RESOLUTION OF SINGULARITIES Contentsfaculty.missouri.edu/~cutkoskys/book7.pdf · Embedded resolution of singularities 42 6. Resolution of singularities of varieties in characteristic

RESOLUTION OF SINGULARITIES

S. DALE CUTKOSKY

Contents

1. Introduction 31.1. Notations 42. Non-singularity and resolution of singularities 42.1. Newton’s method for determining the branches of a plane curve 42.2. Smoothness and non-singularity 72.3. Resolution of singularities 92.4. Normalization 92.5. Local uniformization and generalized resolution problems 103. Curve singularities 143.1. Blowing up a point on A2 143.2. Completion 183.3. Blowing up a point on a non-singular surface 203.4. Resolution of curves embedded in a non-singular surface I 213.5. Resolution of curves embedded in a non-singular surface II 244. Resolution type theorems 284.1. Blow ups of ideals 284.2. Resolution type theorems and corollaries 315. Surface singularities 335.1. Resolution of surface singularities 335.2. Embedded resolution of singularities 426. Resolution of singularities of varieties in characteristic zero 456.1. The operator 4 and other preliminaries for resolution in arbitrary

dimension 466.2. Hypersurfaces of maximal contact and induction in resolution 496.3. Pairs and basic objects 526.4. Basic objects and hypersurfaces of maximal contact 566.5. General basic objects 626.6. Functions on a general basic object 636.7. Resolution theorems for a general basic object 686.8. Resolutions of singularities of algebraic varieties in characteristic zero 767. Resolution of surface singularities in positive characteristic 797.1. Resolution and some invariants 797.2. τ(q) = 2 837.3. τ(q) = 1 867.4. Remarks and further discussion 1008. Local uniformization and resolution of surface singularities 102

1

2 S. DALE CUTKOSKY

8.1. Classification of valuations in algebraic function fields of dimension 2 1028.2. Local uniformization of algebraic function fields of surfaces 1058.3. Resolving systems and the Zariski-Riemann manifold 1139. Ramification of valuations and simultaneous resolution of singularities 11710. Smoothness and non-singularity II 12310.1. Proofs of the basic theorems 12310.2. Non-singularity and uniformizing parameters 12710.3. Higher derivations 12910.4. Upper semi-continuity of νq(I) 131References 134

RESOLUTION OF SINGULARITIES 3

1. Introduction

The notion of singularity is basic to mathematics. In elementary algebra singularityappears as a multiple root of a polynomial. In geometry a point on a space is non-singular if it has a tangent space whose dimension is that of the space. Both notionsof singularity can be detected through the vanishing of derivitives.

Over an algebraically closed field, a variety is non-singular at a point if there existsa tangent space at the point which has the same dimension as that of the variety. Moregenerally, a variety is non-singular at a point if its local ring is a regular local ring. Afundamental problem is to remove a singularity by simple algebraic mappings. Thatis, can a given variety be desingularized by a proper, birational morphism from a non-singular variety? This is always possible in all dimensions, over fields of characteristiczero. We give a complete proof of this in Chapter 6.

We also treat positive characteristic, developing the basic tools needed for thisstudy, and giving a proof of resolution of surface singularities in positive characteristicin Chapter 7.

In Section 2.5 we discuss important open problems, such as resolution of singular-ities in positive characteristic and local monomialization of morphisms.

Chapter 8 gives a classification of valuations in algebraic function fields of surfaces,and a modernization of Zariski’s original proof of local uniformization for surfaces incharacteristic zero.

This book has evolved out of lectures given at the University of Missouri and atthe Chennai School of Mathematics. It can be used as part of a one year introductorysequence in algebraic geometry, and would provide an exciting direction after thebasic notions of schemes and sheaves have been covered. A core course on resolutionis covered in Chapters 2 through 6. The major ideas of resolution have been introducedby the end of Section 6.2, and after reading this far, a student will find the resolutiontheorems of Section 6.8 quite believable, and have a good feel for what goes into theirproof.

Chapters 7 and 8 cover additional topics. These two chapters are independent, andcan be chosen as possible followups to the basic material in the first 5 chapters. Chap-ter 7 gives a proof of resolution of singularities for surfaces in positive characteristic,and Chapter 8 gives a proof of local uniformization and resolution of singularitiesfor algebraic surfaces. This chapter provides an introduction to valuation theory inalgebraic geometry, and to the problem of local uniformization.

Chapter 10 proves foundational results on the singular locus that we need. On afirst reading, I recommend that the reader simply look up the statements as neededin reading the main body of the book. Versions of almost all of these statementsare much easier over algebraically closed fields of characteristic zero, and most of theresults can be found in this case in standard text books in algebraic geometry.

