see

that there are regular parameters (x1, y1) in OB(p),q such that

x = y1(x1 + α), y = y1.

By a similar calculation, if q ∈ π−1(p) and q ∈ U2, there are regular param-eters (x1, y1) in OB(p),q such that

x = x1, y = x1(y1 + β)

for some β ∈ K. If the constant α or β is non-zero, then q is in both U1

and U2. Thus the points in π−1(p) can be expressed (uniquely) in one of theforms

x = x1, y = x1(y1 + α) with α ∈ K, or x = x1y1, y = y1.

Since B(p) is projective over spec(R), it certainly is proper over spec(R).However, it is illuminating to give a direct proof.

Lemma 3.1. B(p) → spec(R) is proper.

Proof. Suppose that V is a valuation ring containing R. Then yx or x

y ∈ V .Say y

x ∈ V . Then R[ yx ] ⊂ V , and we have a morphism

spec(V ) → spec(R[y

x]) ⊂ B(p)

which lifts the morphism spec(V ) → spec(R).

3.1. Blowing up a point on A2 19

More generally, suppose that S = spec(R) is an affine surface over a fieldL, and p ∈ S is a non-singular closed point. After possibly replacing S withan open subset spec(Rf ), we may assume that the maximal ideal of p in Ris mp = (x, y). We can then define the blow up of p in S by

(3.1) π : B(p) = proj(⊕n≥0

mnp ) → S.

We can write B(p) as the union of two affine open subsets:

B(p) = spec(R[x

y]) ∪ spec(R[

y

x]).

π is an isomorphism over S − p, and π−1(p) ∼= P1.Suppose that S is a surface, and p ∈ S is a non-singular point, with ideal

sheaf mp ⊂ OS . The blow-up of p ∈ S is

π : B(p) = proj(⊕n≥0

mnp ) → S.

π is an isomorphism away from p, and if U = spec(R) ⊂ S is an affine openneighborhood of p in S such that

(x, y) = Γ(U,mp) ⊂ R

is the maximal ideal of p in R, then the map π : π−1(U) → U is defined bythe construction (3.1).

Suppose that C ⊂ A2K is a curve. Since K[x, y] is a unique factorization

domain, there exists f ∈ K[x, y] such that V (f) = C.If q ∈ C is a closed point, we will denote the corresponding maximal

ideal of K[x, y] by mq. We set

νq(C) = maxr | f ∈ mrq

(more generally, see Definition A.17). νq(C) is both the multiplicity and theorder of C at q.

Lemma 3.2. q ∈ A2K is a non-singular point of C if and only if νq(C) = 1.

Proof. Let R = K[x, y]mq , and suppose that mq = (x, y). Let T = R/fR.

If νq(C) ≥ 2, then f ∈ m2q , and mqT/m

2qT

∼= mq/m2q has dimension

2 > 1, so that T is not regular and q is singular on C. However, if νq(C) = 1,then we have

f ≡ αx+ βy mod m2q ,

where α, β ∈ K and at least one of α and β is non-zero. Without loss ofgenerality, we may suppose that β 6= 0. Thus

y ≡ −αβx mod m2

q + (f),


and x is a K generator of mqT/m2qT

∼= mq/m2q + (f). We see that

dimK mqT/m2qT = 1

and q is non-singular on C.

Remark 3.3. This lemma is also true in the situation where q is a non-singular point on a non-singular surface S (over a field K) and C is a curvecontained in S. We need only modify the proof by replacing R with theregular local ring R = OS,q, which has regular parameters (x, y) which area K(q) basis of mq/m

2q , and since R is a unique factorization domain, there

is f ∈ R such that f = 0 is a local equation of C at q.

Let p be the origin in A2K , and suppose that νp(C) = r > 0. Let

π : B(p) → A2K be the blow-up of p. The strict transform C of C in B(p) is

the Zariski closure of π−1(C − p) ∼= C − p in B(p). Let E = π−1(p) be theexceptional divisor . Set-theoretically, π−1(C) = C ∪ E.

For some aij ∈ K, we have a finite sum

f =∑

i+j≥r

aijxiyj

with aij 6= 0 for some i, j with i + j = r. In the open subset U2 =spec(K[x1, y1]) of B(p), where

x = x1, y = x1y1,

x1 = 0 is a local equation for E. Also,

f = xr1f1,

wheref1 =

∑i+j≥r

aijxi+j−r1 yj

1.

x1 - f1 since νp(C) = r, so that f1 = 0 is a local equation of the stricttransform C of C in U2.

In the open subset U1 = spec(K[x1, y1]) of B(p), where

x = x1y1, y = y1,

y1 = 0 is a local equation for E and

f = yr1f1,

wheref1 =

∑i+j≥r

aijxi1y

i+j−r1

is a local equation of C in U1.

3.1. Blowing up a point on A2 21

As a scheme, we see that π−1(C) = C + rE. The scheme-theoreticpreimage of C is called the total transform of C. Define the leading form off to be

L =∑

i+j=r

aijxiyj .

Suppose that q ∈ π−1(p). There are regular parameters (x1, y1) at q ofone of the forms

x = x1, y = x1(y1 + α) or x = x1y1, y = y1.

In the first case f1 = fxr1

= 0 is a local equation of C at q, where

f1 =∑

i+j=r

aij(y1 + α)j + x1Ω

for some polynomial Ω. Thus νq(C) ≤ r, and νq(C) = r implies∑i+j=r

aij(y1 + α)j = a0ryr1.

We then see that ∑i+j=r

aijyj1 = a0r(y1 − α)r

andL = a0r(y − αx)r.

In the second case f1 = fyr1

= 0 is a local equation of C at q, where

f1 =∑

i+j=r

aijxi1 + y1Ω

for some series Ω. Thus νq(C) ≤ r, and νq(C) = r implies∑i+j=r

aijxi1 = ar0x

r1

andL = ar0x

r.

We then see that there exists a point q ∈ π−1(p) such that νq(C) = ronly when L = (ax+ by)r for some constants a, b ∈ K. Since r > 0, there isat most one point q ∈ π1(p) where the multiplicity does not drop.

Exercise 3.4. Suppose that K[x1, . . . , xn] is a polynomial ring over a per-fect field K and f ∈ K[x1, . . . , xn] is such that f(0, . . . , 0) = 0. Letm = (x1, . . . , xn), a maximal ideal of K[x1, . . . , xn].

Define ord f(0, . . . , 0, xn) = r if

xrn | f(0, . . . , 0, xn) and xr+1

n - f(0, . . . , 0, xn).


Show that R = (K[x1, . . . , xn]/(f))m is a regular local ring iford f(0, . . . , 0, xn) = 1.

3.2. Completion

Suppose that A is a local ring with maximal ideal m. A coefficient field ofA is a subfield L of A which is mapped onto A/m by the quotient mappingA→ A/m.

A basic theorem of Cohen is that an equicharacteristic complete localring contains a coefficient field (Theorem 27 Section 12, Chapter VIII [92]).This leads to Cohen’s structure theorem (Corollary, loc. cit.), which showsthat an equicharacteristic complete regular local ring A is isomorphic to aformal power series ring over a field. In fact, if L is a coefficient field ofA, and if (x1, . . . , xn) is a regular system of parameters of A, then A is thepower series ring

A = L[[x1, . . . , xn]].

We further remark that the completion of a local ring R is a regular localring if and only if R is regular (cf. Section 11, Chapter VIII [92]).

Lemma 3.5. Suppose that S is a non-singular algebraic surface defined overa field K, p ∈ S is a closed point, π : B = B(p) → S is the blow-up of p, andsuppose that q ∈ π−1(p) is a closed point such that K(q) is separable overK(p). Let R1 = OS,p and R2 = OB,q, and suppose that K1 is a coefficientfield of R1, (x, y) are regular parameters in R1. Then there exist a coefficientfield K2 = K1(α) of R2 and regular parameters (x1, y1) of R2 such that

π∗ : R1 → R2

is the map given by ∑i,j≥0

aijxiyj →

∑i,j≥0

aijxi+j1 (y1 + α)j ,

where aij ∈ K1, or K2 = K1 and

π∗ : R1 → R2

is the map given by ∑i,j≥0

aijxiyj →

∑i,j≥0

aijxi1y

i+j1 .

Proof. We have R1 = K1[[x, y]], and a natural homomorphism

OS → OS,p → R1

3.2. Completion 23

which induces

q ∈ B(p)×S spec(R1) = spec(R1[y

x]) ∪ spec(R1[

x

y]).

Let mq be the ideal of q in B(p)×S spec(R1). If q ∈ spec(R1[ yx ]), then

K(q) ∼= R1[y

x]/mq

∼= K1[y

x]/(f(

y

x))

for some irreducible polynomial f( yx) in the polynomial ring K1[ y

x ]. SinceK(q) is separable overK1

∼= K(p), f is separable. We havemq = (x, f( yx)) ⊂

R1[ yx ].Let α ∈ R1[ y

x ]/mq be the class of yx . We have the residue map φ : R2 → L

where L = R1[ yx ]/mq.

There is a natural embedding of K1 in R2. We have a factorization off(t) = (t − α)γ(t) in L[t], where t − α and γ(t) are relatively prime. ByHensel’s Lemma (Theorem 17, Section 7, Chapter VIII [92]) there is α ∈ R2

such that φ(α) = α and f(t) = (t − α)γ(t) in R2[t], where φ(γ(t)) = γ(t).The subfield K2 = K1[α] of R2 is thus a coefficient field of R2, and mqR2 =(x, y

x −α). Thus R2 = K2[[x, yx −α]]. Set x1 = x, y1 = y

x −α. The inclusion

R1 = K1[[x, y]] → R2 = K2[[x1, y1]]

is natural. A series ∑aijx

iyj

with coefficients aij ∈ K1 maps to the series∑aijx

i+j1 (y1 + α)j .

We now give an example to show that even if a regular local ring containsa field, there may not be a coefficient field of the completion of the ringcontaining that field. Thus the above lemma does not extend to non-perfectfields.

Let K = Zp(t), where t is an indeterminate. Let R = K[x](xp−t). Sup-pose that R has a coefficient field L containing K. Let φ : R→ R/m be theresidue map. Since φ | L is an isomorphism, there exists λ ∈ L such thatλp = t. Thus (xp − t) = (x− λ)p in R. But this is impossible, since (xp − t)is a generator of mR, and is thus irreducible.

Theorem 3.6. Suppose that R is a reduced affine ring over a field K, andA = Rp, where p is a prime ideal of R. Then the completion A = Rp of Awith respect to its maximal ideal is reduced.


When K is a perfect field, this is a theorem of Chevalley (Theorem 31,Section 13, Chapter VIII [92]). The general case follows from Scholie IV7.8.3 (vii) [45].

However, the property of being a domain is not preserved under com-pletion. A simple example is f = y2 − x2 + x3. f is irreducible in C[x, y],but is reducible in the completion C[[x, y]]:

f = y2 − x2 − x3 = (y − x√

1 + x)(y + x√

1 + x).

The first parts of the expansions of the two factors are

y − x− 12x2 + · · ·

andy + x+

12x2 + · · · .

Lemma 3.7 (Weierstrass Preparation Theorem). Let K be a field, andsuppose that f ∈ K[[x1, . . . , xn, y]] is such that

0 < r = ν(f(0, . . . , 0, y)) = maxn | yn divides f(0, . . . , 0, y) <∞.

Then there exist a unit series u in K[[x1, . . . , xn, y]] and non-unit seriesai ∈ K[[x1, . . . , xn]] such that

f = u(yr + a1yr−1 + · · ·+ ar).

A proof is given in Theorem 5, Section 1, Chapter VII [92].A concept which will be important in this book is the Tschirnhausen

transformation, which generalizes the ancient notion of “completion of thesquare” in the solution of quadratic equations.

Definition 3.8. Suppose that K is a field of characteristic p ≥ 0 andf ∈ K[[x1, . . . , xn, y]] has an expression

f = yr + a1yr−1 + · · ·+ ar

with ai ∈ K[[x1, . . . , xn]] and p = 0 or p - r. The Tschirnhausen transfor-mation of f is the change of variables replacing y with

y′ = y +a1

r.

f then has an expression

f = (y′)r + b2(y′)r−2 + · · ·+ br

with bi ∈ K[[x1, . . . , xn]] for all i.

Exercise 3.9. Suppose that (R,m) is a complete local ring and L1 ⊂ R isa field such that R/m is finite and separable over L1. Use Hensel’s Lemma(Theorem 17, Section 7, Chapter VIII [92]) to prove that there exists acoefficient field L2 of R containing L1.

3.3. Blowing up a point on a non-singular surface 25

3.3. Blowing up a point on a non-singular surface

We define the strict transform, the total transform, and the multiplicity (ororder) νp(C) analogously to the definition of Section 3.1 (More generally,see Section 4.1 and Definition A.17).

Lemma 3.10. Suppose that X is a non-singular surface over an alge-braically closed field K, and C is a curve on X. Suppose that p ∈ X andνp(C) = r. Let π : B(p) → X be the blow-up of p, C the strict transform ofC, and suppose that q ∈ π−1(p). Then νq(C) ≤ r, and if r > 0, there is atmost one point q ∈ π−1(p) such that νq(C) = r.

Proof. Let f = 0 be a local equation of C at p. If (x, y) are regularparameters in OX,p, we can write

f =∑

aijxiyj

as a series of order r in OX,p∼= K[[x, y]]. Let

L =∑

i+j=r

aijxiyj

be the leading form of f .If q ∈ π−1(p), then OB(p),q has regular parameters (x1, y1) such that

x = x1, y = x1(y1 + α), or x = x1y1, y = y1.

This substitution induces the inclusion

K[[x, y]] = OX,p → OB(p),q = K[[x1, y1]].

First suppose that (x1, y1) are defined by

x = x1, y = x1(y1 + α).

Then a local equation for C at q is f1 = fxr1, which has the expansion

f1 =∑


for some series Ω. We finish the proof as in Section 3.1.

Suppose that C is the curve y2−x3 = 0 in A2K . The only singular point

of C is the origin p. Let π : B(p) → A2K be the blow-up of p, C the strict

transform of C, and E the exceptional divisor E = π−1(p).On U1 = spec(K[x

y , y]) ⊂ B(p) we have coordinates x1, y1 with x = x1y1,y = y1. A local equation for E on U1 is y1 = 0. Also,

y2 − x3 = y21(1− x3

1y1).

A local equation for C on U1 is 1− x31y1 = 0, which is a unit on E ∩ U1, so

C ∩ E ∩ U1 = ∅.


On U2 = spec(K[x, yx ]) ⊂ B(p) we have coordinates x1, y1 with x = x1,

y = x1y1. A local equation for E on U2 is x1 = 0. Also,

y2 − x3 = x21(y

21 − x1).

A local equation for C on U2 is y21−x1 = 0, which has order ≤ 1 everywhere.

Thus C is non-singular, andC → C

is a resolution of singularities. Finally,

Γ(C,OC) = K[x1, y1]/(y21 − x1) = K[y1] = K[

y

x].

This is the normalization of K[x, y]/(y2 − x3) that we computed in Section2.4.

As a further example, consider the curve C with equation y2 − x5 inA2. The only singular point of C is the origin p. Let π : B(p) → A2 bethe blow-up of p, E be the exceptional divisor, C be the strict transform ofC. There is only one singular point q on C. Regular parameters at q are(x1, y1), where x = x1, y = x1y1. C has the local equation

y21 − x3

1 = 0.

In this example νq(C) = νp(C) = 2, so the multiplicity has not dropped.However, this singularity is resolved after blowing up q, as we calculated

in the previous example.These examples suggest an algorithm to resolve curve singularities. First

blow up all singular points. If the resulting curve is not resolved, blow upthe new singular points, and repeat as long as the curve is singular.

This algorithm ends after a finite number of iterations in a non-singularcurve. In Sections 3.4 and 3.5 we will prove this for curves embedded in anon-singular surface, first over an algebraically closed field K of character-istic zero, and then over an arbitrary field.

3.4. Resolution of curves embedded in a non-singularsurface I

In this section we consider curves embedded in a non-singular surface overan algebraically closed field K of characteristic zero, and prove that theirsingularities can be resolved.

The theorem is proved by passing to the completion of the local ring ofthe surface at a singular point of the curve. This allows us to view a localequation of the curve as a power series in two variables. Our main invariantis the multiplicity of the curve at a point. We know that this multiplicity

3.4. Resolution of curves embedded in a non-singular surface I 27

cannot go up after blowing up, so we must show that it will eventually godown, after enough blowing up.

Theorem 3.11. Suppose that C is a curve on a non-singular surface Xover an algebraically closed field K of characteristic zero. Then there existsa sequence of blow ups of points λ : Y → X such that the strict transformC of C on Y is non-singular.

Proof. Let r = max(νp(C) | p ∈ C. If r = 1, C is non-singular, so we mayassume that r > 1. The set p ∈ C | ν(p) = r is a subset of the singularlocus of C, which is a proper closed subset of the 1-dimensional variety C,so it is a finite set.

The proof is by induction on r.We can construct a sequence of projective morphisms

(3.2) · · · → Xnπn→ · · · π1→ X0 = X,

where each πn+1 : Xn+1 → Xn is the blow-up of all points on the stricttransform Cn of C which have multiplicity r on Cn. If this sequence is finite,then there is an integer n such that all points on the strict transform Cn ofC have multiplicity ≤ r − 1. By induction on r, we can repeat this processto construct the desired morphism Y → X which induces a resolution of C.

We will assume that the sequence (3.2) is infinite, and derive a contra-diction. If it is infinite, then for all n ∈ N there are closed points pn ∈ Cn

which have multiplicity r on Cn and such that πn+1(pn+1) = pn for all n.Let Rn = OXn,pn for all n. We then have an infinite sequence of completionsof quadratic transforms of local rings

R0 → R1 → · · · → Rn → · · · .

Suppose that (x, y) are regular parameters in R0 = OX,p, and f = 0 is alocal equation of C in R0 = K[[x, y]]. f is reduced by Theorem 3.6. After alinear change of variables in (x, y), we may assume that ν(f(0, y)) = r. Bythe Weierstrass preparation theorem,

f = u(yr + a1(x)yr−1 + · · ·+ ar(x)),

where u is a unit series. We now “complete the r-th power of f”. Set

(3.3) y = y +a1(x)r

.