I assume that the reader has some familiarity with algebraic geometry and commu-tative algebra, such as can be obtained from an introductory course on these subjects.This material is covered in books such as Atiyah MacDonald [12] or the basic sectionsof Eisenbud’s book [35], and the first two chapters of Hartshorne’s book on algebraicgeometry [45], or Eisenbud and Harris’s book on Schemes [36].

I thank Professors Seshadri and Ed Dunne for their encouragement to write thisbook, and Laura Ghezzi, Tai Ha and Olga Kashcheyeva for their helpful commentson preliminary versions of the manuscript.

4 S. DALE CUTKOSKY

1.1. Notations. The notations of Hartshorne [45] will be followed, with the followingdifferences and additions.

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

K . Thus an integral va-riety is a “quasi-projective variety” in the classical sense. A curve is a one dimensionalvariety. A surface is a two dimensional variety, and a 3-fold is a three dimensionalvariety. A subvariety Y of a variety X is 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 ideal of Y in X by IY . If W1,W2

are subschemes of a variety of X, we will denote the scheme theoretic intersection ofW1 and W2 by W1 ·W2. This is the subscheme

W1 ·W2 = V (IW1 + IW2) ⊂W.

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

2. Non-singularity and resolution of singularities

2.1. Newton’s method for determining the branches of a plane curve. New-ton’s algorithm for solving f(x, y) = 0 by a fractional power series y = y(x

1m ) can

be thought of as a generalization of the implicit function theorem to general analyticfunctions. We begin with this algorithm because of its simplicity and elegance, andbecause this method contains some of the most important ideas in resolution. We willsee (in Section 2.5) that the algorithm immediately gives a local solution to resolutionof analytic plane curve singularities, and that it can be interpreted to give a globalsolution to resolution of plane curve singularities (in Section 3.5). All of the proofs ofresolution in this book can be viewed as generalizations of Newton’s algorithm, withthe exception of the proof that curve singularities can be resolved by normalization(Theorems 2.12 and 4.3).

Suppose that K is an algebraically closed field of characteristic 0, K[[x, y]] is a ringof power series in two variables and f ∈ K[[x, y]] is a non-unit, such that x 6 |f . Writef =

∑i,j aijx

iyj with aij ∈ K. Let

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

mult (f(0, y)) = min j | a0j 6= 0.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 that δ0 < ∞.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 = p0q0

where q0, p0 are relatively prime positive integers. We make a trans-formation

x = xq01 , y = xp01 (y1 + c1).

Thenf = xr0p0

1 f1(x1, y1)

where

f1(x1, y1) =∑

i+δ0j=δ0r0

aij(c1 + y1)j + x1H(x1, y1). (1)

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

f1 =∑

aij(1)xi1yj1.

Set

δ1 = min

i

r1 − j: j < r1 and aij(1) 6= 0

,

and write δ1 = p1q1

with p1, q1 relatively prime. We can then choose c2 ∈ K for f1, inthe same way that we chose c1 for f , and iterate this process, obtaining a sequenceof transformations

x = xq01 , y = xp01 (y1 + c1)

x1 = xq12 , y1 = xp12 (y2 + c2)

...(2)

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

As our first approximation, we can use our first transformation to solve for y interms 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

y = c1xδ0 + c2x

δ0+δ1q0 + c3x

δ0+δ1q0

+δ2q0q1 + · · · (3)

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 for all i, so itsuffices to show that ri = ri+1 implies δi ∈ N. Without loss of generality, we mayassume that i = 0 and r0 = r1. f1(x1, y1) is given by the expression (1). Set

g(t) = f1(0, t) =∑

i+δ0j=δ0r0

aij(c1 + t)j .

6 S. DALE CUTKOSKY

g(t) has degree r0. Since r1 = r0, we also have mult(g(t)) = r0. Thus g(t) = a0r0tr0 ,

and ∑i+δ0j=δ0r0

aijtj = a0r0(t− c1)r0 .

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

ai,r0−1 6= 0 (4)

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 the smallest possible,and a series

p(t) =∑

biti

such that (3) becomes

y = p(x1m ). (5)

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.

y =∑

bixim (6)

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 into algebraic geometry.Remark 2.2. Our proof of Theorem 2.1 is not valid in positive characteristic, sincewe cannot conclude (4). Theorem 2.1 is in fact false over fields of positive character-istic. See Exercise 2 at the end of this section.

Suppose that f ∈ K[[x, y]] is irreducible, and that we have found a solution 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(x 1m ) divides f in R1 = K[[x

1m , y]].

Let ω be a primitive mth root of unity in K. Since f is invariant under the K-algebraautomorphism φ of R1 determined by x

1m → ωx

1m and y → y, y−p(ωjx 1

m )) | 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.

g =m−1∏j=0

(y − p(ωjx 1

m ))

is invariant under φ, so g ∈ K[[x, y]] and g | f in K[[x, y]], the ring of invariants ofR1 under the action of the group Zm generated by φ. Since f is irreducible,

f = u

m−1∏j=0

(y − p(ωjx 1

m ))

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


Remark 2.3. Some letters of Newton developing this idea are translated (from Latin)in Brieskorn and Knorrer’s book [16].

After we have defined non-singularity, we will return to this algorithm in (7) ofSection 2.5, to see that we have actually constructed a resolution of singularities of aplane curve singularity.

Exercises

1. Construct a Puiseux series solution to f(x, y) = y4−2x3y2−4x5y+x6−x7 = 0over the complex numbers.

2. (Chevalley [20]) Prove that f(x, y) = xyp−xy−1 = 0 does not have a solutionof the form (5) over a field of characteristic p > 0.

2.2. Smoothness and non-singularity.

Definition 2.4. Suppose that X is a scheme. X is non-singular at P ∈ X if OX,Pis a regular local ring.

Recall that a local ring R, with maximal ideal m, is regular if the dimension ofm/m2 as a R/m vector space is equal to the Krull dimension of 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, R = K[x1, . . . , xn]/I.Suppose that P ∈ spec(R) has ideal m in K[x1, . . . , xn] and ideal n in R. LetK(P ) = Rn/nn. We will say that J(f ;x) has rank l at P if the image of the l-thfitting ideal Il(J(f ;x)) of l × l minors of J(f, x) in K(P ) is K(P ) and the image ofthe sth fitting ideal Is(J(f ;x)) in K(P ) is (0) for s > l.

If A is a square matrix, we will denote the determinant of A by | A |.Definition 2.5. Suppose that X is a variety of dimension s over a field K, andP ∈ X. Suppose that U = spec(R) is an affine neighborhood of P such that R ∼=K[x1, . . . , xn]/I with I = (f1, . . . , fm). Then X is smooth over K if J(f ;x) has rankn− s at P .

This definition depends only on P and X and not on any of the choices of U , x orf , as can be seen from a local calculation.

Theorem 2.6. Let K be a field. The set of points in a K-variety X which are smoothover K is an open set of X.

Theorem 2.7. 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-singular point ofX.

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

8 S. DALE CUTKOSKY

In the case when K is algebraically closed, Theorems 2.6 and 2.7 are proven inTheorems I.5.3 and I.5.1 of [45]. Recall that an algebraically closed field is perfect.We will give the proofs of Theorems 2.6 and 2.7 for general fields in Chapter 10.

Corollary 2.8. Suppose that X is a variety over a perfect field K and P ∈ X. ThenX is non-singular at P if and only if X is smooth at P over K.

Proof. This is immediate since an algebraic function field over a perfect field K isalways separably generated over K (Theorem 13, Section 13, Chapter II, [85]).

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

K is an s-dimensional affine variety, where I = (f1, . . . , fm)is a reduced and equidimensional ideal. We interpret the closed points of An

K asthe set of solutions to f1 = · · · = fm = 0 in Kn. We identify a closed pointp = (a1, . . . , an) ∈ V (I) ⊂ An

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

fi ≡ fi(p) + Li mod m2

where

Li =n∑i=1

∂fi∂xi

(p)(xi − ai).

fi(p) = 0 for all i since p is a point of X. The tangent space to X in AnK at the point

p is

Tp(X) = V (L1, . . . , Lm) ⊂ AnK .

We see that dim Tp(X) = n − rank(J(f ;x)(p)). Thus dim Tp(X) ≥ s (by Remark10.9), and X is non-singular at p if and only if dim Tp(X) = s.

Theorem 2.9. Suppose that X is a variety over a field K. Then the set of 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 [45]. Wewill prove Theorem 2.9 when K is perfect in Chapter 10. The general case is provenin the Corollary to Theorem 11 [78].

Exercise

Consider the curve y2 − x3 = 0 in A2K , over an algebraically closed field K of

characteristic 0 or p > 3.

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

−3(x− 1) + 2(y − 1) = 0.

b. Show that at the point p2 = (0, 0), the tangent space Tp2(X) is the entireplane A2

K .c. 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 2.10. A resolution of singularities of X is a proper birational morphismφ : Y → X such that Y is a non-singular variety.

A birational morphism is a morphism φ : Y → X of varieties such that there is adense open subset U of X such that φ−1(U) → U is an isomorphism. If X and Yare integral and K(X) and K(Y ) are the respective function fields of X and Y , φ isbirational if and only if φ∗ : K(X)→ K(Y ) is an isomorphism.