Then

(3.4) yr + a1(x)yr−1 + · · ·+ ar(x) = yr + b2(x)yr−2 + · · ·+ br(x)

for some series bi(x). We may thus assume that

(3.5) f = yr + a2(x)yr−2 + · · ·+ ar(x).


Since f is reduced, we must have some ai(x) 6= 0. Set

(3.6) n = min

mult(ai(x))i

.

We have n ≥ 1, since ν(f) = r.

OX1,p1 has regular parameters x1, y1 such that

1. x = x1, y = x1(y1 + α) with α ∈ K, or2. x = x1y1, y = y1

In case 2,f = yr

1f1,

where

f1 = 1 +a2(x1y1)

y21

+ · · ·+ ar(x1y1)yr1

is a local equation of the strict transform C1 of C at p1. f1 is a unit, so thatνp1(C1) = 0. Thus case 2 cannot occur. In case 1,

f = xr1f1

where

f1 = (y1 + α)r +a2(x1)x2

1

(y1 + α)r−2 + · · ·+ ar(x1)xr

1

.

f1 = 0 is a local equation at p1 for C1. If α 6= 0, then νp1(C1) ≤ r − 1, sothis case cannot occur.

If α = 0, and νp1(C1) = r, then f1 has the form of (3.5), with n decreasedby 1.

Thus for some index i with i ≤ n we have that pi must have multiplicity< r on Ci, a contradiction to the assumption that (3.2) has infinite length.

The transformation of (3.3) is called a Tschirnhausen transformation(Definition 3.8). The Tschirnhausen transformation finds a (formal) curve ofmaximal contact H = V (y) for C at p. The Tschirnhausen transformationwas introduced as a fundamental method in resolution of singularities byAbhyankar.

Definition 3.12. Suppose that C is a curve on a non-singular surface Sover a field K, and p ∈ C. A non-singular curve H = V (g) ⊂ spec(OS,p) isa formal curve of maximal contact for C at p if whenever

Sn → Sn−1 → · · · → S1 → spec(OS,p)

is a sequence of blow-ups of points pi ∈ Si, such that the strict transformCi of C in Si contains pi with νpi(Ci) = νp(C) for i ≤ n, then the stricttransform Hi of H in Si contains pi.

3.5. Resolution of curves embedded in a non-singular surface II 29

Exercise 3.13.

1. Resolve the singularities by blowing up:a. x2 = x4 + y4,b. xy = x6 + y6,c. x3 = y2 + x4 + y4,d. x2y + xy3 = x4 + y4,e. y2 = xn,f. y4 − 2x3y2 − 4x5y + x6 − x7.

2. Suppose that D is an effective divisor on a non-singular surface S.That is, there exist curves Ci on S and numbers ri ∈ N such thatID = Ir1

C1· · · Irn

Cn. D has simple normal crossings (SNCs) on S if

for every p ∈ S there exist regular parameters (x, y) in OS,p suchthat if f = 0 is a local equation for D at p, then f = unit xayb inOS,p.

Find a sequence of blow-ups of points making the total trans-form of y2 − x3 = 0 a SNCs divisor.

3. Suppose that D is an effective divisor on a non-singular surface S.Show that there exists a sequence of blow-ups of points

Sn → · · · → S

such that the total transform π∗(D) is a SNCs divisor.

3.5. Resolution of curves embedded in a non-singularsurface II

In this section we consider curves embedded in a non-singular surface, de-fined over an arbitrary field K.

Lemma 3.14. Suppose that K is a field, S is a non-singular surface overK, C is a curve on S and p ∈ C is a closed point. Suppose that π : B =B(p) → S is the blow-up of p, C is the strict transform of C on B andq ∈ π−1(p) ∩ C. Then

νq(C) ≤ νp(C),

and ifνq(C) = νp(C),

then K(p) = K(q).

Proof. Let R = OS,p, (x, y) be regular parameters in R. Since there is anatural embedding of π−1(p) in B ×S spec(R), it follows that

q ∈ B ×S spec(R) = spec(R[x

y]) ∪ spec(R[

y

x]).


Without loss of generality, we may assume that q ∈ spec(R[ yx ]). Let K1

∼=K(p) be a coefficient field of R, m ⊂ R[ y

x ] the ideal of q. Since R[ yx ]/(x, y)

∼= K1[ yx ], there exists an irreducible monic polynomial h(t) ∈ K1[t] such that

mR[y

x] = (x, h(

y

x)).

Let f ∈ R be such that f = 0 is a local equation of C, and let r = νp(C).We have an expansion in R,

f =∑

i+j≥r

aijxiyj

with aij ∈ K1. Thus f = xrf1, where f1 = 0 is a local equation of the stricttransform C of C in spec(R[ y

x ]) and

f1 =∑

i+j=r

aij(y

x)j + xΩ

in R[ yx ]. Let n = νq(C). Then

h(y

x)n divides

∑i+j=r

aij(y

x)j

in K1[ yx ]. Also,

deg(∑

i+j=r

aij(y

x)j) ≤ r

in K1[ yx ] implies n ≤ r and νq(C) = r implies deg(h( y

x)) = 1, so thath = y

x − α with α ∈ K1 and (by Lemma 3.5) there exist regular parameters(x1, y1) in mq ⊂ OB(p),q such that

OS,p = K1[[x, y]] → K1[[x1, y1]] = OB(p),q

is the natural K1∼= K(p) algebra homomorphism

x = x1, y = x1(y1 + α).

Theorem 3.15. Suppose that C is a curve which is a subvariety of a non-singular surface X over an infinite field K. Then there exists a sequence ofblow-ups of points λ : Y → X such that the strict transform C of C on Y isnon-singular.

Proof. Let r = maxνp(C) | p ∈ C. If r = 1, C is non-singular, so we mayassume that r > 1. The set p ∈ C | νp(C) = r is a subset of the singularlocus of C, which is a proper closed subset of the 1-dimensional variety C,so it is a finite set.

The proof is by induction on r.


We can construct a sequence of projective morphisms

(3.7) · · · → Xnπn→ · · · → X1

π1→ X0 = X,

where each πn+1 : Xn+1 → Xn is the blow-up of all points on the stricttransform Cn of C which have multiplicity r on Cn. If this sequence isfinite, then there is an integer n such that all points on the strict transformCn of C have multiplicity ≤ r − 1. By induction on r, we can then repeatthis process to construct the desired morphism Y → X which induces aresolution of C.

We will assume that the sequence (3.7) is infinite, and derive a contra-diction. If it is infinite, then for all n ∈ N there are closed points pn ∈ Cn

which have multiplicity r on Cn and are such that πn+1(pn+1) = pn. LetRn = OXn,pn for all n. We then have an infinite sequence of completions ofquadratic transforms (blow-ups of maximal ideals) of local rings

R = R0 → R1 → · · · → Rn → · · · .

We will define

δpi ∈1r!

N

such that

(3.8) δpi = δpi−1 − 1

for all i ≥ 1. We can thus conclude that pi has multiplicity < r on thestrict transform Ci of C for some natural number i ≤ δp + 1. From thiscontradiction it will follow that (3.7) is a sequence of finite length.

Suppose that f ∈ R0 = OX,p is such that f = 0 is a local equation of Cand (x, y) are regular parameters in R = OX,p such that r = mult(f(0, y)).We will call such (x, y) good parameters for f . Let K ′ be a coefficient fieldof R. There is an expansion

f =∑

i+j≥r

aijxiyj

with aij ∈ K ′ for all i, j and a0r 6= 0. Define

(3.9) δ(f ;x, y) = min

i

r − j| j < r and aij 6= 0

.

We have δ(f ;x, y) ≥ 1 since (x, y) are good parameters We thus have anexpression (with δ = δ(f ;x, y))

f =∑

i+jδ≥rδ

aijxiyj = Lδ +

∑i+jδ>rδ

aijxiyj ,


where

(3.10) Lδ =∑

i+jδ=rδ

aijxiyj = a0ry

r + Λ

is such that a0r 6= 0 and Λ is not zero.Suppose that (x, y) are fixed good parameters of f . Define

(3.11)

δp = supδ(f ;x, y1) | y = y1+n∑

i=1

bixi with n ∈ N and bi ∈ K ′ ∈ 1

r!N∪∞.

We cannot have δp = ∞, since then there would exist a series

y = y1 +∞∑i=1

bixi

such that δ(f ;x, y1) = ∞, and thus there would be a unit series γ in R suchthat

f = γyr1.

But then r = 1 since f is reduced in R, a contradiction. We see then thatδp ∈ 1

r!N.After possibly making a substitution

y = y1 +n∑

i=1

bixi

with bi ∈ K ′, we may assume that δp = δ(f ;x, y).Let δ = δp.Since νp1(C1) = r, by Lemma 3.14, we have an expression

R1 = K ′[[x1, y1]],

where R→ R1 is the natural K ′-algebra homomorphism such that either

x = x1, y = x1(y1 + α)

for some α ∈ K ′, orx = x1y1, y = y1.

We first consider the case where x = x1y1, y = y1. A local equation ofC1 (in spec(R1)) is f1 = 0, where

f1 =f

yr1

=∑

i+j=r

aijxi1 + y1Ω = a0r + x1g + y1h

for some Ω, g, h ∈ R1. Thus f1 is a unit in R1, a contradiction.


Now consider the case where x = x1, y = x1(y1 + α) with 0 6= α ∈ K ′.A local equation of C1 is

f1 =f

xr1

=∑

i+j=r


for some Ω ∈ R1. Since νp1(C1) = r,∑i+j=r

aij(y1 + α)j = a0ryr1.

Substituting t = y1 + α, we have∑i+j=r

aijtj = a0r(t− α)r.

If we now substitute t = yx and multiply the series by xr, we obtain the

leading form L of f ,

L =∑

i+j=r

aijxiyj = a0r(y − αx)r = a0ry

r + · · ·+ (−1)ra0rαrxr.

Comparing with (3.10), we see that r = rδ and δ = 1, so that Lδ = L. Butwe can replace y with y−αx to increase δ, a contradiction to the maximalityof δ. Thus νp1(C1) < r, a contradiction.

Finally, consider the case x = x1, y = x1y1. Set δ′ = δ − 1. A localequation of C1 is

f1 = fxr1

=∑

i+jδ≥rδ

aijxi+j−r1 yj

1

=∑

i+jδ′=rδ′

ai−j+r,jxi1y

j1 +

∑i+jδ′>rδ′

ai−j+r,jxi1y

j1,

where i = i+ j − r. Since

Lδ′(x1, y1) =1xr

1

Lδ(x1, x1y1)

has at least two non-zero terms, we see that δ(f1;x1, y1) = δ′ = δ − 1.We will now show that δp1 = δ(f1;x1, y1). Suppose not. Then we can

make a substitution

y1 = y′1 −∑

bixi1 = y′1 − bxd

1 + higher order terms in x1

with 0 6= b ∈ K ′ such that

δ(f1;x1, y′1) > δ(f1;x1, y1).

Then we have an expression∑i+jδ′=rδ′

ai−j+r,jxi1(y

′1 − bxd

1)j = a0r(y′1)

r +∑

i+jδ′>rδ′

bijxi1(y

′1)

j ,


so that δ′ = d ∈ N, and∑i+jδ′=rδ′

ai−j+r,jxi1(y

′1 − bxd

1)j = a0r(y′1)

r.

Thus ∑i+jδ=rδ

aijxi+j−r1 yj

1 =∑

i+jδ′=rδ′

ai−j+r,jxi1y

j1 = a0r(y1 + bxδ′

1 )r.

If we now multiply these series by xr1, we obtain

Lδ =∑

i+jδ=rδ

aijxiyj = a0r(y + bxδ)r.

But we can now make the substitution y′ = y − bxδ and see that

δ(f ;x, y′) > δ(f ;x, y) = δp,

a contradiction, from which we conclude that δp1 = δ′ = δp − 1. We canthen inductively define δpi for i ≥ 0 by this procedure so that (3.8) holds.The conclusions of the theorem now follow.

Remark 3.16.

1. This proof is a generalization of the algorithm of Section 2.1. Amore general version of this, valid in arbitrary two-dimensional reg-ular local rings, can be found in [5] or [73].

2. Good parameters (x, y) for f which achieve δp = δ(f ;x, y) are suchthat y = 0 is a (formal) curve of maximal contact for C at p (Defi-nition 3.12).

Exercise 3.17.

1. Prove that the δp defined in formula (3.11) is equal to

δp = supδ(f ;x, y) | (x, y) are good parameters for f.

Thus δp does not depend on the initial choice of good parameters,and δp is an invariant of p.

2. Suppose that νp(C) > 1. Let π : B(p) → X be the blow-up of p, Cbe the strict transform of C. Show that there is at most one pointq ∈ π−1(p) such that νq(C) = νp(C).

3. The Newton polygon N(f ;x, y) is defined as follows. Let I be theideal in R = K[[x, y]] generated by the monomials xαyβ such thatthe coefficient aαβ of xαyβ in f(x, y) =

∑aαβx

αyβ is not zero. Set

P (f ;x, y) = (α, β) ∈ Z2|xαyβ ∈ I.

Now define N(f ;x, y) to be the smallest convex subset of R2 suchthat N(f ;x, y) contains P (f ;x, y) and if (α, β) ∈ N(f ;x, y), (s, t) ∈


R2+, then (α + s, β + t) ∈ N(f ;x, y). Now suppose that (x, y)

are good parameters for f (so that ν(f(0, y)) = ν(f) = r). Then(0, r) ∈ N(f ;x, y). Let the slope of the steepest segment ofN(f ;x, y) be s(f ;x, y). We have 0 ≥ s(f ;x, y) ≥ −1, since aαβ = 0if α+ β < r. Show that

δ(f ;x, y) = − 1s(f ;x, y)

.

Chapter 4

Resolution TypeTheorems

4.1. Blow-ups of ideals

Suppose that X is a variety, and J ⊂ OX is an ideal sheaf. The blow-up ofJ is

π : B = B(J ) = proj(⊕n≥0

J n) → X.

B is a variety and π is proper. If X is projective then B is projective. π isan isomorphism over X − V (J ), and JOB is a locally principal ideal sheaf.

If U ⊂ X is an open affine subset, and R = Γ(U,OX), I = Γ(U,J ) =(f1, . . . , fm) ⊂ R, then

π−1(U) = B(I) = proj(⊕n≥0

In).

If X is an integral scheme, we have

proj(⊕n≥0

In) =m⋃

i=1

spec(R[f1

fi, · · · , fm

fi]).

Suppose that W ⊂ X is a subscheme with ideal sheaf IW on X.The total transform of W , π∗(W ), is the subscheme of B with the ideal

sheaf

Iπ∗(W ) = IWOB.

Set-theoretically, π∗(W ) is the preimage of W , π−1(W ).

37

38 4. Resolution Type Theorems

Let U = X − V (J ). The strict transform W of W is the Zariski closureof π−1(W ∩ U) in B(J ).

Lemma 4.1. Suppose that R is a Noetherian ring, I, J ⊂ R are ideals andI = q1 ∩ · · · ∩ qm is a primary decomposition, with primes pi =

√qi. Then

∞⋃n=0

(I : Jn) =⋂

i|J 6⊂pi

qi.

The proof of Lemma 4.1 follows easily from the definition of a primaryideal. Geometrically, Lemma 4.1 says that

⋃∞n=0(I : Jn) removes the pri-

mary components qi of I such that V (qi) ⊂ V (J).

Thus we see that the strict transform W of W has the ideal sheaf

IW =∞⋃

n=0

(IWOB : J nOB).

For q ∈ B,

IW ,q = f ∈ OB,q | fJ nq ⊂ IWOB,q for some n ≥ 0.

The strict transform has the property that

W = B(JOW ) = proj(⊕n≥0

(JOW )n)

This is shown in Corollary II.7.15 [47].

Theorem 4.2 (Universal Property of Blowing Up). Suppose that I is anideal sheaf on a variety V and f : W → V is a morphism of varieties suchthat IOW is locally principal. Then there is a unique morphism g : W →B(I) such that f = π g.

This is proved in Proposition II.7.14 [47].

Theorem 4.3. Suppose that C is an integral curve over a field K. Considerthe sequence

(4.1) · · · → Cnπn→ · · · → C1

π1→ C,

where Cn+1 → Cn is obtained by blowing up the (finitely many) singularpoints on Cn. Then this sequence is finite. That is, there exists n such thatCn is non-singular.

Proof. Without loss of generality, C = spec(R) is affine. Let R be theintegral closure of R in the function field K(C) of C. R is a regular ring.Let C = spec(R). All ideals in R are locally principal (Proposition 9.2, page94 [13]). By Theorem 4.2, we have a factorization

C → Cn → · · · → C1 → C

4.1. Blow-ups of ideals 39

for all n. Since C → C is finite, Ci+1 → Ci is finite for all i, and thereexist affine rings Ri such that Ci = spec(Ri). For all n we have sequencesof inclusions

R→ R1 → · · · → Rn → R.

Here Ri 6= Ri+1 for all i, since a maximal ideal m in Ri is locally principalif and only if (Ri)m is a regular local ring. Since R is finite over R, we havethat (4.1) is finite.

Corollary 4.4. Suppose that X is a variety over a field K and C is anintegral curve on X. Consider the sequence

· · · → Xnπn→ Xn−1 → · · · → X1

π1→ X,

where πi+1 is the blow-up of all points of X which are singular on the stricttransform Ci of C on Xi. Then this sequence is finite. That is, there existsan n such that the strict transform Cn of C is non-singular.

Proof. The induced sequence

· · · → Cn → · · · → C1 → C

of blow-ups of points on the strict transform Ci of C is finite by Theorem4.3.

Theorem 4.5. Suppose that V and W are varieties and V → W is aprojective birational morphism. Then V ∼= B(I) for some ideal sheaf I ⊂OW .

This is proved in Proposition II.7.17 [47].Suppose that Y is a non-singular subvariety of a varietyX. The monoidal

transform of X with center Y is π : B(IY ) → X. We define the exceptionaldivisor of the monoidal transform π (with center Y ) to be the reduced divisorwith the same support as π∗(Y ).

In general, we will define the exceptional locus of a proper birationalmorpism φ : V → W to be the (reduced) closed subset of V on which φ isnot an isomorphism.

Remark 4.6. If Y has codimension greater than or equal to 2 in X, thenthe exceptional divisor of the monoidal transform π : B(IY ) → X withcenter Y , is the exceptional locus of π. However, if X is singular and notlocally factorial, it may be possible to blow up a non-singular codimension 1subvariety Y of X to get a monoidal transform π such that the exceptionallocus of π is a proper closed subset of the exceptional divisor of π. For anexample of this kind, see Exercise 4.7 at the end of this section. If X isnon-singular and Y has codimension 1 in X, then π is an isomorphism and


we have defined the exceptional divisor to be Y . This property will be usefulin our general proof of resolution in Chapter 6.

Exercise 4.7.

1. Let K be an algebraically closed field, and X be the affine surface

X = spec(K[x, y, z]/(xy − z2)).

Let Y = V (x, z) ⊂ X, and let π : B → X be the monoidal trans-form centered at the non-singular curve Y . Show that π is a res-olution of singularities. Compute the exceptional locus of π, andcompute the exceptional divisor of the monoidal transform π of Y .

2. Let K be an algebraically closed field, and X be the affine 3-fold

X = spec(K[x, y, z, w]/(xy − zw).

Show that the monodial transform π : B → X with center Y isa resolution of singularities in each of the following cases, and de-scribe the exceptional locus and exceptional divisor (of the monoidaltransform):

a. Y = V ((x, y, z, w)),b. Y = V ((x, z)),c. Y = V ((y, z)).

Show that the monoidal transforms of cases b and c factor themonoidal transform of case a.

4.2. Resolution type theorems and corollaries

Resolution of singularities. Suppose that V is a variety. A resolution ofsingularities of V is a proper birational morphism φ : W → V such that Wis non-singular.

Principalization of ideals. Suppose that V is a non-singular variety, andI ⊂ OV is an ideal sheaf. A principalization of I is a proper birationalmorphism φ : W → V such that W is non-singular and IOW is locallyprincipal.

Suppose that X is a non-singular variety, and I ⊂ OX is a locallyprincipal ideal. We say that I has simple normal crossings (SNCs) at p ∈X if there exist regular parameters (x1, . . . , xn) in OX,p such that Ip =xa1

1 · · ·xann OX,p for some natural numbers a1, . . . , an.

4.2. Resolution type theorems and corollaries 41

Suppose that D is an effective divisor on X. That is, D = r1E1 + · · ·+rnEn, where Ei are irreducible codimension 1 subvarities of X, and ri arenatural numbers. D has SNCs if ID = Ir1

E1· · · Irn

Enhas SNCs.

Suppose that W ⊂ X is a subscheme, and D is a divisor on X. We saythat W has SNCs with D at p if

1. W is non-singular at p,

2. D is a SNC divisor at p,

3. there exist regular parameters (x1, . . . , xn) in OX,p and r ≤ n suchthat IW,p = (x1, . . . , xr) (or IW,p = OX,p) and

ID,p = xa11 · · ·xan

n OX,p

for some ai ∈ N.

Embedded resolution. Suppose that W is a subvariety of a non-singularvariety X. An embedded resolution of W is a proper morphism π : Y → Xwhich is a product of monoidal transforms, such that π is an isomorphism onan open set which intersects every component of W properly, the exceptionaldivisor of π is a SNC divisor D, and the strict transform W of W has SNCswith D.

Resolution of indeterminacy. Suppose that φ : W → V is a rationalmap of proper K-varieties such that W is non-singular. A resolution ofindeterminacy of φ is a proper non-singular K-variety X with a birationalmorphism ψ : X →W and a morphism λ : X → V such that λ = φ ψ.

Lemma 4.8. Suppose that resolution of singularities is true for K-varietiesof dimension n. Then resolution of indeterminacy is true for rational mapsfrom K-varieties of dimension n.

Proof. Let φ : W → V be a rational map of proper K-varieties, where Wis non-singular. Let U be a dense open set of W on which φ is a morphism.Let Γφ be the Zariski closure of the image in W ×V of the map U →W ×Vdefined by p 7→ (p, φ(p)). By resolution of singularities, there is a properbirational morphism X → Γφ such that X is non-singular.

Theorem 4.9. Suppose that K is a perfect field, resolution of singularities istrue for projective hypersurfaces over K of dimension n, and principalizationof ideals is true for non-singular varieties of dimension n over K. Thenresolution of singularities is true for projective K-varieties of dimension n.


Proof. Suppose V is an n-dimensional projective K-variety. If V1, . . . , Vr

are the irreducible components of V , we have a projective birational mor-phism from the disjoint union of the Vi to V . Thus it suffices to assume thatV is irreducible. The function field K(V ) of V is a finite separable extensionof a rational function field K(x1, . . . , xn) (Chapter II, Theorem 30, page 104[92]). By the theorem of the primitive element,

K(V ) ∼= K(x1, . . . , xn)[xn+1]/(f).

We can clear the denominator of f so that

f =∑

ai1...in+1xi11 · · ·x

in+1

n+1

is irreducible in K[x1, . . . , xn+1]. Let

d = maxi1 + · · · in+1 | ai1...in+1 6= 0.

SetF =

∑ai1...in+1X

d−(i1+···+in+1)0 Xi1

1 · · ·Xin+1

n+1 ,

the homogenization of f . Let V be the variety defined by F = 0 in Pn+1.Then K(V ) ∼= K(V ) implies there is a birational rational map from V toV . That is, there is a birational morphism φ : V → V , where V is a denseopen subset of V . Let Γφ be the Zariski closure of (a, φ(a)) | a ∈ V in V × V . We have birational projection morphisms π1 : Γφ → V andπ2 : Γφ → V . Γφ is the blow-up of an ideal sheaf J on V (Theorem 4.5).By resolution of singularities for n-dimensional hypersurfaces, we have aresolution of singularities f : W ′ → V . By principalization of ideals in non-singular varieties of dimension n, we have a principalization g : W → W ′

for JOW ′ . By the universal property of blowing up (Theorem 4.2), wehave a morphism h : W → Γφ. Hence π1 h : W → V is a resolution ofsingularities.

Corollary 4.10. Suppose that C is a projective curve over an infinite perfectfield K. Then C has a resolution of singularities.

Proof. By Theorem 3.15, resolution of singularities is true for projectiveplane curves over K. All ideal sheaves on a non-singular curve are locallyprincipal, since the local rings of points are Dedekind local rings.

Note that Theorem 4.3 and Corollary 4.4 are stronger results than Corol-lary 4.10. However, the ideas of the proofs of resolution of plane curves inTheorems 3.11 and 3.15 extend to higher dimensions, while Theorem 4.3does not. As a consequence of embedded resolution of curve singularities ona surface, we will prove principalization of ideals in non-singular varieties ofdimension two. This simple proof is by Abhyankar (Proposition 6, [73]).

4.2. Resolution type theorems and corollaries 43

Theorem 4.11. Suppose that S is a non-singular surface over a field K,and J ⊂ OS is an ideal sheaf. Then there exists a finite sequence of blow-upsof points T → S such that JOT is locally principal.

Proof. Without loss of generality, S is affine. Let

J = Γ(J ,OS) = (f1, . . . , fm).

By embedded resolution of curve singularities on a surface (Exercise 3.13 ofSection 3.4) applied to

√f1 · · · fm ∈ S, there exists a sequence of blow-ups

of points π1 : S1 → S such that f1 · · · fm = 0 is a SNC divisor everywhereon S1.

Let p1, . . . , pr be the finitely many points on S1 where JOS1 is notlocally principal (they are contained in the singular points of V (

√JOS1)).

Suppose that p ∈ p1, . . . , pr. By induction on the number of generators ofJ , we may assume that JOS1,p = (f, g). After possibly multiplying f and gby units in OS1,p, there are regular parameters (x, y) in OS1,p such that

f = xayb, g = xcyd.

Set tp = (a − c)(b − d). (f, g) is principal if and only if tp ≥ 0. By ourassumption, tp < 0. Let π2 : S2 → S1 be the blow-up of p, and suppose thatq ∈ π−1(p). We will show that tq > tp.

The only two points q where (f, g) may not be principal have regularparameters (x1, y1) such that

x = x1, y = x1y1 or x = x1y1, y = y1.

If x = x1, y = x1y1, then

f = xa+b1 yb

1, g = xc+d1 yd

1 ,

tq = (a+ b− (c+ d))(b− d) = (b− d)2 + (a− c)(b− d) > tp.

The case when x = x1y1, y = y1 is similar.By induction on mintp | JOS1,p is not principal we can principalize J

after a finite number of blow-ups of points.

Chapter 5

Surface Singularities

5.1. Resolution of surface singularities

In this section, we give a simple proof of resolution of surface singularities incharacteristic zero. The proof is by the good point algorithm of Abhyankar([7], [62], [73]).

Theorem 5.1. Suppose that S is a projective surface over an algebraicallyclosed field K of characteristic 0. Then there exists a resolution of singu-larities

Λ : T → S.

Theorem 5.1 is a consequence of Theorems 5.2, 4.9 and 4.11.

Theorem 5.2. Suppose that S is a hypersurface of dimension 2 in a non-singular variety V of dimension 3, over an algebraically closed field K ofcharacteristic 0. Then there exists a sequence of blow-ups of points andnon-singular curves contained in the strict transform Si of S,

Vn → Vn−1 → · · · → V1 → V,

such that the strict transform Sn of S on Vn is non-singular.

The remainder of this section will be devoted to the proof of Theorem5.2. Suppose that V is a non-singular three-dimensional variety over analgebraically closed field K of characteristic 0, and S ⊂ V is a surface.

For a natural number t, define (Definitions A.17 and A.20)

Singt(S) = p ∈ V | νp(S) ≥ t.

By Theorem A.19, Singt(S) is Zariski closed in V .

45

46 5. Surface Singularities

Letr = maxt | Singt(S) 6= ∅

be the maximal multiplicity of points of S. There are two types of blow-upsof non-singular subvarieties on a non-singular three-dimensional variety, ablow-up of a point, and a blow-up of a non-singular curve.

We will first consider the blow-up π : B(p) → V of a closed point p ∈ V .Suppose that U = spec(R) ⊂ V is an affine open neighborhood of p, and phas ideal mp = (x, y, z) ⊂ R. Then

π−1(U) = proj(⊕

mnp ) = spec(R[

y

x,z

x]) ∪ spec(R[

x

y,z

y]) ∪ spec(R[

x

z,y

z]).

The exceptional divisor is E = π−1(p) ∼= P2.At each closed point q ∈ π−1(p), we have regular parameters (x1, y1, z1)

of the following forms:

x = x1, y = x1(y1 + α), z = x1(z1 + β),

with α, β ∈ K, x1 = 0 a local equation of E, or

x = x1y1, y = y1, z = y1(z1 + α)

with α ∈ K, y1 = 0 a local equation of E, or

x = x1z1, y = y1z1, z = z1,

z1 = 0 a local equation of E.We will now consider the blow-up π : B(C) → V of a non-singular curve

C ⊂ V . If p ∈ V and U = spec(R) ⊂ V is an open affine neighborhood of pin V such that mp = (x, y, z) and the ideal of C is I = (x, y) in R, then

π−1(U) = proj(⊕

In) = spec(R[x

y]) ∪ spec(R[

y

x]).

Also, π−1(p) ∼= P1 and π−1(C ∩ U) ∼= (C ∩ U) × P1. Let E = π−1(C) bethe exceptional divisor. E is a projective bundle over C. At each pointq ∈ π−1(p), we have regular parameters (x1, y1, z1) such that

x = x1, y = x1(y1 + α), z = z1,

where α ∈ K, x1 = 0 is a local equation of E, or

x = x1y1, y = y1, z = z1,

where y1 = 0 is a local equation of E.In this section, we will analyze the blow-up

π : B(W ) = B(IW ) → V

of a non-singular subvariety W of V above a closed point p ∈ V , bypassing to a formal neighborhood spec(OV,p) of p and analyzing the mapπ : B(IW,p) → spec(OV,p). We have a natural isomorphism B(IW,p) ∼=

5.1. Resolution of surface singularities 47

B(IW ) ×V spec(OV,p). Observe that we have a natural identification ofπ−1(p) with π−1(p).

We say that an ideal I ⊂ OV,p is algebraic if there exists an ideal J ⊂OV,p such that I = J . This is equivalent to the statement that there existsan ideal sheaf I ⊂ OV such that Ip = I.

If I ⊂ OV,p is algebraic, so that there exists an ideal sheaf I ⊂ OV suchthat Ip = I, then we can extend the blow-up π : B(I) → spec(OV,p) to ablow-up π : B(I) → V .

The maximal ideal mpOV is always algebraic. However, the ideal sheafI of a non-singular (formal) curve in spec(OV,p) may not be algebraic.

One example of a formal, non-algebraic curve is

I = (y − ex) ⊂ C[[x, y]] = OA2C,0.

A more subtle example, which could occur in the course of this section, isthe ideal sheaf of the irreducible curve

J = (y2 − x2 − x3, z) ⊂ R = K[x, y, z],

which we studied after Theorem 3.6. In R = K[[x, y, z]] we have regularparameters

x = y − x√

1 + x, y = y + x√

1 + x, z = z.

ThusJR = (xy, z) = (x, z) ∩ (y, z) ⊂ R = K[[x, y, z]].

In this example, we may be tempted to blow-up one of the two formalbranches x = 0, z = 0 or y = 0, z = 0, but the resulting blown-up schemewill not extend to a blow-up of an ideal sheaf in A3

K .A situation which will arise in this section when we will blow up a formal

curve which will actually be algebraic is given in the following lemma.

Lemma 5.3. Suppose that V is a non-singular three-dimensional variety,p ∈ V is a closed point and π : V1 = B(p) → V is the blow-up of p with ex-ceptional divisor E = π−1(p) ∼= P2. Let R = OV,p. We have a commutativediagram of morphisms of schemes

B = B(mpR) → V1 = B(p)π ↓ ↓ π

spec(R) → V

such that π−1(mp) → π−1(p) = E is an isomorphism of schemes. Supposethat I ⊂ R is any ideal, and I ⊂ OB is the strict transform of I. Then thereexists an ideal sheaf J on V1 such that JOB = IEOB + I.


Proof. IOE is an ideal sheaf on E, so there exists an ideal sheaf J ⊂ OV1

such that IE ⊂ J and J /IE∼= IOE . Thus J has the desired property.

Lemma 5.4. Suppose that V is a non-singular three-dimensional variety,S ⊂ V is a surface, C ⊂ Singr(S) is a non-singular curve, π : B(C) → V

is the blow-up of C, and S is the strict transform of S in B(C). Supposethat p ∈ C. Then νq(S) ≤ r for all q ∈ π−1(p), and there exists at most onepoint q ∈ π−1(p) such that νq(S) = r. In particular, if E = π−1(C), theneither Singr(S)∩E is a non-singular curve which maps isomorphically ontoC, or Singr(S) ∩ E is a finite union of points.

Proof. By the Weierstrass preparation theorem and after a Tschirnhausentransformation (Definition 3.8), a local equation f = 0 of S in OS,p =K[[x, y, z]] has the form

(5.1) f = zr + a2(x, y)zr−2 + · · ·+ ar(x, y).

But f ∈ (I(r)C,p) = Ir

C,p implies ∂f∂z ∈ Ir−1

C,p , and r!z = ∂r−1f∂zr−1 ∈ IC,p. Thus

z ∈ IC,p. After a change of variables in x and y, we may assume thatIC,p = (x, z). Then f ∈ Ir

C,p implies xi | ai for all i. If q ∈ π−1(p), thenOB(C),q has regular parameters (x1, y, z1) such that

x = x1z1, z = z1

orx = x1, z = x1(z1 + α)

for some α ∈ K.In the first case, a local equation of the strict transform of S is a unit.

In the second case, the strict transform of S has a local equation

f1 = (z1 + α)r +a2

x21

(z1 + α)r−2 + · · ·+ ar

xr1

.

We have ν(f1) ≤ r, and ν(f1) < r if α 6= 0.

Lemma 5.5. Suppose that p ∈ Singr(S) is a point, π : B(p) → V is theblow-up of p, S is the strict transform of S in B(p), and E = π−1(p). Thenνq(S) ≤ r for all q ∈ π−1(p), and either Singr(S)∩E is a non-singular curveor Singr(S) ∩ E is a finite union of points.

Proof. By the Weierstrass preparation theorem and after a Tschirnhausentransformation, a local equation f = 0 of S in OS,p = K[[x, y, z]] has theform

(5.2) f = zr + a2(x, y)zr−2 + · · ·+ ar(x, y).


If q ∈ π−1(p) and νq(S) ≥ r, then OB(C),q has regular parameters (x1, y, z1)such that

x = x1y1, y = y1, z = y1z1.

orx = x1, y = x1(y1 + α), z = x1z1

for some α ∈ K, and ν1(S) = r. Thus Singr(S) ∩ E is contained in theline which is the intersection of E with the strict transform of z = 0 inB(C)×V spec(OV,p).

Definition 5.6. Singr(S) has simple normal crossings (SNCs) if

1. all irreducible components of Singr(S) (which could be points orcurves) are non-singular, and

2. if p is a singular point of Singr(S), then there exist regular param-eters (x, y, z) in OV,p such that ISingr(S),p = (xy, z).

Lemma 5.7. Suppose that Singr(S) has simple normal crossings, W is apoint or an irreducible curve in Singr(S), π : V ′ = B(W ) → V is the blow-up of W and S′ is the strict transform of S on V ′. Then Singr(S′) hassimple normal crossings.

Proof. This follows from Lemmas 5.4 and 5.5 and a simple local calculationon V ′.