A morphism of varieties φ : Y → X is proper if for every valuation ring V withmorphism α : spec(V ) → X, there is a unique morphism β : spec(V ) → Y such thatφ β = α. If X and Y are integral, and K(X) is the function field of X, then weonly need consider valuation rings V such that K ⊂ V ⊂ K(X) in the definition ofproperness.

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

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

Birational proper morphisms of non-singular varieties (over a field of characteristiczero) can be factored by alternating sequences of blow ups and blow downs of non-singular subvarieties ([76], [10]). Locally, along a valuation, it is possible to factor abirational morphism of n-dimensional varieties as a product φn−2 · · ·φ1 where eachφi is a product of blow ups followed by a product of blow downs ([23], [24]).

We can extend our definition of a resolution of singularities to arbitrary schemes.A reasonable category to consider is excellent (or quasi-excellent) schemes (definedin IV.7.8 [43] and on page 260 of [61]). The definition of excellence is extremelytechnical, but the idea is to give minimal conditions ensuring that the the singularlocus is preserved by natural base extensions such as completion. There are examplesof non-excellent schemes which admit a resolution of singularities. Rotthaus [68] givesan example of a regular local ring R of dimension 3 containing a field which is notexcellent. 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. LetS be the integral closure of R in L. Since R is affine, S is finite over R, so that S isaffine (c.f. Theorem 9, Chapter V [85]). We say that spec(S) is the normalization ofspec(R).

Suppose that V is a valuation ring of L such that R ⊂ V . V is integrally closedin L (although V need not be Noetherian). Thus S ⊂ V . The morphism spec(V )→spec(R) lifts to a morphism spec(V )→ spec(S), so that spec(S)→ spec(R) is proper.

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

For a general (reduced but not necessarily integral) variety X, the normalizationof X is the disjoint union of the normalizations of the irreducible components of X.Theorem 2.11. (Serre) A local ring A is integrally closed if and only if A is R1 (Apis regular if ht(p) ≤ 1) and S2 (depth Ap ≥ 2 if ht(p) ≥ 2, depth Ap = 1 if ht(p) = 1).

A proof of this theorem is given in Theorem 23.8 [61].

10 S. DALE CUTKOSKY

Theorem 2.12. Suppose that X is a 1-dimensional variety over a field K. Then thenormalization of X is a resolution of singularities.

Proof. Let X be the normalization of X. All local rings OX,p of points p ∈ X arelocal rings of dimension 1 which are integrally closed, so Theorem 2.11 implies theyare regular.

As an example, consider the curve singularity y2 − x3 = 0 in A2, with affine ringR = K[x, y]/(y2−x3). In the quotient field L of R we have the relation ( yx )2−x = 0.Thus y

x is integral over R. Since R[ yx ] = K[ yx ] is a regular ring, it must be the integralclosure 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 dimension largerthan 1. As an example, consider the surface singularity X defined by z2 − xy = 0in A3

K , 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

K[x, y, z]/(z2 − xy) is R1. Since it is a complete intersection, it is S2, so that X isnormal, but singular.

Kawasaki [54] has proven that under extremely mild assumptions a Noetherianscheme admits a Macaulayfication (all local rings have maximal depth). In general,Cohen-Macaulay schemes are far from being non-singular, but they do share manygood homological properties with non-singular schemes.

A scheme which does not admit a resolution of singularities is given by the exampleof Nagata (Example 3, Appendix [63]) of a one dimensional noetherian domain Rwhose integral closure R is not a finite R module. We will show that such a ringcannot have a resolution of singularities. Suppose that there exists a resolution ofsingularities φ : Y → spec(R). That is, Y is a regular scheme and φ is proper andbirational. Let L be the quotient field of R. Y is a normal scheme by Theorem 2.11.Thus Y has an open cover by affine open sets U1, . . . , Us such that Ui ∼= spec(Ti), withTi = R[gi1, . . . , git] where gij ∈ L, and each Ti is integrally closed. The integrallyclosed subring

T = Γ(Y,OY ) = ∩si=1Ti

of L is a finite R module since φ is proper (Theorem III, 3.2.1 [42] or Corollary II.5.20[45] if φ is assumed to be projective). Thus T = R which is impossible since R is nota finite R module.

Exercises1. LetK = Zp(t) where p is a prime, t is an indeterminate. LetR = K[x, y]/(xp+yp − t), X = spec(R). Prove that X is non-singular, but there are no pointsof X which are smooth over K.