Definition 5.8. A closed point p ∈ S is a pregood point if, in a neighborhoodof p, Singr(S) is either empty, a non-singular curve through p, or a a unionof two non-singular curves intersecting transversally at p (satisfying 2 ofDefinition 5.6).

Definition 5.9. p ∈ S is a good point if p is pregood and for any sequence

Xn → Xn−1 → · · · → X1 → spec(OV,p)

of blow-ups of non-singular curves in Singr(Si), where Si is the strict trans-form of S ∩ spec(OV,p) on Xi, then q is pregood for all closed points q ∈Singr(Sn). In particular, Singr(Sn) contains no isolated points.

A point which is not good is called bad.

Lemma 5.10. Suppose that all points of Singr(S) are good. Then thereexists a sequence of blow ups

V ′ = Vn → · · · → V1 → V

of non-singular curves contained in Singr(Si), where Si is the strict trans-form of S in Vi, such that Singr(S′) = ∅ if S′ is the strict transform of Son V ′.


Proof. Suppose that C is a non-singular curve in Singr(S). Let π1 : V1 =B(C) → V be the blow-up of C, and let S1 be the strict transform of S.If Singr(S1) 6= ∅, we can choose another non-singular curve C1 in Singr(S1)and blow-up by π2 : V2 = B(C1) → V1. Let S2 be the strict transform ofS1. We either reach a surface Sn such that Singr(Sn) = ∅, or we obtain aninfinite sequence of blow-ups

· · · → Vn → Vn−1 → · · · → V

such that each Vi+1 → Vi is the blow-up of a curve Ci in Singr(Si), whereSi is the strict transform of S on Vi. Each curve Ci which is blown up mustmap onto a curve in S by Lemma 5.5. Thus there exists a curve γ ⊂ Ssuch that there are infinitely many blow-ups of curves mapping onto γ inthe above sequence. Let R = OV,γ , a two-dimensional regular local ring.IS,γ is a height 1 prime ideal in this ring, and P = Iγ,γ is the maximal idealof R. We have

dimR+ trdegK R/P = 3

by the dimension formula (Theorem 15.6 [66]). Thus R/P has transcendencedegree 1 over K. Let t ∈ R be the lift of a transcendence basis of R/P overK. Then K[t] ∩ P = (0), so the field K(t) ⊂ R. We can write R = AQ,where A is a finitely generated K(t)-algebra (which is a domain) and Q isa maximal ideal in A. Thus R is the local ring of a non-singular point qon the K(t) surface spec(A). q is a point of multiplicity r on the curve inspec(A) with ideal sheaf IS,γ in R.

The sequence

· · · → Vn ×V spec(R) → Vn−1 ×V spec(R) → · · · → spec(R)

consists of infinitely many blow-ups of points on a K(t)-surface, which areof multiplicity r on the strict transform of the curve spec(R/IS,γ). But thisis impossible by Theorem 3.15.

Lemma 5.11. The number of bad points on S is finite.

Proof. Let

B0 = isolated points of Singr(S) ∪ singular points of Singr(S).

Singr(S)−B0 is a non-singular 1-dimensional subscheme of V . Let

π1 : V1 = B(Singr(S)−B0) → V −B0

be its blow-up. Let S1 be the strict transform of S, and let

B1 = isolated points of Singr(S1) ∪ singular points of Singr(S1).

We can iterate to construct a sequence

· · · → (Vn −Bn) → · · · → V,


where πn : Vn −Bn → V − Tn are the induced surjective maps, with

Tn = B0 ∪ π1(B1) ∪ · · · ∪ πn(Bn).

Let Sn be the strict transform of S on Vn.let C ⊂ Singr(S) be a curve. spec(OS,C) is a curve singularity of mul-

tiplicity r, embedded in a non-singular surface over a field of transcendencedegree 1 over K (as in the proof of Lemma 5.10). πn induces by base change

Sn ×S spec(OS,C) → spec(OS,C),

which corresponds to an open subset of a sequence of blow-ups of pointsover spec(OS,C). So for n >> 0, Sn ×S spec(OS,C) is non-singular. Thusthere are no curves in Singr(Sn) which dominate C. For n >> 0 thereare thus no curves in Singr(Sn) which dominate curves of Singr(S), so thatSingr(Sn) ∩ (Vn − Bn) is empty for large n, and all bad points of S arecontained in a finite set Tn.

Theorem 5.12. Let

(5.3) · · · → Vn → Vn−1 → · · · → V1 → V

be the sequence where πn : Vn → Vn−1 is the blow-up of all bad points on thestrict transform Sn−1 of S. Then this sequence is finite, so that it terminatesafter a finite number of steps with a Vm such that all points of Singr(Sm)are good.

We now give the proof of Theorem 5.12.Suppose there is an infinite sequence of the form of (5.3). Then there is

an infinite sequence of points pn ∈ Singr(Sn) such that πn(pn) = pn−1 forall n. We then have an infinite sequence of homomorphisms of local rings

R0 = OV,p → R1 = OV1,p1 → · · · → Rn = OVn,pn → · · · .By the Weierstrass preparation theorem and after a Tshirnhausen transfor-mation (Definition 3.8) there exist regular parameters (x, y, z) in R0 suchthat there is a local equation f = 0 for S in R0 of the form

f = zr + a2(x, y)zr−2 + · · ·+ ar(x, y)

with ν(ai(x, y)) ≥ i for all i. Since νpi(Si) = r for all i, we have regularparameters (xi, yi, zi) in Ri for all i and a local equation fi = 0 for Si suchthat

xi−1 = xi, yi−1 = xi(yi + αi), zi−1 = xi−1zi−1

with αi ∈ K, or

xi−1 = xiyi, yi−1 = yi, zi−1 = yi−1zi−1,

andfi = zr

i + a2,i(xi, yi)zr−2i + · · · ar,i(xi, yi),


where

aji(xi, yi) =

aj,i−1(xi,xi(yi+αi))

xji

or aj,i−1(xiyi,yi)

yji

for all j. Thus the sequence

K[[x, y] → K[[x1, y1]] → · · ·is a sequence of blow-ups of a maximal ideal, followed by completion. InK[[xi, yi]] we have relations

aj = γj,ixcj,i

i ydj,i

i aj,i−1

for all i and 2 ≤ j ≤ r, where γji is a unit. By embedded resolution of curvesingularities (Exercise 3.13), there exists m0 such that

∏rj=2 aj(x, y) = 0 is

a SNC divisor in Ri for all i ≥ m0. Thus for i ≥ m0 we have an expression

(5.4) fi = zri + a2i(xi, yi)xa2i

i yb2ii zr−2

i + · · ·+ ari(xi, yi)xarii ybri

i ,

where the aji are units (or zero) in Ri and aji + bji ≥ j if aji 6= 0.We observe that, by the proof of Theorem A.19 and Remark A.21,

ISingr(Si),pi=√J,

where J is the ideal in Ri generated by

∂j+k+lfi

∂xji∂y

ki ∂z

li

| 0 ≤ j + k + l < r.

Thus ISingr(Si),pimust be one of (xi, yi, zi), (xi, zi), (yi, zi) or (xiyi, zi).

Singr(Si) has SNCs at pi unless√J = (xiyi, zi) is the completion of an

ideal of an irreducible (singular) curve on Vi. Let E = π−1i (pi−1) ∼= P2.

By construction, we have that xi = 0 or yi = 0 is a local equation of Eat pi−1. Without loss of generality, we may suppose that xi = 0 is a localequation of E. zi = 0 is a local equation of the strict transform of z = 0 inVi ×V spec(R), so that (xi, zi) is the completion at qi of the ideal sheaf ofa subscheme of E. Thus there is a curve C in Vi such that IC,pi = (xi, zi)by Lemma 5.3. The ideal sheaf J = (ISingr(Si) : IC) in OVi is such that(J ∩ IC)pi

∼= ISingr(Si),pi, so that (xiyi, zi) is not the completion of an ideal

of an irreducible curve on Vi, and Singr(Si) has simple normal crossings atpi.

We may thus assume that m0 is sufficently large that Singr(Si) has SNCseverywhere on Vi for i ≥ m0.

We now make the observation that we must have bji 6= 0 and aki 6= 0for some j and k if i ≥ m0. If we did have bji = 0 for all j (with a similaranalysis if aji = 0 for all j) in (5.4), then, since

ISingr(Si),pi=√J,


where J is the ideal in Ri generated by

∂j+k+lfi

∂xji∂y

kj ∂z

li

| 0 ≤ j + k + l < r,

we have (with the condition that bji = 0 for all j) that√J = (xi, zi). Since

Singr(Si) has SNCs in Vi, there exists a non-singular curve C ⊂ Singr(Si)such that xi = zi = 0 are local equations of C in Ri. Let λ : W → Vi be theblow-up of C. If q ∈ λ−1(pi), we have regular parameters (u, v, y) in OW,q

such thatxi = u, zi = u(v + β)

orxi = uv, zi = v.

If f ′ = 0 is a local equation of the strict transform S′ of Si in W , we havethat νq(f ′) < r except possibly if xi = u, zi = uv. Then

f ′ = vr + a2iua2i−2vr−2 + · · ·+ ariu

ari−r.

We see that f ′ is of the form of (5.4), with bji = 0 for all j, but

minaji

j| 2 ≤ j ≤ r

has decreased by 1. We see, using Lemma 5.4, that after a finite numberof blow-ups of non-singular curves in Singr of the strict transform of S, themultiplicity must be < r at all points above pi. Thus pi is a good point, acontradiction to our assumption. Thus bji 6= 0 and aki 6= 0 in (5.4) for somej and k if i ≥ m0.

If follows that, for i ≥ m0, we must have

xi = xi+1, yi = xi+1yi+1, zi = xi+1zi+1

orxi = xi+1yi+1, yi = yi+1, zi = yi+1zi+1.

Thus in (5.4), we must have for 2 ≤ j ≤ r,

(5.5)aj,i+1 = aji + bji − j, bj,i+1 = bji oraj,i+1 = aji, bj,i+1 = aj,i + bji − j

respectively, for all i ≥ m0.Suppose that ζ ∈ R. Then the fractional part of ζ is

ζ = t

if ζ = a+ t with a ∈ Z and 0 ≤ t < 1.

Lemma 5.13. For i ≥ m0, pi is a good point on Vi if there exists j suchthat aji 6= 0 and


1.aji

j≤ aki

k,

bjij≤ bki

k

for all k with 2 ≤ k ≤ r and aki 6= 0, while

2. aji

j

+

bjij

< 1.

Proof. Since νpi(Si) = r, aji + bji ≥ j. Condition 2 thus implies that eitheraji ≥ j or bji ≥ j. Without loss of generality, aji ≥ j. Then condition 1implies that aki ≥ k for all k, and

ISingr(Si),pi⊂ (xi, zi)Ri.

Since Singr(Si) has SNCs, there exists a non-singular curve C ⊂ Singr(Si)such that IC,pi = (xi, zi). Let π : V ′ = B(C) → Vi be the blow-up of C,and let S′ be the strict transform of S on V ′. By Lemma 5.7, Singr(S′)has SNCs. A direct local calculation shows that there is at most one pointq ∈ π−1(pi) such that νq(S′) = r, and if such a point q exists, then OV ′,q

has regular parameters (x′, y′, z′) such that

xi = x′, yi = y′, zi = x′z′.

A local equation f ′ = 0 of S′ at q is

f ′ = (z′)r + a2i(x′)a2i−2(y′)b2i(z′)r−2 + · · ·+ ari(x′)ari−r(y′)bri .

Thus we have an expression of the form (5.4), and conditions 1 and 2 of thestatement of this lemma hold for f ′. After repeating this procedure a finitenumber of times we will construct V → Vi such that νa(S) < r for all pointsa on the strict transform S of S such that a maps to pi, since aji

j + bji

j mustdrop by 1 every time we blow up.

Lemma 5.14. Let

δj,k,i =(aji

j− aki

k

) (bjij− bki

k

).

Thenδj,k,i+1 ≥ δjki,

and if δjki < 0 then

δjk,i+1 − δjki ≥1r4.

Proof. We may assume that the first case of (5.5) holds. Then

δj,k,i+1 = δj,k,i +(bjij− bki

k

)2

.


If δj,k,i < 0 we must have bji

j − bkik 6= 0. Thus(

bjij− bki

k

)2

≥ 1j2k2

≥ 1r4.

Corollary 5.15. There exists m1 such that i ≥ m1 implies condition 1 ofLemma 5.13 holds.

Proof. There exists m1 such that i ≥ m1 implies δjki ≥ 0 for all j, k. Thenthe pairs (aki

k ,bkik ) with 2 ≤ k ≤ r are totally ordered by ≤, so there exists

a minimal element (aji

j ,bji

j ).

Lemma 5.16. There exists an i′ ≥ m1 such that we have condition 2 ofLemma 5.13,

aji′

j+

bji′

j < 1.

Proof. A calculation shows thataji

j

+

bjij

≥ 1

implies aj,i+1

j

+

bj,i+1

j

≤

aji

j

+

bjij

− 1r.

With the above i′, pi′ is a good point, since pi′ satisfies the conditionsof Lemma 5.13. Thus the conclusions of Theorem 5.12 must hold.

Now we give a the proof of Theorem 5.2. Let r be the maximal mul-tiplicity of points of S. By Theorem 5.12, there exists a finite sequenceof blow-ups of points Vm → V such that all points of Singr(Sm) are goodpoints, where Sm is the strict transform of S on Vm. By Lemma 5.10, thereexists a sequence of blow-ups of non-singular curves Vn → Vm such thatSingr(Sn) = ∅, where Sn is the strict transform of S on Vn. By descendinginduction on r we can reach the case where Sing2(Sn) = ∅, so that Sn isnon-singular.

Remark 5.17. Zariski discusses early approaches to resolution in [91].Some early proofs of resolution of surface singularities are by Albanese [12],Beppo Levi [61], Jung [57] and Walker [82]. Zariski gave several proofs ofresolution of algebraic surfaces over fields of characteristic zero, includingthe first algebraic proof [86], which we present later in Chapter 8.


Exercise 5.18.

1. Suppose that S is a surface which is a subvariety of a non-singularthree-dimensional variety V over an algebraically closed field K ofcharacteristic 0, and p ∈ S is a closed point.

a. Show that a Tschirnhausen transformation of a suitable localequation of S at p gives a formal hypersurface of maximalcontact for S at p.

b. Show that if f = 0 is a local equation of S at p, and (x, y, z)are regular parameters at p such that ν(f) = ν(f(0, 0, z)),then ∂r−1f

∂zr−1 = 0 is a hypersurface of maximal contact (A.20)for f = 0 in a neighborhood of the origin.

2. Follow the algorithm to reduce the multiplicity of f = zr+xayb = 0,where a+ b ≥ r.

5.2. Embedded resolution of singularities

In this section we will prove the following theorem, which is an extension ofthe main resolution theorem, Theorem 5.2 of the previous section.

Theorem 5.19. Suppose that S is a surface which is a subvariety of a non-singular three-dimensional variety V over an algebraically closed field K ofcharacteristic 0. Then there exists a sequence of monoidal transforms overthe singular locus of S such that the strict transform of S is non-singular,and the total transform π∗(S) is a SNC divisor.

Definition 5.20. A resolution datum R = (E0, E1, S, V ) is a 4-tuple whereE0, E1, S are reduced effective divisors on the non-singular three-dimensionalvariety V such that E = E0 ∪ E1 is a SNC divisor.

R is resolved at p ∈ S if S is non-singular at p and E ∪ S has SNCs atp. For r > 0, let

Singr(R) = p ∈ S | νp(S) ≥ r and R is not resolved at p.

Observe that Singr(R) = Singr(S) if r > 1, and Singr(R) is a proper Zariskiclosed subset of S for r ≥ 1.

For p ∈ S, let

η(p) = the number of components of E1 containing p.

We have 0 ≤ η(p) ≤ 3. For r, t ∈ N, let

Singr,t(R) = p ∈ Singr(R) | η(p) ≥ t.

5.2. Embedded resolution of singularities 57

Singr,t(R) is a Zariski closed subset of Singr(R) (and of S). For the rest ofthis section we will fix

r = ν(R) = ν(S,E) = maxνp(S) | p ∈ S,R is not resolved at p,

t = η(R) = the maximum number of components of E1

containing a point of Singr(R).

Definition 5.21. A permissible transform of R is a monoidal transformπ : V ′ → V whose center is either a point of Singr,t(R) or a non-singularcurve C ⊂ Singr,t(R) such that C makes SNCs with E.

Let F be the exceptional divisor of π. Then π∗(E)∪F is a SNC divisor.We define the strict transform R′ of R to be R′ = (E′

0, E′1, S

′), where S′ isthe strict transform of S, and we also define

E′0 = π∗(E0)red + F,

E′1 = strict transform of E1 if ν(S′, π∗((E0 + E1)red + F ) = ν(R),

E′0 = ∅, E′

1 = π∗(E0 + E1)red + F if ν(S′, π∗((E0 + E1)red + F ) < ν(R).

Lemma 5.22. With the notation of the previous definition,

ν(R′) ≤ ν(R)

andν(R′) = ν(R) implies η(R′) ≤ η(R).

The proof of Lemma 5.22 is a generalization of Lemmas 5.3 and 5.4.

Definition 5.23. p ∈ Singr,t(R) is a pregood point if the following twoconditions hold:

1. In a neighborhood of p, Singr,t(R) is either a non-singular curvethrough p, or two non-singular curves through p intersecting trans-versally at p (satisfying 2 of Definition 5.6).

2. Singr,t(R) makes SNCs with E at p. That is, there exist regularparameters x, y, z in OV,p such that the ideal sheaf of each com-ponent of E and each curve in Singr,t(R) containing p is generatedby a subset of x, y, z at p.

p ∈ S is a good point if p is a pregood point and for any sequence

π : Xn → Xn−1 → · · · → X1 → spec(OV,p)

of permissible monodial transforms centered at curves in Singr,t(Ri), whereRi is the transform of R on Xi, then q is a pregood point for all q ∈ π−1(p).

A point p ∈ Singr,t(R) is called bad if p is not good.

Lemma 5.24. The number of bad points in Singr,t(R) is finite.


Lemma 5.25. Suppose that all points of Singr,t(R) are good. Then thereexists a sequence of permissible monoidal transforms, π : V ′ → V , cen-tered at non-singular curves contained in Singr,t(Si), where Si is the stricttransform of S on the i-th monoidal transform, such that if R′ is the stricttransform of R on V ′, then either

ν(R′) < ν(R)

orν(R′) = ν(R) and η(R′) < η(R).

The proofs of the above two lemmas are similar to the proofs of Lemmas5.11 and 5.10 respectively, with the use of embedded resolution of planecurves.

Theorem 5.26. Suppose that R is a resolution datum such that E0 = ∅.Let

(5.6) · · · → Vnπn→ Vn−1 → · · · π1→ V0 = V

be the sequence where πn is the monoidal transform centered at the unionof bad points in Singr,t(Ri), where Ri is the transform of R on Vi. Thenthis sequence is finite, so that it terminates in a Vm such that all points ofSingr,t(Rm) are good.

Proof. Suppose that (5.6) is an infinite sequence. Then there exists aninfinite sequence of points pn ∈ Vn such that πn+1(pn+1) = pn for all n andpn is a bad point. Let Rn = OVn,pn for n ≥ 0. Since the sequence (5.6) isinfinite, its length is larger than 1, so we may assume that t ≤ 2.

Let f = 0 be a local equation of S at p0, and let gk, 0 ≤ k ≤ t, be localequations of the components of E1 containing p0. In R0 = OV,p there areregular parameters (x, y, z) such that ν(f(0, 0, z)) = r and ν(gk(0, 0, z)) = 1for all k. This follows since for each equation this is a non-trivial Zariski opencondition on linear changes of variables in a set of regular parameters. Bythe Weierstrass preparation theorem, after multiplying f and the gk by unitsand performing a Tschirnhausen transformation on f , we have expressions

(5.7)f = zr +

∑rj=2 aj(x, y)zr−j ,

gk = z − φk(x, y), (or gk is a unit)

in R0. z = 0 is a local equation of a formal hypersurface of maximal contactfor f = 0. Thus we have regular parameters (x1, y1, z1) in R1 = OV1,p1

defined byx = x1, y = x1(y1 + α), z = x1z1,

with α ∈ K, orx = x1y1, y = y1, z = y1z1.

5.2. Embedded resolution of singularities 59

The strict transforms f1 of f and gk1 of gk have the form of (5.7) withaj replaced by aj

xj1

(or aj

yj1

), φk with φkx1

(or φky1

). After a finite number of

monoidal transforms centered at points pj with ν(fj) = r, we have forms inRi = OVi,pi for the strict transforms of f and gk:

(5.8) fi = zri +

∑rj=2 aj(xi, yi)x

aji

i ybji

i zr−ji ,

gki = zrk + φki(xi, yi)x

ckii ydki

i ,

such that aj , φki are either units or zero (as in the proof of Theorem 5.12).We further have that local equations of the exceptional divisor Ei

0 of Vi → Vare one of xi = 0, yi = 0 or xiyi = 0. This last case can only occur if t < 2.Note that our assumption η(pi) = t implies gki is not a unit in Ri for1 ≤ k ≤ t.

We can further assume, by an extension of the argument in the proof ofTheorem 5.12, that all irreducible curves in Singr,t(Ri) are non-singular.

By Lemma 5.14, we can assume that the set

T =(

aji

j,bjij

), | aj 6= 0

∪ (cki, dki) | φki 6= 0

is totally ordered. Now by Lemma 5.16 we can further assume thataj0i

j0

+

bj0i

j0

< 1,

where (aj0i

j0,

bj0i

j0) is the minimum of the (aji

j ,bji

j ). Recall that, if t > 0,∏ti=1 gi = 0 makes SNCs with the exceptional divisor Ei

0 of Vi → V . Theseconditions imply that, after possibly interchanging the gki, interchangingxi, yi and multiplying xi, yi by units in Ri, that the gki can be described byone of the following cases at pi:

1. t = 0.

2. t = 1,a. g1i = zi + xλ

i , λ ≥ 1.b. g1i = zi + xλ

i yµ, λ ≥ 1, µ ≥ 1.

c. g1i = zi

3. t = 2,a. g1i = zi, g2i = zi + xi.b. gi1 = zi + xi, gi2 = zi + εxi, where ε is a unit in Ri such that

the residue of ε in the residue field of Ri is not 1.c. gi1 = zi + xi, gi2 = zi + εxλ

i yµi , where ε is a unit in Ri, λ ≥ 1

and λ+ µ ≥ 2. If µ > 0 we can take ε = 1.

We can check explicitly by blowing up permissible curves that pi is agood point. In this calculation, if t = 0, this follows from the proof of


Theorem 5.12, recalling that xi = 0, yi = 0 or xiyi = 0 are local equationsof E0 at pi. If t = 1, we must remember that we can only blow up a curveif it lies on g1 = 0, and if t = 2, we can only blow up the curve g1 = g2 = 0.Although we are constructing a sequence of permissible monoidal transformsover spec(OVi,pi), this is equivalent to constructing such a sequence overspec(OVi,pi), as all curves blown up in this calculation are actually algebraic,since all irreducible curves in Singr,t(Ri) are non-singular.

We have thus found a contradiction, by which we conclude that thesequence (5.6) has finite length.

The proof of Theorem 5.19 now follows easily from Theorem 5.26, Lemma5.25 and descending induction on (r, t) ∈ N×N, with the lexicographic order.

Remark 5.27. This algorithm is outlined in [73] and [62].

Exercise 5.28. Resolve the following (or at least make (ν(R), η(R)) dropin the lexicographic order):

1. R = (∅, E1, S), where S has local equation f = z = 0, and E1 isthe union of two components with respective local equations g1 =z + xy, g2 = z + x;

2. R = (∅, E1, S), where S has local equation f = z3 + x7y7 = 0 andE1 is the union of two components with respective local equationsg1 = z + xy, g2 = z + y(x2 + y3).

Chapter 6

Resolution ofSingularities inCharacteristic Zero

The first proof of resolution in arbitrary dimension (over a field of charac-teristic zero) was by Hironaka [52]. The proof consists of a local proof, anda complex web of inductions to produce a global proof. The local proof usessophisticated methods in commutative algebra to reduce the problem of re-ducing the multiplicity of the strict transform of an ideal after a permissiblesequence of monoidal transforms to the problem of reducing the multiplic-ity of the strict transform of a hypersurface. The use of the Tschirnhausentransformation to find a hypersurface of maximal contact lies at the heart ofthis proof. The order ν and the invariant τ (of Chapter 7) are the principalinvariants considered.

De Jong’s theorem [36], referred to at the end of Chapter 7, can berefined to give a new proof of resolution of singularities of varieties of char-acteristic zero ([9], [16], [70]). The resulting theorem is not as strong as theclassical results on resolution of singularities in characteristic zero (Section6.8), as the resulting resolution X ′ → X may not be an isomorphism overthe non-singular locus of X.

In recent years several different proofs of canonical resolution of singu-larities have been constructed (Bierstone and Milman [14], Bravo, Encinasand Villamayor [17], Encinas and Villamayor [40], [41], Encinas and Hauser[39], Moh [67], Villamayor [79], [80], and Wlodarczyk [84]). These papers

61

62 6. Resolution of Singularities in Characteristic Zero

have significantly simplified Hironaka’s original proofs in [47] and [55], andrepresent a great advancement of the subject.

In this chapter we prove the basic theorems of resolution of singularitiesin characteristic 0. Our proof is based on the algorithms of [40] and [41].The final statements of resolution are proved in Section 6.8, and in Exercise6.42 at the end of that section.

Throughout this chapter, unless explicitely stated otherwise, we willassume that all varieties are over a field K of characteristic zero. Supposethat X1 and X2 are subschemes of a variety W . We will denote the reducedset-theoretic intersection of X1 and X2 by X1∩X2. However, we will denotethe scheme-theoretic intersection of X1 and X2 by X1 ·X2. X1 ·X2 is thesubscheme of W with ideal sheaf IX1 + IX2 .

6.1. The operator 4 and other preliminaries

Definition 6.1. Suppose that K is a field of characteristic zero, R =K[[x1, . . . , xn]] is a ring of formal power series overK, and J = (f1, . . . , fr) ⊂R is an ideal. Define

4(J) = 4R(J) = (f1, . . . , fr) +(∂fi

∂xj| 1 ≤ i ≤ r, 1 ≤ j ≤ n

).

4(J) is independent of the choice of generators fi of J and regular param-eters xj in R.

Lemma 6.2. Suppose that W is a non-singular variety and J ⊂ OW is anideal sheaf. Then there exists an ideal 4(J) = 4W (J) ⊂ OW such that

4(J)OW,q = 4(J)

for all closed points q ∈W .

Proof. We define 4(J) as follows. We can cover W by affine open subsetsU = spec(R) which satisfy the conclusions of Lemma A.11, so that Ω1

R/K =dy1R⊕ · · · ⊕ dynR. If Γ(U, J) = (f1, . . . , fr), we define

Γ(U,4(J)) = (fi,∂fi

∂yj| 1 ≤ i ≤ r, 1 ≤ j ≤ n).

This definition is independent of all choices made in this expression.Suppose that q ∈W is a closed point, and x1, . . . , xn are regular param-

eters in OW,q. Let U = spec(R) be an affine neighborhood of q as above.Let A = OW,q. Then dy1, . . . , dyn and dx1, . . . , dxn are A-bases of Ω1

A/k

by Lemma A.11 and Theorem A.10. Thus the determinant

|( ∂yi

∂xj)|(q) 6= 0

6.1. The operator 4 and other preliminaries 63

from which the conclusions of the lemma follow.

Given J ⊂ OW and q ∈W , we define νJ(q) = νq(J) (Definition A.17). IfX is a subvariety of W , we denote νX = νIX

. We can define 4r(J) for r ∈ Ninductively, by the formula 4r(J) = 4(4r−1(J)). We define 40(J) = J .

Lemma 6.3. Suppose that W is a non-singular variety, (0) 6= J ⊂ OW isan ideal sheaf, and q ∈W (which is not necessarily a closed point). Then:

1. νq(J) = b > 0 if and only if νq(4(J)) = b− 1.

2. νq(J) = b > 0 if and only if νq(4b−1(J)) = 1.

3. νq(J) ≥ b > 0 if and only if q ∈ V (4b−1(J)).4. νJ : W → N is an upper semi-continuous function.

Proof. The lemma follows from Theorem A.19.

If f : X → I is an upper semi-continuous function, we define max f tobe the largest value assumed by f on X, and we define a closed subset ofX,

Max f = q ∈ X | f(q) = max f.We have Max νJ = V (4b−1(J)) if b = max νJ .

Suppose that X is a closed subset of a n-dimensional variety W . Weshall denote by

(6.1) R(1)(X) ⊂ X

the union of irreducible components of X which have dimension n− 1.

Lemma 6.4. Suppose that W is a non-singlar variety, J ⊂ OW is anideal sheaf, b = max νJ and Y ⊂ Max νJ is a non-singular subvariety. Letπ : W1 → W be the monoidal transform with center Y . Let D be theexceptional divisor of π. Then:

1. There is an ideal sheaf J1 ⊂ OW1 such that

J1 = I−bD J

and ID - J1.2. If q ∈W1 and p = π(q), then

νJ1(q) ≤ νJ(p).

3. max νJ ≥ max νJ1.

Proof. Suppose that q ∈W1 and p = π(q). Let A be the Zariski closure ofq in W1, and B the Zariski closure of p in W . Then A → B is proper.By upper semi-continuity of νJ , there exists a closed point x ∈ B such thatνJ(x) = νJ(p). Let y ∈ π−1(x) ∩ A be a closed point. By semi-continuity


of νJ1, we have νJ1

(q) ≤ νJ1(y), so it suffices to prove 2 when q and p are

closed points.By Corollary A.5, there is a regular system of parameters y1, . . . , yn in

OW,p such that y1 = y2 = · · · = yr = 0 are local equations of Y at p. ByRemark A.18, we may assume that K = K(p) = K(q). Then we can makea linear change of variables in y1, . . . , yn to assume that there are regularparameters y1, . . . , yn in OW1,q such that

y1 = y1, y2 = y1y2, . . . , yr = y1yr, yr+1 = yr+1, . . . , yn = yn.

y1 = 0 is a local equation of D. By Remark A.18, if suffices to verify theconclusions of the theorem in the complete local rings

R = OW,p = K[[y1, . . . , yn]

andS = OW1,q = K[[y1, . . . , yn]].

Y ⊂ Max νJ implies νη(J) = b, where η is the general point of the componentof Y whose closure contains p. Suppose that f ∈ JR. Let t = νR(f) ≥ b.Then there is an expansion

f =∑

ai1,...,inyi11 · · · y

inn

in R, with ai1,...,in ∈ K and ai1,...,in = 0 if i1 + · · · + ir < t. Further, thereexist i1, . . . , in with i1 + · · ·+ in = t such that ai1,...,in 6= 0.

In S, we havef = yt

1f1,

wheref1 =

∑ai1,...,iny

i1+···+ir−t1 · · · yin

n

is such that y1 does not divide f1. Thus we have νS(f1) ≤ νR(f). Inparticular, we have yb

1 | JS; and since there exists f ∈ R with νR(f) = b,yb+1

1 - JS. ThusJ1 = I−b

D J

satisfies 1 andνJ1

(q) ≤ νJ(p) = b.

Suppose that W is a non-singular variety and J ⊂ OW is an ideal sheaf.Suppose that Y ⊂ W is a non-singular subvariety. Let Y1, . . . , Ym be thedistinct irreducible (connected) components of Y . Let π : W1 → W bethe monoidal transform with center Y and exceptional divisor D. Let D =D1+· · ·+Dm, where Di are the distinct irreducible (connected) components

6.1. The operator 4 and other preliminaries 65

of D, indexed so that π(Di) = Yi. Then by an argument as in the proof ofLemma 6.4, there exists an ideal sheaf J1 ⊂ OW1 such that

(6.2) JOW1 = Ic1D1· · · Icm

DmJ1,

where J1 is such that for 1 ≤ i ≤ m we have IDi- J1 and ci = νηi(J), with

ηi the generic point of Yi. J1 is called the weak transform of J .There is thus a function c on W1, defined by c(q) = 0 if q 6∈ D and

c(q) = ci if q ∈ Di, such that

(6.3) JOW1 = IcDJ1.

In the situation of Lemma 6.4, J1 has properties 1 and 2 of that lemma.With the notation of Lemma 6.4, suppose that J = IX is the ideal sheaf

of a subvariety X of W . The weak transform X of X is the subscheme ofW1 with ideal sheaf J1. We see that the weak transform X of X is mucheasier to calculate than the strict transform X of X. We have inclusions

X ⊂ X ⊂ π∗(X).

Example 6.5. Suppose that X is a hypersurface on a variety W , r =max νX and Y ⊂ Max νX is a non-singular subvariety. Let π : W1 → Wbe the monoidal transform of W with center Y and exceptional divisor E.In this case the strict transform X and the weak transform X are the samescheme, and π∗(X) = X + rE.

Example 6.6. Let X ⊂ W = A3 be the nodal plane curve with idealIX = (z, y2−x3) ⊂ k[x, y, z]. Let m = (x, y, z), and let π : B = B(m) → A3

be the blow-up of the point m. The strict transform X of X is non-singular.The most interesting point in X is the point q ∈ π−1(m) with regular

parameters (x1, y1, z1) such that

x = x1, y = x1y1, z = x1z1.

Then

Iπ∗(X),q = (x1z1, x21y

21 − x3

1),

IX,q = (z1, x1y21 − x2

1),

IX,q = (z1, y21 − x1).

In this example, X, X and π∗(X) are all distinct.


6.2. Hypersurfaces of maximal contact and induction inresolution

Suppose that W is a non-singular variety, J ⊂ OW is an ideal sheaf andb ∈ N. Define

Sing(J, b) = q ∈W | νJ(q) ≥ b.

Sing(J, b) is Zariski closed in W by Lemma 6.3. Suppose that Y ⊂ Sing(J, b)is a non-singular (but not necessarily connected) subvariety of W . Let π1 :W1 → W be the monodial transform with center Y , and let D1 = π−1(Y )be the exceptional divisor.

Recall (6.3) that the weak transform J1 of J is defined by

JOW1 = J cD1J1,

where c is locally constant on connected components of D1. We have c ≥ bby upper semi-continuity of νJ .

We can now define J1 by

(6.4) JOW1 = J bD1J1,

so that

J1 = J c−bD1

J1.

Suppose that

(6.5) Wnπn→Wn−1 → · · · →W1

π1→W

is a sequence of monoidal transforms such that each πi is centered at anon-singular subvariety Yi ⊂ Sing(Ji, b), where Ji is defined inductively by

Ji−1OWi = IbDiJi

and Di is the exceptional divisor of πi. We will say that (6.5) is a resolutionof (W,J, b) if Sing(Jn, b) = ∅.

Definition 6.7. Suppose that r = max νJ , q ∈ Sing(J, r). A non-singularcodimension 1 subvariety H of an affine neighborhood U of q in W is calleda hypersurface of maximal contact for J at q if

1. Sing(J, r) ∩ U ⊂ H, and

2. if

Wn → · · · →W1 →W

is a sequence of monoidal transforms of the form of (6.5) (withb = r), then the strict transform Hn of H on Un = Wn ×W U issuch that Sing(Jn, r) ∩ Un ⊂ Hn.

6.2. Hypersurfaces of maximal contact and induction in resolution 67

With the assumptions of Definition 6.7, a non-singular codimension 1subvariety H of U = spec(OW,q) is called a formal hypersurface of maximalcontact for J at q if 1 and 2 of Definition 6.7 hold, with U = spec(OW,q).

If X ⊂W is a codimension 1 subvariety, with r = max νX , q ∈ max νX ,then our definition of hypersurface of maximal contact for J = IX coincideswith the definition of a hypersurface of maximal contact for X given inDefinition A.20. In this case, the ideal sheaf Jn in (6.5) is the ideal sheaf ofthe strict transform Xn of X on Wn.