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

2.5. Local uniformization and generalized resolution problems. Suppose thatf is irreducible in K[[x, y]]. Then the Puiseux series (5) of Section 2.1 determines aninclusion

R = K[[x, y]]/(f(x, y))→ K[[t]] (7)


with x = tm, y = p(t) the series of (5). Since the m chosen in (5) of Section 2.1 isthe smallest possible, R and K[[t]] have the same quotient field. (7) is an explicitrealization of the normalization of the curve singularity germ f(x, y) = 0, and thusspec(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 ∈ L by

ν(h(t)) = n ∈ Z

if h(t) = tnu where u is a unit in K[[t]]. This valuation can be understood in R bythe Puiseux series (6).

More generally, we can consider an algebraic function field L over a field K. Sup-pose that V is a valuation ring of L containing K. The problem of local uniformizationis to find a regular local ring, essentially of finite type over K and with quotient fieldL such that the valuation ring V dominates R (R ⊂ V and the intersection of themaximal ideal of V with R is the maximal ideal of R).

If L has dimension 1 over K then the valuation rings V of L which contain K areprecisely the local rings of points on the unique non-singular projective curve C withfunction field L (c.f. Section I.6 [45]). The Newton method of Section 2.1 can beviewed as a solution to the local uniformization problem for complex analytic curves.

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

Zariski proved local uniformization for two dimensional function fields over analgebraically closed field of characteristic zero in [79]. He was able to patch togetherlocal solutions to prove the existence of a resolution of singularities for algebraicsurfaces over an algebraically closed field of characteristic zero. He later was able toprove local uniformization for algebraic function fields of characteristic zero in [80].This method leads to an extremely difficult patching problem in higher dimensions,which Zariski was able to solve in dimension 3 in [83], but still remains open in higherdimensions. Zariski’s proof of local uniformization can be considered as an extensionof the Newton method to general valuations. In Chapter 8, we present Zariski’s proofof resolution of surface singularities through local uniformization.

Local uniformization has been proven for two dimensional function fields in positivecharacteristic by Abhyankar. Abhyankar has proven resolution of singularities inpositive characteristic for surfaces and for three dimensional varieties [1],[3],[4].

There has recently been a resurgence of interest in local uniformization in positivecharacteristic ([47], [55], [71], [72]).

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

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

X/K ,OX) is a vector field. If p ∈ X is a closed point, and x1, . . . , xn areregular 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.

12 S. DALE CUTKOSKY

There is an effective divisor FD and an ideal sheaf JD such that V (JD) has codi-mension ≥ 2 in X and ID = OX(−FD)JD. The goal of resolution of vector fields is tofind a proper morphism π : Y → X so that if D′ = π−1(D), then the ideal ID′ ⊂ OYis a simple as possible.

This was accomplished by Seidenberg for vector fields on non-singular surfaces overan algebraically closed field of characteristic zero in [69]. The best statement thatcan be attained is that the order of JD′ at p (Definition 10.17), νp(JD′) ≤ 1 for allp ∈ Y . The basic invariant considered in the proof is the order νp(JD). While this isan upper semi-continuous function on Y , it can go up under a monoidal transform.However, we have that νq(JD′) ≤ νp(JD) + 1 if π : Y → X is the blow up of p, andq ∈ π−1(p). This should be compared with the classical resolution problem, wherea basic result is that order cannot go up under a permissible monoidal transform(Lemma 6.4). Resolution of vector fields for smooth surfaces over a perfect field hasbeen proven by Cano [17]. Cano has also proven a local theorem for resolution forvector fields over fields of characteristic zero [18], which implies local resolution (alonga valuation) of a vector field. The statement is that we can achieve νq(JD′) ≤ 1 (atleast locally along a valuation) after a morphism π : Y → X. There is an analogousproblem for differential forms. A recent paper on this is [19].

We can also consider resolution problems for morphisms f : Y → X of varieties.The natural question to ask is if it is possible to perform monoidal transforms (blowups of non-singular subvarieties) over X and Y to produce a morphism which is amonomial mapping.Definition 2.13. Suppose that Φ : X → Y is a dominant morphism of non-singularirreducible K-varieties (where K is a field of characteristic zero). Φ is monomialif 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) ofnon-negative integers (which necessarily has rank m) such that

y1 = xa111 · · ·xa1n

n...

ym = xam11 · · ·xamnn

Definition 2.14. Suppose that Φ : X → Y is a dominant morphism of integral K-varieties. A morphism Ψ : X1 → Y1 is a monomialization of Φ [25] if there aresequences of blow ups of non-singular subvarieties α : X1 → X and β : Y1 → Y , anda morphism Ψ : X1 → Y1 such that the diagram

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 to a surface

(over an algebraically closed field of characteristic 0) then there is a monomializationof Φ [25]. A generalized multiplicity is defined in this paper which can go up, causinga very high complexity in the proof. An extension of this result to strongly preparedmorphisms from n-folds to surfaces is proven in [31]. It is not known if monomializa-tion is true even for birational morphisms of varieties of dimension ≥ 3, although it


is true locally along a valuation, from the following Theorem 2.15. Theorem 2.15 isproven when the quotient field of S is finite over the quotient field of R in [23]. Theproof for general field extensions is in [26].

Theorem 2.15. (Theorem 1.1 [24], Theorem 1.1 [26]) Suppose that R ⊂ S are regularlocal rings, essentially of finite type over a field K of characteristic zero.

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

x1 = ya111 .....ya1n

n δ1...

xm = yam11 .....yamnn δm.

(8)

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

The standard theorems on resolution of singularities allow one to easily find R′

and S′ such that (8) holds, but, in general, we will not have the essential conditionrank(aij) = m. The difficulty of the problem is to achieve this condition.

This result gives very simple structure theorems for the ramification of valuationsin characteristic zero function fields [33]. We discuss some of these results in Chapter9. A generalization of monomialization in characteristic p function fields of algebraicsurfaces is obtained in [32] and especially in [33].

We point out that while it seems possible that Theorem 2.15 does hold in positivecharacteristic, there are simple examples in positive characteristic where a monomi-alization does not exist. The simplest example is the map of 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 properness (Hironaka, Chapter0, Section 6 of [48] and Chapter 8 of [24]).

The construction of a monomialization by quasi-complete varieties follows fromTheorem 2.15. Theorem 2.16 is proven for generically finite morphisms in [24] andfor arbitrary morphisms in Theorem [26].

Theorem 2.16. (Theorem 1.2 [24] ,Theorem 1.2 [26]) Let K be a field of character-istic zero, Φ : X → Y a dominant morphism of proper K-varieties. Then there arebirational morphisms of non-singular quasi-complete K-varieties α : X1 → X andβ : 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 subvarieties.That is, for every z ∈ X1, there exist affine neighborhoods V1 of z, V of x = α(z),such that α : V1 → V is a finite product of monoidal transforms, and there exist affine

14 S. DALE CUTKOSKY

neighborhoods W1 of Ψ(z), W of y = β(Ψ(z)), such that β : W1 → W is a finiteproduct of monoidal transforms.

A monoidal transform of a non-singular K-scheme S is the map T → S induced byan open subset T of proj(⊕In), where I is the ideal sheaf of a non-singular subvarietyof S.

The proof of Theorem 2.16 follows from Theorem 2.15, by patching a finite numberof local solutions. The resulting schemes may not be separated.

It is an extremely interesting question to determine if a monomialization exists forall morphisms of varieties (over a field of characteristic zero). That is, the conclusionsof Theorem 2.16 hold, but with the stronger conditions that α and β are products ofmonoidal transforms on proper varieties X1 and Y1.

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

setsU2 − 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 we get amorphism

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

where p denotes the origin of U0. Upon localization of the above maps, we getisomorphisms

K[x, y]y ∼= K[s, t]t and K[x, y]x ∼= K[u, v]u.Thus π : U1−V (t) ∼= U0−V (y), π : U2−V (u) ∼= U0−V (x), and π is an isomorphismover U0 − V (x, y) = U0 − p.

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).

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 thatB(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 associated ideal mq isa maximal ideal of R1 = K[xy , y] which contains (x, y)R1 = yR1. Thus mq = (y, xy−α)for some α ∈ K. If we set y1 = y, x1 = x

y−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 parameters(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. Thusthe points in π−1(p) can be expressed (uniquely) in one of the forms

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 . Sayyx ∈ 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).

More generally, suppose that S = spec(R) is an affine surface over a field L, andp ∈ S is a non-singular closed point. After possibly replacing S with an open subsetspec(Rf ), we may assume that the maximal ideal of p in R is mp = (x, y). We canthen define the blow up of p in S by

π : B(p) = proj(⊕n≥0

mnp )→ S. (9)

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 open neigh-borhood of p in S such that

(x, y) = Γ(U,mp) ⊂ R

16 S. DALE CUTKOSKY

is the maximal ideal of p in R. Then the map π : π−1(U) → U is defined by theconstruction (9).

Suppose that C ⊂ A2K is a curve. Since K[x, y] is an integral 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 setνq(C) = max r | f ∈ mr

q(more generally, see Definition 10.17).Lemma 3.2. q ∈ A2

K 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 ∈ m2

q, and mqT/m2qT∼= mq/m

2q 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 of generality,we may suppose that β 6= 0. Thus

y ≡ −αβx mod m2

q + (f)

and x is a K generator of mqT/m2qT∼= mq/m

2q+(f). We see that dimK mqT/m

2qT = 1

and q is non-singular on C.