Now suppose that X ⊂ W is a singular hypersurface, and K is alge-braically closed. Let

r = maxνX > 1

be the set of points of maximal multiplicity on X. Let q ∈ Sing(IX , r) be aclosed point of maximal multiplicity r.

The basic strategy of resolution is to construct a sequence

Wn → · · · →W1 →W

of monoidal transforms centered at non-singular subvarieties of the locus ofpoints of multiplicity r on the strict transform of X so that we eventuallyreach a situation where all points of the strict transform of X have multiplic-ity < r. We know that under such a sequence the multiplicity can never goup, and always remains ≤ r (Lemma 6.4). However, getting the multiplicityto drop to less that r everywhere is much more difficult.

Notice that the desired sequence Wn → W1 will be a resolution of theform of (6.5), with (Ji, b) = (IXi , r), where Xi is the strict transform of Xon Wi.

Let q ∈ Sing(IX , r) be a closed point. After Weierstrass preparation andperforming a Tschirnhausen transformation, there exist regular parameters(x1, . . . , xn, y) in R = OW,q such that there exists f ∈ R such that f = 0 isa local equation of X at q, and

(6.6) f = yr +r∑

i=2

ar(x1, . . . , xn)yr−i.

Set H = V (y) ⊂ spec(R). Then H = spec(S), where S = K[[x1, . . . , xn]].We observe that H is a (formal) hypersurface of maximal contact for X

at q. The verification is as follows.We will first show that H contains a formal neighborhood of the locus

of points of maximal multiplicity on X at q. Suppose that Z ⊂ Sing(IX , r)is a subvariety containing q. Let J = IZ,q.

Write J = P1∩· · ·∩Pr, where the Pi are prime ideals of the same heightin R. Let Ri = RPi . By semi-continuity of νX , we have f ∈ P r

i RPi for all


i. Since ∂∂y : RPi → RPi , we have (r − 1)!y = ∂r−1f

∂yr−1 ∈ JRPi for all i. Thusy ∈

⋂JRPi = J and Z ∩ spec(R) ⊂ H.

Now we will verify that if Y ⊂ Sing(IX , r) is a non-singular subvariety,π : W1 → W is the monoidal transform with center Y , X1 is the stricttransform of X and q1 is a closed point of X1 such that π(q1) = q whichhas maximal multiplicity r, then q1 is on the strict transform H1 of H (overthe formal neighborhood of q). Suppose that q1 is such a point. That is,q1 ∈ π−1(q) ∩ Sing(IX1 , r). We have shown above that y ∈ IY . Withoutchanging the form of (6.6), we may then assume that there is s ≤ n such thatx1 = . . . = xs = y = 0 are (formal) local equations of Y at q, and there isa formal regular system of parameters x1(1), . . . , xn(1), y(1) in OW1,q1 suchthat

(6.7)

x1 = x1(1),xi = x1(1)xi(1) for 2 ≤ i ≤ s,y = x1(1)y(1),xi = xi(1) for s < i ≤ n.

Since Y ⊂ Sing(IX , r), we have qi ∈ (x1, . . . , xs)i for all i. Then we have a(formal) local equation f1 = 0 of X1 at q1, where

(6.8)

f1 =f

xr1

= y(1)r +r∑

i=2

ai

xi1

y(1)r−i

= y(1)r +r∑

i=2

ai(1)(x1(1), . . . , xn(1))y(1)r−i

and y1(1) = 0 is a local equation of H1 at q1, while x1(1) = 0 is a localequation of the exceptional divisor of π. With our assumption that f1 alsohas multiplicity r, f1 has an expression of the same form as (6.6), so byinduction, H is a (formal) hypersurface of maximal contact.

We see that (at least over a formal neighborhood of q) we have reducedthe problem of resolution of X to some sort of resolution problem on the(formal) hypersurface H.

We will now describe this resolution problem. Set

I = (ar!22 , . . . , a

r!rr ) ⊂ S = K[[x1, . . . , xn]].

Observe that the scheme of points of multiplicity r in the formal neighbor-hood of q on X coincides with the scheme of points of multiplicity ≥ r! ofI,

Sing(fR, r) = Sing(I, r!).However, we may have νR(I) > r!.

Let us now consider the effect of the monodial transform π of (6.7)on I. Since H is a hypersurface of maximal contact, π certainly induces

6.2. Hypersurfaces of maximal contact and induction in resolution 69

a morphism H1 → H, where H1 is the strict transform of H. By directverification, we see that the transform I1 of I (this transform is defined by(6.4)) satisfies

I1S1 =1xr!

1

IS1,

where S1 = K[[x1(1), . . . , xn(1)]] = OH1,q1 , and this ideal is thus, by (6.8),the ideal

I1S1 = (a2(1)r!2 , . . . , ar(1)

r!r ),

which is obtained from the coefficients of f1, our local equation of X1, in theexact same way that we obtained the ideal I from our local equation f ofX. We further observe that νS1(I1S1) ≥ r! if and only if νq1(X1) ≥ r. Wesee then that to reduce the multiplicity of the strict transform of X over q,we are reduced to resolving a sequence of the form (6.5), but over a formalscheme.

The above is exactly the procedure which is followed in the proof ofresolution for curves given in Section 3.4, and the proof of resolution forsurfaces in Section 5.1.

When X is a curve (dim W = 2) the induced resolution problem on theformal curve H is essentially trivial, as the ideal I is in fact just the idealgenerated by a monomial, and we only need blow up points to divide outpowers of the terms in the monomial.

When X is a surface (dim W = 3), we were able to make use of anotherinduction, by realizing that resolution over a general point of a curve in thelocus of points of maximal multiplicity reduces to the problem of resolutionof curves (over a non-closed field). In this way, we were able to reduceto a resolution problem over finitely many points in the locus of points ofmaximal multiplicity on X. The potentially worrisome problem of extendinga resolution of an object over a formal germ of a hypersurface to a globalproper morphism over W was not a great difficulty, since the blowing upof a point on the formal surface H always extends trivially, and the onlyother kind of blow-ups we had to consider were the blow-ups of non-singularcurves contained in the locus of maximal multiplicity. This required a littleattention, but we were able to fairly easily reduce to a situation were theseformal curves were always algebraic. In this case we first blew up points toreduce to the situation where the ideal I was locally a monomial ideal. Thenwe blew up more points to make it a principal monomial ideal. At this point,we finished the resolution process by blowing up non-singular curves in thelocus of maximal multiplicity on the strict transform of the surface X, whichamounts to dividing out powers of the terms in the principal monomial ideal.

In Section 5.2, we extended this algorithm to obtain embedded resolutionof the surface X. This required keeping track of the exceptional divisors


which occur under the resolution process, and requiring that the centers ofall monoidal transforms make SNCs with the previous exceptional divisors.We introduced the η invariant, which counts the number of components ofthe exceptional locus which have existed since the multiplicity last dropped.

In this chapter we give a proof of resolution in arbitrary dimension (overfields of characteristic zero) which incorporates all of these ideas into ageneral induction statement. The fact that we must consider some kindof covering by local hypersurfaces which are not related in any obvious wayis incorporated in the notion of General Basic Object.

In the algorithm of this chapter Tschirnhausen is replaced by a methodof Giraud for finding algebraic hypersurfaces of maximal contact for an idealJ of order b, by finding a hypersurface in 4b−1(J).

6.8. Resolution of singularities in characteristic zero 99

6.8. Resolution of singularities in characteristic zero

Recall our conventions on varieties in the notation part of Section 1.1.

Theorem 6.36 (Principalization of Ideals). Suppose that I is an ideal sheafon a non-singular variety W over a field of characteristic zero. Then thereexists a sequence of monodial transforms

π : W1 →W

which is an isomorphism away from the closed locus of points where I is notlocally principal, such that IOW1 is locally principal.

Proof. We can factor I = I1I2, where I1 is an invertible sheaf and V (I2)is the set of points where I is not locally principal. Then we apply Theorem6.35 to J0 = I2.

Theorem 6.37 (Embedded Resolution of Singularities). Suppose that X isan algebraic variety over a field of characteristic zero which is embedded in anon-singular variety W . Then there exists a birational projective morphism

π : W1 →W

such that π is a sequence of monodial transforms, π is an embedded resolu-tion of X, and π is an isomorphism away from the singular locus of W .

Proof. Set X0 = X, W0 = W , J0 = IX0 ⊂ OW0 . Let

(WN , EN ) → (W0, E0)

be the strong principalization of J0 constructed in Theorem 6.35, so thatthe weak transform of J0 on WN is JN = OWN

.With the notation of the proofs of Theorem 6.34 and Theorem 6.35, ifX0

has codimension r in the d-dimensional variety W0, then for q ∈ Reg(X0),

pd0(q) = (1, (1, 0), . . . , (1, 0), 0, . . . , 0) ∈ I ′d,

where there are r copies of (1, 0) followed by d − r zeros. The function pd0

is thus constant on the non-empty open set Reg(X0) of non-singular pointsof X0. Let this constant be c ∈ I ′d. By property 4 of Theorem 6.34 thereexists a unique index r ≤ N − 1 such that max pd

r = c. Since Wr → W0 isan isomorphism over the dense open set Reg(X0), the strict transform Xr ofX0 must be the union of the irreducible components of the closed set Max pd

r

of Wr. Since Max pdr is a permissible center, Xr is non-singular and makes

simple normal crossings with the exceptional divisor Er.


Theorem 6.38 (Resolution of Singularities). Suppose that X is an algebraicvariety over a field of characteristic zero. Then there is a resolution ofsingularities

π : X1 → X

such that π is a projective morphism which is an isomorphism away fromthe singular locus of X.

Proof. This is immediate from Theorem 6.37, after choosing an embeddingof X into a projective space W .

Theorem 6.39 (Resolution of Indeterminacy). Suppose that K is a fieldof characteristic zero and φ : W → V is a rational map of projective K-varieties. Then there exist a projective birational morphism π : W1 → Wsuch that W1 is non-singular and a morphism λ : W1 → V such that λ =φ π. If W is non-singular, then π is a product of monoidal transforms.

Proof. By Theorem 6.38, there exists a resolution of singularities ψ : W1 →W . After replacing W with W1 and φ with ψ φ, we may assume that Wis non-singular. Let Γφ be the graph of φ, with projections π1 : Γφ → Wand π2 : Γφ → V . As π1 is birational and projective, there exists an idealsheaf I ⊂ OW such that Γφ

∼= B(I) is the blow-up of I (Theorem 4.5).By Theorem 6.36, there exists a principalization π : W1 → W of I. Theuniversal property of blowing up (Theorem 4.2) now shows that there is amorphism λ : W1 → V such that λ = φ π.

Theorem 6.40. Suppose that π : Y → X is a birational morphism ofprojective non-singular varieties over a field K of characteristic 0. Then

H i(Y,OY ) ∼= H i(X,OX) ∀ i.

Proof. By Theorem 6.39, there exists a projective morphism f : Z → Ysuch that g = π f is a product of blow-ups of non-singular subvarieties,

g : Z = Zngn→ Zn−1

gn−1→ . . .g2→ Z1

g1→ Z0 = X.

We have ([65] or Lemma 2.1 [31])

Rigj∗OZj =

0, i > 0,OZj−1 , i = 0.

Thus, by the Leray spectral sequence,

Rig∗OZ =

0, if i > 0,OX , if i = 0,

(6.9)

andg∗ : H i(X,OX) ∼= H i(Z,OZ)


for all i. Since g∗ = f∗ π∗, we conclude that π∗ is one-to-one. To show thatπ∗ is an isomorphism we now only need to show that f∗ is also one-to-one.

Theorem 6.40 also gives a projective morphism γ : W → Z such thatβ = f γ is a product of blow-ups of non-singular subvarieties, so we have

β∗ : H i(Y,OY ) ∼= H i(W,OW )

for all i. This implies that f∗ is one-to-one, and the theorem is proved.

For the following theorem, which is stronger than Theorem 6.38, we givea complete proof only the case of a hypersurface. In the embedded resolutionconstructed in Theorem 6.37, which induces the resolution of 6.38, if X isnot a hypersurface, there may be monodial transforms Wi+1 → Wi whosecenters are not contained in the strict transform Xi ofX, so that the inducedmorphism Xi+1 → Xi of strict transforms of X will in general not be amonodial transform.

Theorem 6.41 (A Stronger Theorem of Resolution of Singularities). Sup-pose that X is an algebraic variety over a field of characteristic zero. Thenthere is a resolution of singularities

π : X1 → X

such that π is a sequence of monodial transforms centered in the closed setsof points of maximum multiplicity.

Proof. In the case of a hypersurface X = X0 embedded in a non-singularvariety W = W0, the proof given for Theorem 6.37 actually produces aresolution of the kind asserted by this theorem, since the resolution sequenceis constructed by patching together resolutions of the simple basic objects(Wi, (J i, b), Ei), where J i is the weak transform of the ideal sheaf J0 of X0

in W0. Since J0 is a locally principal ideal, and each transformation inthese sequences is centered at a non-singular subvariety of MaxJ i, the weaktransform J i is actually the strict transform of J0.

For the case of a general variety X, we choose an embedding of X ina projective space W . We then modify the above proof by consideringthe Hilbert-Samuel function of X. It can be shown that there is a simplebasic object (W0, (K0, b), E0) such that Max νK0 is the maximal locus ofthe Hilbert-Samuel function. In the construction of pd

i of Theorem 6.35 weuse νK0 in place of νJi

. Then the resolution of this basic object induces asequence of monodial transformations of X such that the maximum of theHilbert-Samuel function has dropped.

More details about this process are given, for instance, in [55], [14] and[79].


Exercise 6.42.

1. Follow the algorithm of this chapter to construct an embeddedresolution of singularities of:

a. The singular curve y2 − x3 = 0 in A2.b. The singular surface z2 − x2y3 = 0 in A3.c. The singular curve z = 0, y2 − x3 = 0 in A3.

2. Give examples where the fdi achieve the 3 cases of the proof of

Theorem 6.34.3. Suppose that K is a field of characteristic zero and X is an integral

finite type K-scheme. Prove that there exists a proper birationalmorphism π : Y → X such that Y is non-singular.

4. Suppose that K is a field of characteristic zero and X is a reducedfinite type K-scheme. Prove that there exists a proper birationalmorphism π : Y → X such that Y is non-singular.

5. Let notation be as in the statement of Theorem 6.35.a. Suppose that q ∈ V (J0) ⊂ W0 is a closed point. Let U =

spec(OW0,q), Di = Di∩U for 1 ≤ i ≤ r and E = D1, . . . , Dr.Consider the function pd

0 of Theorem 6.35. Show that pd0(q)

depends only on the local basic object (U,E) and the ideal(J0)q ⊂ OW0,q. In particular, pd

0(q) is independent of the K-algebra isomorphism of OX,q which takes J0OX,q to J0OX,q

and Di to Di for 1 ≤ i ≤ r.b. Suppose that Θ : W0 → W0 is a K-automorphism such that

Θ∗(J0) = J0. Show that pd0(Θ(q)) = pd

0(q) for all q ∈ V (J0).c. Suppose that Θ : W0 → W0 is a K-automorphism, and Y ⊂W0 is a non-singular subvariety such that Θ(Y ) = Y . Letπ1 : W1 →W be the monoidal transform centered at Y . Showthat Θ extends to a K-automorphism Θ : W1 →W1 such thatΘ(π1(q)) = π1(Θ(q)) for all q ∈W1 and Θ(D) = D if D is theexceptional divisor of π1.

6. Suppose that (W0, E0) is a pair and J0 ⊂ OW0 is an ideal sheaf,with notation as in Theorem 6.35. Consider the sequence in 2 ofTheorem 6.35. Suppose that G is a group of K-automorphisms ofW0 such that Θ(Di) = Di for 1 ≤ i ≤ r and Θ∗(J0) = J0. Showthat G extends to a group of K-automorphisms of each Wi in thesequence 2 of Theorem 6.35 such that Θ(Yi) = Yi for all Θ ∈ G andeach πi is G-equivariant. That is,

πi(Θ(q)) = Θ(πi(q))

for all q ∈ Wi. Furthermore, the functions pdi of Theorem 6.35

satisfy pdi (Θ(q)) = pd

i (q) for all q ∈Wi.


7. (Equivariant resolution of singularities) Suppose that X is a varietyover a field K of characteristic zero. Suppose that G is a group ofK-automorphisms of X. Show that there exists an equivariantresolution of singularities π : X1 → X of X. That is, G extendsto a group of K-automorphisms of X so that π(Θ(q)) = Θ(π(q))for all q ∈ X1. More generally, deduce equivariant versions of allthe other theorems of Section 6.8, and of parts 3 and 4 of Exercise6.42).

Chapter 7

Resolution of Surfacesin PositiveCharacteristic

105

Chapter 8

Local Uniformizationand Resolution ofSurfaces

133

Chapter 9

Ramification ofValuations andSimultaneousResolution

155

Appendix. Smoothnessand Non-singularity II

In this appendix, we prove the theorems on the singular locus stated inSection 2.2, and prove theorems on upper semi-continuity of order neededin our proofs of resolution. The proofs in Section A.1 are based on Zariski’soriginal proofs in [85].

A.1. Proofs of the basic theorems

Suppose that P ∈ AnK = spec(K[x1, . . . , xn]) has height n − r. Let mP ⊂

OAn,P be the ideal of P . The differential

d : OAn → Ω1An = OAndx1 ⊕ · · · ⊕ OAndxn

is defined by

d(u) =∂u

∂x1dx1 + · · ·+ ∂u

∂xndxn.

d induces a linear transformation of K(P )-vector spaces

dP : mP /(mP )2 → Ω1An,P ⊗K(P ).

Define D(P ) to be the image of dP . Since OAn,P is a regular local ring,dimK(P )mP /m

2P = n−r. Thus dimK(P )D(P ) ≤ n−r, and dimK(P )D(P ) =

n− r if and only ifdP : mP /(mP )2 → D(P )

is a K(P )-linear isomorphism.

Theorem A.1. Suppose that P ∈ AnK = spec(K[x1, . . . , xn]) is a closed

point. Then dimK(P )D(P ) = n if and only if K(P ) is separable over K.