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

2q, and since R is

a UFD, there is f ∈ R such that f = 0 is a local equation of C at q.Let p be the origin in A2

K , 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 closureof π−1(C − p) ∼= C − p in B(p). Let E = π−1(p) be the exceptional divisor. Settheoretically, 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]) ofB(p) where

x = x1, y = x1y1

x1 = 0 is a local equation for E.f = xr1f1

wheref1 =

∑i+j≥r

aijxi+j−r1 yj1.

x1 6 | f1 since νp(C) = r, so that f1 = 0 is a local equation of the strict transform Cof 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

where

f1 =∑i+j≥r

aijxi1yi+j−r1

is a local equation of C in U1.As a scheme, we see that π−1(C) = C + rE. The scheme theoretic preimage of C

is called the total transform of C. Define the leading form of f to be

L =∑i+j=r

aijxiyj

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

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

In the first case we have 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

and

L = a0r(y − αx)r.

In the second case we have 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

and

L = ar0xr.

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

18 S. DALE CUTKOSKY

3.2. Completion. Suppose that A is a local ring with maximal ideal m. A coefficientfield of A 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 local ring containsa coefficient field (Theorem 27, Section 12, Chapter VIII [85]). This leads to Cohen’sstructure theorem (Corollary, Section 12, Chapter VIII [85]), which shows that anequicharacteristic complete regular local ring A is isomorphic to a formal power seriesring over a field. In fact, if L is a coefficient field of A, and if (x1, . . . , xn) is a regularsystem of parameters of A, then A is the power series ring

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

We further remark that the completion of a local ring R is a regular local ring if andonly if R is regular (c.f. Section 11, Chapter VIII [85]).Lemma 3.4. Suppose that S is a non-singular algebraic surface defined over a fieldK, p ∈ S is a closed point, π : B = B(p) → S is the blow up of p, and suppose thatq ∈ π−1(p) is a closed point such that K(q) is separable over K(p). Let R1 = OS,pand R2 = OB,q, and suppose that K1 is a coefficient field of R1, (x, y) are regularparameters in R1. Then there exists a coefficient field K2 = K1(α) of R2 and regularparameters (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

aijxi1yi+j1

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

OS → OS,p → R1

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[ yx ]. Since K(q) isseparable over K1

∼= K(p), f is separable. mq = (x, f( yx )) ⊂ R1[ yx ].Let α ∈ R1[ yx ]/mq be the class of y

x . We have the residue map φ : R2 → L whereL = R1[ yx ]/mq.

There is a natural embedding of K1 in R2. We have a factorization of f(t) =(t − α)γ(t) in L[t] where t − α and γ(t) are relatively prime. By Hensel’s Lemma(Theorem 17, Section 7, Chapter VIII [85]) there is α ∈ R2 such that φ(α) = α andf(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, yx − α). Thus R2 = K2[[x, yx − α]]. Setx1 = 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 contains a field,there may not be a coefficient field of the completion of the ring containing that field.Thus the above lemma does not extend to non-perfect fields.

Let K = Zp(t) where t is an indeterminate. Let R = K[x](xp−t). Suppose that Rhas a coefficient field L containing K. Let φ : R → R/m be the residue map. Sinceφ | L is an isomorphism, there exists λ ∈ L such that λp = 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 3.5. Suppose that R is a reduced affine ring over a field K, and A = Rpwhere p is a prime ideal of R. Then the completion A = Rp of A with respect to itsmaximal ideal is reduced.

When K is a perfect field, this is a theorem of Chevalley (Theorem 31, Section 13,Chapter VIII [85]). The general case follows from Scholie IV 7.8.3 (vii) [43].

However, the property of being a domain is not preserved under completion. Asimple example is f = y2− x2 + x3. f is irreducible in C[x, y], but is reducible in thecompletion C[[x, y]].

f = y2 − x2 − x3 = (y − x√

1 + x)(y + x√

1 + x).

The first part of the expansions of the two factors are

y − x− 12x2 + · · ·

and

y + x+12x2 + · · · .

Lemma 3.6. (Weierstrass Preparation Theorem) Let K be a field, and suppose thatf ∈ K[[x1, . . . , xn, y]] is such that

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

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

f = u(yr + a1yr−1 + · · ·+ ar).

A proof is given in Theorem 5, Section 1, Chapter VII [85].A concept which will be important in this book is the Tschirnhausen transforma-

tion, which generalizes the ancient notion of “completion of the square” in the solutionof quadratic equations.