163

164 Appendix. Smoothness and Non-singularity II

Proof. Suppose that P ∈ spec(K[x1, . . . , xn]) is a closed point. Let mP ⊂OAn,P be the ideal of P . Let αi be the image of xi in K(P ) for 1 ≤ i ≤ n.Let f1(z1) ∈ K[z1] be the minimal polynomial of α1 over K. We can theninductively define fi(z1, . . . , zi) in the polynomial ring K[z1, . . . , zi] so thatfi(α1, . . . , αi−1, zi) is the minimal polynomial of αi over K(α1, . . . , αi−1) for2 ≤ i ≤ n. Let I be the ideal

(f1(x1), f2(x1, x2), . . . , fn(x1, . . . , xn)) ⊂ OAn,P .

By construction I ⊂ mP and K[x1, . . . , xn]/I ∼= K(P ), so we have I =mP . Thus f1(x1), f2(x1, x2), . . . , fn(x1, . . . , xn) is a regular system of pa-rameters in OAn

K ,P , and the images of f1(x1), f2(x1, x2), . . . , fn(x1, . . . , xn)in mP /m

2P form a K(P )-basis. Thus dimK(P )D(P ) = n if and only if

dP (f1), . . . , dP (fn) are linearly independent over K(P ). This holds if andonly if J(f ;x) has rank n at P , which in turn holds if and only if the deter-minant

|J(f ;x)| =n∏

i=1

∂fi

∂xi

is not zero in K(P ). This condition holds if and only if K(α1, . . . , αi) isseparable over K(α1, . . . , αi−1) for 1 ≤ i ≤ n, which is true if and only ifK(P ) is separable over K.

Corollary A.2. Suppose that P ∈ AnK = spec(K[x1, . . . , xn]) is a closed

point such that K(P ) is separable over K. Let mP ⊂ OAn,P be the ideal ofP , and suppose that u1, . . . , un ∈ mP . Then u1, . . . , un is a regular systemof parameters in OAn

K ,P if and only if J(u;x) has rank n at P .

Proof. This follows since dP : mP /m2P → D(P ) is an isomorphism if and

only if K(P ) is separable over K.

Lemma A.3. Suppose that A is a regular local ring of dimension b with max-imal ideal m, and p ⊂ A is an equidimensional ideal such that dim(A/p) = a.Then

dimA/m(p+m2/m2) ≤ b− a,

and dimA/m(p+m2/m2) = b− a if and only if A/p is a regular local ring.

Proof. Let A′ = A/p, m′ = mA′. k = A/m ∼= A′/m′. There is a shortexact sequence of k-vector spaces

(A.1) 0 → p/p∩m2 ∼= p+m2/m2 → m/m2 → m′/(m′)2 = m/(p+m2) → 0.

The lemma now follows, since

dimk m′/(m′)2 ≥ a

and A′ is regular if and only if dimk m′/(m′)2 = a (Corollary 11.5 [13]).

A.1. Proofs of the basic theorems 165

We now give two related results, Lemma A.4 and Corollary A.5.

Lemma A.4. Suppose that A is a regular local ring of dimension n withmaximal ideal m, p ⊂ A is a prime ideal such that A/p is regular, anddim(A/p) = n−r. Then there exist regular parameters u1, . . . , un in A suchthat p = (u1, . . . , ur) and ur+1, . . . , un map to regular parameters in A/p.

Proof. Consider the exact sequence (A.1). Since A and A′ are regular,there exist regular parameters u1, . . . , un in A such that ur+1, . . . , un mapto regular parameters in A′, and u1, . . . , ur are in p. (u1, . . . , ur) is thus aprime ideal of height r contained in p. Thus p = (u1, . . . , ur).

Corollary A.5. Suppose that Y is a subvariety of dimension t of a varietyX of dimension n. Suppose that q ∈ Y is a non-singular closed point ofboth Y and X. Then there exist regular parameters u1, . . . , un in OX,q suchthat IY,q = (u1, . . . , un−t), and un−t+1, . . . , un map to regular parameters inOY,q. If vn−t+1, . . . , vn are regular parameters in OY,q, there exist regularparameters u1, . . . , un as above such that ui maps to vi for n− t+1 ≤ i ≤ n.

Lemma A.6. Let I = (f1, . . . , fm) ⊂ K[x1, . . . , xn] be an ideal. Supposethat P ∈ V (I) and J(f ;x) has rank n at P . Then K(P ) is separable alge-braic over K, and n of the polynomials f1, . . . , fm form a system of regularparameters in OAn

K ,P .

Proof. Let m be the ideal of P in K[x1, . . . , xn]. After possibly reindexingthe fi, we may assume that |J(f1, . . . , fn;x)| 6∈ m. By the Nullstellensatz,m is the intersection of all maximal ideals n of K[x1, . . . , xn] containing m.Thus there exists a closed point Q ∈ An

K with maximal ideal n containingm such that |J(f1, . . . , fn;x)| 6∈ n, so that J(f1, . . . , fn;x) has rank n at n.We thus have dimK(Q)D(Q) = n, so that K(Q) is separable over K and(f1, . . . , fn) is a regular system of parameters in OAn,Q by Theorem A.1 andCorollary A.2. Since I ⊂ m ⊂ n, we have P = Q.

Lemma A.7. Suppose that P ∈ AnK = spec(K[x1, . . . , xn]) with ideal

m ⊂ K[x1, . . . , xn]

is such that m has height n − r and t1, . . . , tn−r ∈ mm ⊂ OAn,P . Letαi be the image of xi in K(P ) for 1 ≤ i ≤ r. Then the determinant|J(t1, . . . , tn−r;xr+1, . . . , xn)| is non-zero at P if and only if α1, . . . , αr isa separating transcendence basis of K(P ) over K and t1, . . . , tn−r is a reg-ular system of parameters in OAn,P .

Proof. Suppose that |J(t1, . . . , tn−r;xr+1, . . . , xn)| is non-zero at P . Thereexist s0, s1, . . . , sn−r ∈ K[x1, . . . , xn] such that si ∈ K[x1, . . . , xn] for all i,s0 6∈ m and si

s0= ti for 1 ≤ i ≤ n− r. Then

|J(s1, . . . , sn−r;xr+1, . . . , xn)|


is non-zero at P . Let K∗ = K(α1, . . . , αr), let n be the kernel of

K∗[xr+1, . . . , xn] → K(P ),

and let s∗i for 1 ≤ i ≤ n− r be the image of si under

K[x1, . . . , xn] → K∗[xr+1, . . . , xn].

n is the ideal of a point Q ∈ An−rK∗ . We have K(P ) = K∗(Q), and at Q,

|J(s∗1, . . . , s∗n−r;xr+1, . . . , xn)| 6= 0.

By Lemma A.6, K∗(Q) is separable algebraic over K∗ and (s∗1, . . . , s∗n−r)

is a regular system of parameters in OAn−rK∗ ,Q = K∗[xr+1, . . . , xn]n. Since

trdegK K(P ) = r, α1, . . . , αr is a separating transcendence basis of K(P )over K. Thus K∗ ⊂ K[x1, . . . , xn]m and

K[x1, . . . , xn]m = K∗[xr+1, . . . , xn]n,

so that t1, . . . , tn−r is a regular system of parameters in

OAnK ,P = K[x1, . . . , xn]m.

Now suppose that α1, . . . , αr is a separating transcendence basis of K(P )over K and t1, . . . , tn−r is a regular system of parameters in OAn

K ,P =K[x1, . . . , xn]m. Let K∗ = K(α1, . . . , αr), and let n be the kernel of

K∗[xr+1, . . . , xn] → K(P ).

n is the ideal of a point Q ∈ An−rK∗ . Then K∗(Q) = K(P ) and K∗(Q) is

separable over K∗.We have a natural inclusion K∗ ⊂ K[x1, . . . , xn]m. Thus

K[x1, . . . , xn]m = K∗[xr+1, . . . , xn]n,

and t1, . . . , tn−r is a regular system of parameters in K∗[xr+1, . . . , xn]n. ByCorollary A.2, |J(t1, . . . , tn−r;xr+1, . . . , xn)| 6= 0 at Q, and thus it is also6= 0 at P .

Theorem A.8. Suppose that P ∈ AnK = spec(K[x1, . . . , xn]) and t1, . . . , tn−r

are regular parameters in OAn,P . Then rank(J(t;x)) = n − r at P if andonly if K(P ) is separably generated over K.

Proof. Let αi, 1 ≤ i ≤ n, be the images of xi in K(P ), and let m be theideal of P in K[x1, . . . , xn].

Suppose that rank(J(t;x)) = n− r at P . After possibly reindexing thexi and tj , we have |J(t1, . . . , tn−r;xr+1, . . . , xn)| 6= 0 at P . Now K(P ) isseparably generated over K by Lemma A.7.

Suppose that K(P ) is separably generated over K. Let ζ1, . . . , ζr bea separating transcendence basis of K(P ) over K. There exists Ψj(x) ∈K[x1, . . . , xn] for 0 ≤ j ≤ r with Ψ0(x) 6∈ m such that ζj = Ψj(α)

Ψ0(α) . Let

A.1. Proofs of the basic theorems 167

n be the kernel of the K-algebra homomorphism K[x1, . . . , xn+r] → K(P )defined by xi 7→ αi for 1 ≤ i ≤ n, xi 7→ ζi−n if n < i. n is the ideal ofa point Q ∈ An+r

K . Let Φ0(x),Φ1(x), . . . ,Φn−r(x) ∈ K[x1, . . . , xn] be suchthat Φ0 6∈ m and

ti =Φi(x)Φ0(x)

for 1 ≤ i ≤ n− r.

Let Φn−r+j(x) = xn+jΨ0(x)−Ψj(x) for 1 ≤ j ≤ r. Let I ⊂ K[x1, . . . , xn+r]be the ideal generated by Φ1, . . . ,Φn; thus I ⊂ n. We will show thatΦ1, . . . ,Φn are in fact a regular system of parameters in

OAn+rK ,Q = K[x1, . . . , xn+r]n.

It suffices to show that if F (x) ∈ n, then there exists

A(x) ∈ K[x1, . . . , xn+r]− n

such that AF ∈ I. Given such an F , we have an expansion

Ψ0(x)sF (x) =r∑

j=1

Bj(x)Φn−r+j(x) +G(x1, x2, . . . , xn),

where Bj(x) ∈ K[x1, . . . , xn+r] and s is a non-negative integer. Now F ∈ nimplies G ∈ m. Thus there exists an expansion

G =n−r∑i=1

fiti

with fi ∈ K[x1, . . . , xn]m, and there exist h ∈ K[x1, . . . , xn] − m andCi ∈ K[x1, . . . , xn] such that

hG =r∑

i=1

CiΦi(x)

and hΨ0(x)sF (x) ∈ I.Since Φ1, . . . ,Φn is a regular system of parameters in K[x1, . . . , xn+r]n

and ζ1, . . . , ζr is a separating transcendence basis of K(Q) over K, it followsthat

|J(Φ1, . . . ,Φn;x1, . . . , xn)|is non-zero at Q by Lemma A.7. Thus J(Φ1, . . . ,Φn−r;x1, . . . , xn) is ofrank n− r at P , and by Lemma A.7, α1, . . . , αn must contain a separatingtranscendence basis of K(P ) over K.

Proof that smoothness at a point is well defined

Suppose that X is a variety of dimension s over a field K and P ∈ X. InDefinition 2.6 we gave a definition for smoothness of X over K at P which


depended on choices of an affine neighborhood U = spec(R) of P and apresentation R ∼= K[x1, . . . , xn]/I with I = (f1, . . . , fm).

We now show that this definition is well defined, so that it is independentof all choices U , f and x.

Consider the sheaf of differentials Ω1X/K . We have, by the “second fun-

damental exact sequence” (Theorem 25.2 [66]), a presentation

(A.2) (K[x1, . . . , xn])m J(f ;x)→ Ω1K[x1,...,xn]/K ⊗R→ Ω1

R/K → 0.

Tensoring (A.2) with K(P ), we see that Ω1R/K ⊗K(P ) has rank s if and

only if J(f ;x) has rank n− s at P .

Proof of Theorem 2.8

Let s = dimX. There exists an affine neighborhood U = spec(R)of P in X such that R = K[x1, . . . , xn]/I, I = (f1, . . . , fm). Let mP ∈spec(K[x1, . . . , xn]) be the ideal of P in An

K . Let

t = dim(K[x1, . . . , xn]/mP ).

Suppose that K(P ) is separably generated over K and P is a non-singular point of X. By Theorem A.8, dP : mP /m

2P → D(P ) is an isomor-

phism. By Lemma A.3, dimK(P ) dP (I) = n−s (take A = OAn,P , p = ImP inthe statement of the lemma, so that b = n− t, a = s− t), which is equivalentto the condition that J(f ;x) has rank n − s at P . Thus X is smooth overK at P .

Suppose that X is smooth at P , so that J(f ;x) has rank n − s at P .Then dimK(P ) dP (I) = n− s, so that dimK(P )(I +m2

P )/m2P ≥ n− s. Thus

P is a non-singular point of X by Lemma A.3.

Remark A.9. With the notation of Theorem 2.8 and it’s proof, for anypoint P ∈ V (I), we have

dimK(P )(I +m2P )/m2

P ≤ n− s

by Lemma A.3, so that J(f ;x) has rank ≤ n− s at P .

Proof of Theorem 2.7

It suffices to prove that the locus of smooth points of X lying on anopen affine subset U = spec(R) of X of the form of Definition 2.6 is open.Let A = In−s(J(f ;x)). Then P ∈ U − V (A) if and only if J(f ;x) has rank

A.2. Non-singularity and uniformizing parameters 169

greater than or equal to n−s at P , which in turn holds if and only if J(f ;x)has rank n− s at P by Remark A.9.

Proof of Theorem 2.10 when K is perfect

Suppose thatK is perfect. The openness of the set of non-singular pointsof X follows from Theorem 2.7 and Corollary 2.9.

Let η ∈ X be the generic point of an irreducible component of X. ThenOX,η is a field which is a regular local ring. Thus η is a non-singular point.We conclude that the non-singular points of X are dense in X.

A.2. Non-singularity and uniformizing parameters

Theorem A.10. Suppose that X is a variety of dimension r over a fieldK, and P ∈ X is a closed point such that X is non-singular at P and K(P )is separable over K.

1. Suppose that y1, . . . , yr are regular parameters in OX,P . Then

Ω1X/K,P = dy1OX,P ⊕ · · · ⊕ dyrOX,P .

2. Let m be the ideal of P in OX,P . Suppose that f1, . . . , fr ∈ m ⊂OX,P and df1, . . . , dfr generate Ω1

X/K,P as an OX,P module. Thenf1, . . . , fr are regular parameters in OX,P .

Proof. We can restrict to an affine neighborhood of X, and assume that

X = spec(K[x1, . . . , xn]/I).

The conclusions of part 1 of the theorem follow from Theorem A.8 whenX = An

K .In the general case of 1, by Corollary A.5 there exist regular param-

eters (y1, . . . , yn) in A = K[x1, . . . , xn]m (where m is the ideal of P inspec(K[x1, . . . , xn]) such that IA = (yr+1, . . . , yn) and (y1, . . . , yr) map tothe regular parameters (y1, . . . , yr) in B = A/IA. By the second funda-mental exact sequence (Theorem 25.2 [66]) we have an exact sequence ofB-modules

IA/I2A→ Ω1A/K ⊗A B → Ω1

B/K → 0.

This sequence is thus a split short exact sequence of free B-modules, withΩ1

B/K∼= dy1B⊕· · ·⊕dyrB, since IA/I2A ∼= yr+1B⊕· · ·⊕ynB and Ω1

A/K⊗A

B ∼= dy1B ⊕ · · · ⊕ dynB.Suppose that the assumptions of 2 hold. Let R = OX,P . Let y1, . . . , yr

be regular parameters in R. Then dy1, . . . , dyr is a basis of the R-module


Ω1X/K,P by part 1 of this theorem. There are aij ∈ R such that

fi =r∑

j=1

aijyj , 1 ≤ j ≤ r.

We have expressions

dfi =r∑

j=1

aijdyj + (r∑

j=1

daijyj).

Let wi =∑r

j=1 aijdyj for 1 ≤ i ≤ r. By Nakayama’s Lemma we have thatthe wi generate Ω1

X/K,P as an R-module. Thus |(aij)| 6∈ m and (aij) isinvertible over R, so that f1, . . . , fr generate m.

Lemma A.11. Suppose that X is a variety of dimension r over a perfectfield K. Suppose that P ∈ X is a closed point such that X is non-singularat P , and y1, . . . , yr are regular parameters in OX,P . Then there exists anaffine neighborhood U = spec(R) of P in X such that y1, . . . , yr ∈ R, thenatural inclusion S = K[y1, . . . , yr] → R induces a morphism π : U → Ar

K

such that

(A.3) Ω1R/K = dy1R⊕ · · · ⊕ dyrR,

DerK(R,R) = HomR(Ω1R/K , R) =

∂

∂y1R⊕ · · · ⊕ ∂

∂yrR

(with ∂∂yi

(dyj) = δij), and the following condition holds. Suppose that a ∈ Uis a closed point, b = π(a), and (x1, . . . , xr) are regular parameters in OAr

K ,b.Then (x1, . . . , xr) are regular parameters in OX,a.

y1, . . . , yr are called uniformizing parameters on U .

Proof. By Theorem A.10, there exists an affine neighborhood U = spec(R)of P in X such that y1, . . . , yr ∈ R and dy1, . . . , dyr is a free basis of Ω1

R/K .Consider the natural inclusion S = K[y1, . . . , yr] → R, with induced mor-phism π : spec(R) → spec(S). Suppose that a ∈ U is a closed point withmaximal ideal m in R, and b = π(a), with maximal ideal n in S. Supposethat (x1, . . . , xr) are regular parameters in B = Sn. Let A = Rm. By con-struction, Ω1

S/K is generated by dy1, . . . , dyr as an S-module. Thus Ω1A/K is

generated by Ω1B/K as an A-module. By 1 of Theorem A.10, dx1, . . . , dxr

generate Ω1B/K as a B-module, and hence they generate Ω1

A/K as an A-module. By 2 of Theorem A.10, x1, . . . , xr is a regular system of parametersin A.

A.3. Higher derivations 171

A.3. Higher derivations

Suppose that K is a field and A is a K-algebra. A higher derivation of Aover K of length m is a sequence

D = (D0, D1, . . . , Dm)

if m <∞, and a sequence

D = (D0, D1, . . .)

if m = ∞, of K-linear maps Di : A→ A such that D0 = id and

(A.4) Di(xy) =i∑

j=1

Dj(x)Dj−i(y)

for 1 ≤ i ≤ m and x, y ∈ A.Let t be an indeterminate. D = (D0, D1, . . .) is a higher derivation of

A over K (of length ∞) precisely when the map Et : A→ A[[t]] defined byEt(f) =

∑∞i=0Di(f)ti is a K-algebra homomorphism with D0(f) = f , and

D = (D0, D1, . . . , Dm) is a higher derivation of A over K of length m pre-cisely when the map Et : A→ A[t]/(tm+1) defined by Et(f) =

∑mi=0Di(f)ti

is a K-algebra homomorphism with D0(f) = f .

Example A.12. Suppose that A = K[y1, . . . , yr] is a polynomial ring overa field K. For 1 ≤ i ≤ r, let εj be the vector in Nr with a 1 in the j-th coefficient and 0 everywhere else. For f =

∑ai1,...,iry

i11 · · · yir

r ∈ A andm ∈ N, define

D∗mεj

(f) =∑

ai1,...,ir

(ijm

)yi11 · · · y

ij−mj · · · yir

r .

Then D∗j = (D∗

0, D∗εj, D∗

2εj, . . .) is a higher derivation of A over K of length

∞ for 1 ≤ j ≤ r. Observe that the D∗j are iterative; that is,

Diεj Dkεj=

(i+ k

i

)D(i+k)εj

for all i, k. Define

D∗i1,...,ir = D∗

irεrD∗

ir−1εr−1 · · · D∗

i1ε1

for (i1, . . . , ir) ∈ Nr. The D∗i1,...,ir

commute with each other.

The D∗i1,...,ir

are called Hasse-Schmidt derivations [56]. If K has charac-teristic zero, then

D∗i1,...,ir =

1i1!i2! · · · ir!

∂i1+···+ir

∂yi11 · · · ∂y

irr

.


Lemma A.13. Suppose that X is a variety of dimension r over a perfectfield K. Suppose that P ∈ X is a closed point such that X is non-singularat P , y1, . . . , yr are regular parameters in OX,P , and U = spec(R) is anaffine neighborhood of P in X such that the conclusions of Lemma A.11hold. Then there are differential operators Di1,...,ir on R, which are uniquelydetermined on R by the differential operators D∗

i1,...,iron S of Example A.12,

such that if f ∈ OX,P and

f =∑

ai1,...,iryi11 · · · y

irr

in OX,P∼= K(P )[[y1, . . . , yr]], where we identify K(P ) with the coefficient

field of OX,P containing K, then ai1,...,ir = Di1,...,ir(f)(P ) is the residue ofDi1,...,ir(f) in K(P ) for all indices i1, . . . , ir.

Proof. If K has characteristic zero, we can take

(A.5) Di1,...,ir =1

i1! · · · ir!∂i1+···+ir

∂yi11 · · · ∂y

irr

.

Suppose that K has characteristic p ≥ 0. Consider the inclusion ofK-algebras

S = K[y1, . . . , yr] → R

of Lemma A.11, with induced morphism

π : U = spec(R) → V = spec(S).

Let D∗j for 1 ≤ j ≤ r be the higher order derivation of S over K defined in

Example A.12. We have Ω1R/S = 0 by 1 of Theorem A.10, (A.3) and the “first

fundamental exact sequence” (Theorem 25.1 [66]). Thus π is non-ramifiedby Corollary IV.17.4.2 [45]. By Theorem IV.18.10.1 [45] and PropositionIV.6.15.6 [45], π is etale.

We will now prove that D∗j extends uniquely to a higher derivation Dj of

R over K for 1 ≤ j ≤ r. It suffices to prove that (D∗0, D

∗εj, . . . , D∗

mεj) extends

to a higher derivation of R over K for all m. The extension of D∗0 = id

to D0 = id is immediate. Suppose that an extension (D0, D1, . . . , Dm) of(D∗

0, D∗εj, . . . , D∗

mεj) has been constructed. Then we have a commutative

diagram of K-algebra homomorphisms

Rg→ R[t]/(tm+1)

↑ ↑S

f→ R[t]/(tm+2)

where

g(a) =m∑

i=0

Diεj (a)tj ,

A.3. Higher derivations 173

f(a) =m+1∑i=0

D∗iεj

(a)tj .

Since (tm+1)R[t]/(tm+2) is an ideal of square zero, and S → R is formallyetale (Definitions IV.17.3.1 and 17.1.1 [45]), there exists a unique K-algebrahomomorphism h making a commutative diagram

Rg→ R[t]/(tm+1)

↑h ↑

Sf→ R[t]/(tm+2)

Thus there is a unique extension of (D∗0, D

∗εj, . . . , D∗

(m+1)εj) to a higher

derivation of R.Set Di1,...,ir = Dirεr · · · Di1ε1 .Let A = OX,P . Since K(P ) is separable over K, and by Exercise 3.9 of

Section 3.2, we can identify the residue field K(P ) of A with the coefficientfield of A which contains K. Thus we have natural inclusions

K[y1, . . . , yr] ⊂ A ⊂ A ∼= K(P )[[y1, . . . , yr]].

We will now show that Dεjextends uniquely to a higher derivation of A

over K for 1 ≤ j ≤ r. Let m be the maximal ideal of A. Let t be anindeterminate. Then A[[t]] is a complete local ring with maximal ideal

n = mA[[t]] + tA[[t]].

Define Et : A → A[[t]] by Et(f) =∑∞

n=0Dnεj (f)tn. Et is a K-algebrahomomorphism, since Dεj

is a higher derivation. Et is continuous in them-adic topology, as Et(ma) ⊂ na for all a. Thus Et extends uniquely toa K-algebra homomorphsim A → A[[t]]. Thus Dεj

extends uniquely to ahigher derivation of A over K.

We can thus extend Di1,...,ir uniquely to A. Since K(P ) is algebraic overK, and Di1,...,ir extends D∗

i1,...,ir, it follows that Di1,...,ir acts on A as stated

in the theorem.

Remark A.14. Suppose that the assumptions of Lemma A.13 hold, andthat K is an algebraically closed field. Then the following properties hold.

1. Suppose that Q ∈ U is a closed point and a is the correspondingmaximal ideal in R. Then a = (y1, . . . , yr), where yi = yi−αi withαi = yi(Q) ∈ K for 1 ≤ i ≤ r, and K[[y1, . . . , yr]] = Ra = OU,Q.

2. The differential operators Di1,...,in on R are such that if f ∈ R, anda is a maximal ideal in R, then, with the notation of 1,

f =∑

ai1,...,iryi11 · · · y

irr


in K[[y1, . . . , yr]] = Ra, where ai1,...,ir = Di1,...,ir(f)(Q) is theresidue of Di1,...,ir(f) in K(Q) for all indices i1, . . . , ir.

Proof. The ideal of π(Q) in S is (y1−α1, . . . , yr−αr) with αi = yi(Q) ∈ K.Let yi = yi − αi for 1 ≤ i ≤ r. We have a = (y1, . . . , yr) by Theorem A.102, and 1 follows. The derivations ∂

∂y1, . . . , ∂

∂yrare exactly the derivations

∂∂y1

, . . . , ∂∂yr

on S of Example A.12, so they extend to the differential op-erators D∗

i1,...,iron S of Example A.12. Now 2 follows from Lemma A.13

applied to Q and y1, . . . , yr.

Lemma A.15. Let notation be as in Lemmas A.11 and A.13. In particular,we have a fixed closed point P ∈ U = spec(R) and differential operatorsDi1,...,ir on R. Let I ⊂ R be an ideal. Define an ideal Jm,U ⊂ R by

Jm,U = Di1,...,ir(f) | f ∈ I and i1 + · · ·+ ir < m.

Suppose that Q ∈ U is a closed point, with corresponding ideal mQ in R,and suppose that (z1, . . . , zr) is a regular system of parameters in mQ.

Let Di1,...,ir be the differential operators constructed on OW,Q by the con-struction of Lemmas A.11 and A.13. Then

(Jm,U )mQ = Di1,...,ir(f) | f ∈ I and i1 + · · ·+ ir < m.

Proof. When K has characteristic zero, this follows directly from differen-tiation and (A.5).

Assume that K has positive characteristic. By Lemma A.13 and Theo-rem IV.16.11.2 [45], U is differentiably smooth over spec(K) and Di1,...,ir |i1, . . . , ir < m is an R-basis of the differential operators of order less thanm on R. Since dz1, . . . , dzr is a basis of Ω1

RmQ/K (by Lemma A.11),

Di1,...,ir | i1, . . . , ir < m is an RmQ-basis of the differential operators oforder less than m on RmQ by Theorem IV.16.11.2 [45]. The lemma fol-lows.

A.4. Upper semi-continuity of νq(I)

Lemma A.16. Suppose that W is a non-singular variety over a perfectfield K and K ′ is an algebraic field extension of K. Let W ′ = W ×K K ′,with projection π1 : W ′ → W . Then W ′ is a non-singular variety overK ′. Suppose that q ∈ W is a closed point. Then there exists a closed pointq′ ∈W ′ such that π1(q′) = q. Furthermore,

1. Suppose that (f1, . . . , fr) is a regular system of parameters in OW,q.Then (f1, . . . , fr) is a regular system of parameters in OW ′,q′.

A.4. Upper semi-continuity of νq(I) 175

2. If I ⊂ OW,q is an ideal, then

(IOW ′,q′) ∩ OW,q = I.

Proof. The fact that W ′ is non-singular follows from Theorem 2.8 and thedefinition of smoothness. Let q′ ∈W ′ be a closed point such that π1(q′) = q.

We first prove 1. Suppose that U = spec(R) is an affine neighborhoodof q in W . There is a presentation R = K[x1, . . . , xn]/I. We further havethat π−1

1 (U) = spec(R′), with R′ = K ′[x1, . . . , xn]/I(K ′[x1, . . . , xn]). Let mbe the ideal of q in K[x1, . . . , xn], and n the ideal of q′ in K ′[x1, . . . , xn].By the exact sequence of Lemma A.3, there exists a regular system ofparameters f1, . . . , fn in K[x1, . . . , xn]m, such that Im = (f r+1, . . . , fn)m

and f i maps to fi for 1 ≤ i ≤ r. By Corollary A.2, the determinant|J(f1, . . . , fn;x1, . . . , xn)| is non-zero at q, so it is non-zero at q′. By Corol-lary A.2, f1, . . . , fn is a regular system of parameters in K ′[x1, . . . , xn]n.Thus f1, . . . , fr is a regular system of parameters in R′

n = OW ′,q′ .We will now prove 2. Since n maps to a maximal ideal of OW,q ⊗K K ′,

OW ′,q′ = (OW,q ⊗K K ′)n

( n denotes completion with respect to nOW,q ⊗K K ′). Thus OW ′,q is faith-fully flat over OW,q, and (IOW ′,q′)∩OW,q = I (by Theorem 7.5 (ii) [66]).

Suppose that (R,m) is a regular local ring containing a field K of char-acteristic zero. Let K ′ be the residue field of R. Suppose that J ⊂ R is anideal. We define the order of J in R to be

νR(J) = maxb | J ⊂ mb.

If J is a locally principal ideal, then the order of J is its multiplicity. How-ever, order and multiplicity are different in general.

Definition A.17. Suppose that q is a point on a variety W and J ⊂ OW

is an ideal sheaf. We denote

νq(J) = νOW,q(JOW,q).

If X ⊂W is a subvariety, we denote

νq(X) = νq(IX).

Remark A.18. Let notation be as in Lemma A.16 (except we allow K tobe an arbitrary, not necessarily perfect field). Observe that if q ∈ W andq′ ∈W ′ are such that π(q′) = q and J ⊂ OW is an ideal sheaf, then

νOW,q(Jq) = νOW,q

(JqOW,q) = νOW ′,q′ (JqOW ′,q′) = νOW ′,q′(JqOW ′,q′).


Suppose that X is a noetherian topological space and I is a totallyordered set. f : X → I is said to be an upper semi-continuous function iffor any α ∈ I the set

q ∈ X | f(q) ≥ αis a closed subset of X.

Suppose that X is a subvariety of a non-singular variety W . Define

νX : W → N

by νX(q) = νq(X) for q ∈W . The order νq(X) is defined in Definition A.17.

Theorem A.19. Suppose that K is a perfect field, W is a non-singularvariety over K and I is an ideal sheaf on W . Then

νq(I) : W → N

is an upper semi-continuous function.

Proof. Suppose that m ∈ N. Suppose that U = spec(R) is an open affinesubset of W such that the conclusions of Lemma A.11 hold on W . SetI = Γ(U, I).

With the notation of Lemmas A.11 and A.13, define an ideal Jm,U ⊂ Rby

Jm,U = Di1,...,ir(g) | g ∈ I and i1 + · · · ir < m.By Lemma A.15, this definition is independent of the choice of y1, . . . , yr sat-isfying the conclusions of Lemma A.11, so our definition of Jm,U determinesa sheaf of ideals Jm on W .

We will show that

V (Jm) = η ∈W | νη(I) ≥ m,from which upper semi-continuity of νq(I) follows.

Suppose that η ∈ W is a point. Let Y be the closure of η in W , andsuppose that P ∈ Y is a closed point such that Y is non-singular at P .

By Corollary A.5, Lemma A.11 and Lemma A.15, there exists an affineneighborhood U = spec(R) of P in W such that there are y1, . . . , yr ∈ Rwith the properties that y1 = . . . = yt = 0 (with t ≤ r) are local equationsof Y in U and y1, . . . , yr satisfy the conclusions of Lemma A.11.

For g ∈ IP , consider the expansion

g =∑

ai1,...,iryi11 · · · y

irr

in OW,P∼= K(P )[[y1, . . . , yr]], where we have identified K(P ) with the coef-

ficient field of OW,P containing K and (with the notation of Lemma A.13)

ai1,...,ir = Di1,...,ir(g)(P ).

A.4. Upper semi-continuity of νq(I) 177

Then η ∈ V (Jm) is equivalent to Di1,...,ir(g) ∈ (y1, . . . , yt) wheneverg ∈ IP and i1 + · · ·+ ir < m, which is equivalent to

ai1,...,ir = Di1,...,ir(g)(P ) = 0

if g ∈ IP and i1 + · · ·+ it < m, which in turn holds if and only if

g ∈ ImY,P ∩ OW,P = Im

Y,P

for all g ∈ IP , where the last equality is by Lemma A.16. Localizing OW,P

at η, we see that νη(I) ≥ m if and only if η ∈ V (Jm).

Definition A.20. Suppose that W is a non-singular variety over a perfectfield K and X ⊂W is a subvariety.

For b ∈ N, define

Singb(X) = q ∈W | νq(W ) ≥ b.

Singb(X) is a closed subset of W by Theorem A.19. Let

r = maxνX(q) | q ∈W.

Suppose that q ∈ Singr(X). A subvariety H of an affine neighborhood U ofq in W is called a hypersurface of maximal contact for X at q if

1. Singr(X) ∩ U ⊂ H, and2. if

Wnπn→ · · · →W1

π1→W

is a sequence of monoidal transforms such that for all i, πi is cen-tered at a non-singular subvariety Yi ⊂ Singr(Xi), where Xi is thestrict transform of X on Wi, then the strict transform Hn of H onUn = Wn ×W U is such that Singr(Xn) ∩ Un ⊂ Hn.

With the above assumptions, a non-singular codimension one subvarietyH of U = spec(OW,q) is called a formal hypersurface of maximal contact forX at q if 1 and 2 above hold for U = spec(OW,q).

Remark A.21. Suppose that W is a non-singular variety over a perfectfield K and X ⊂W is a subvariety, q ∈W . Let T = OW,q, and define

Singb(OX,q) = P ∈ spec(T ) | ν(IX,qTP ) ≥ b.

Then Singb(OX,q) is a closed set. If J ⊂ OW is a reduced ideal sheafwhose support is Singb(X), then JqT is a reduced ideal whose support isSingb(OX,q).

Proof. Let U = spec(R) be an affine neighborhood of q in W satisfying theconclusions of Lemma A.13 (with p = q), and let notation be as in LemmaA.13 (with p = q). With the further notation of Theorem A.19 (and I = IX)we see that Γ(U, J) =

√Jb,U . JqT is a reduced ideal by Theorem 3.6.


Suppose that P ∈ Singb(OX,q). Then νTP(g) ≥ b for all g ∈ IX,q.

Thus g ∈ P (b) ⊂ T , and there exists h ∈ T − P such that gh ∈ P b. Nowinduction in the formula (A.4) shows that Di1,...,ir(gh) ∈ P b−i1,...,ir andDi1,...,ir(g) ∈ P (b−i1−···−ir) ⊂ P if i1 + · · ·+ ir < b. Thus JqT ⊂ P .

Now suppose that P ∈ spec(T ) and JqT ⊂ P . Let S = OW,q. IfJq = P1∩· · ·∩Pm is a primary decomposition, then all primes Pi are minimalsince Jq is reduced. By Theorem 3.6 there are primes Pij in T and a functionσ(i) such that PiT = Pi1 ∩ · · · ∩ Piσ(i) is a minimal primary decomposition.Thus JqTPij = PiTPij = PijTPij for all i, j. We have

⋂ij Pij ⊂ P , so Pij ⊂ P

for some i, j. Moreover, IX,q ⊂ P bi SPi implies IX,q ⊂ P b

ijTPij , which in turnimplies IX,q ⊂ P bTPij . Thus νTP

(IX,qTP ) ≥ b and P ∈ Singb(OX,q).

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