20 S. DALE CUTKOSKY

Definition 3.7. Suppose that K is a field of characteristic p ≥ 0 and f ∈ K[[x1, . . . , xn, y]]has an expression

f = yr + a1yr−1 + · · ·+ ar

with ai ∈ K[[x1, . . . , xn]] and p = 0 or p 6 | r. The Tschirnhausen transformation off 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

Suppose that (R,m) is a complete local ring and L1 ⊂ R is a field such that R/mis finite and separable over L1. Use Hensel’s Lemma (Theorem 17, Section 7, ChapterVIII [85]) to prove that there exists a coefficient field L2 of R containing L1.

3.3. Blowing up a point on a non-singular surface. We define the strict trans-form, the total transform and νp(C) analogously to the definition of Section 3.1 (Moregenerally, see Section 4.1 and Definition 10.17).Lemma 3.8. Suppose that X is a non-singular surface over an algebraically closedfield K, and C is a curve on X. Suppose that p ∈ X and νp(C) = r. Let π : B(p)→ X

be the blow up of p, C be the strict transform of C, and suppose that q ∈ π−1(p). Thenνq(C) ≤ r, and if r > 0, there is at most 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 regular parameters inOX,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 OX,p 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),p = 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 =∑

aijxi+j−r1 (y1 + α)j + x1Ω

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 be the strict transform of C,

E be the exceptional divisor E = π−1(p).On U1 = spec(K[xy , y]) ⊂ B(p) we have coordinates x1, y1 with x = x1y1, y = y1.

A local equation for E on U1 is y1 = 0.

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.

y2 − x3 = x21(y2

1 − 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.

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 Section 2.4.As a further example, consider the curve C with equation y2−x5 in A2. The only

singular point of C is the origin p. Let π : B(p)→ A2 be the blow up of p, E be theexceptional divisor, C be the strict transform of C. There is only one singular pointq on C. Regular parameters at q are (x1, y1) where x = x1, y = x1y1. C has the localequation

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 up the new singularpoints, and repeat as long as the curve is singular.

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

3.4. Resolution of curves embedded in a non-singular surface I. In this sec-tion we consider curves embedded in a non-singular surface, over an algebraicallyclosed field K of characteristic zero, and prove that their singularities can be resolved.

The theorem is proven by passing to the completion of the local ring of the surfaceat a singular point of the curve. This allows us to view a local equation of the curveas a powerseries in two variables. Our main invariant is the multiplicity of the curveat a point. We know that this multiplicity can not go up after blowing up, so we mustshow that it will eventually go down, after enough blowing up.

22 S. DALE CUTKOSKY

Theorem 3.9. Suppose that C is a curve on a non-singular surface X over analgebraically closed field K of characteristic zero. Then there exists a sequence ofblow ups of points λ : Y → X such that the strict transform C of C on Y is non-singular.

Proof. Let r = max(νp(C) | p ∈ C. If r = 1, C is non-singular, so we may assumethat r > 1. The set p ∈ C | ν(p) = r is a subset of the singular locus of C, whichis 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

· · · → Xnπn→ · · · π1→ X0 = X (10)

where each πn+1 : Xn+1 → Xn is the blow up of all points on the strict transform Cnof C which have multiplicity r on Cn. If this sequence is finite, then there is an integern such that all points on the strict transform Cn of C have multiplicity ≤ r − 1. Byinduction on r, we can repeat this process then to construct the desired morphismY → X which induces a resolution of C.

We will assume that the sequence (10) is infinite, and derive a contradiction. If itis infinite, then for all n ∈ N there are closed points pn ∈ Cn which have multiplicityr on Cn and such that πn+1(pn+1) = pn for all n. Let Rn = OXn,pn for all n. Wethen have an infinite sequence of completions of quadratic transforms of local rings

R0 → R1 → · · · → Rn → · · ·

Suppose that (x, y) are regular parameters in R0 = OX,p, and f = 0 is a localequation of C in R0 = K[[x, y]]. f is reduced by Theorem 3.5. After a linear change ofvariables in (x, y), we may assume that ν(f(0, y)) = r. By the Weierstrass preparationtheorem,

f = u(yr + a1(x)yr−1 + · · ·+ ar(x))

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

y = y +a1(x)r

(11)

Then

yr + a1(x)yr−1 + · · ·+ ar(x) = yr + b2(x)yr−2 + · · ·+ br(x). (12)

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

f = yr + a2(x)yr−2 + · · ·+ ar(x). (13)

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

n = min

mult(ai(x))i

(14)

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 = yr1f1

where

f1 = 1 +a2(x1y1)

y21

+ · · ·+ br(x1y1)yr1

is a local equation of the strict transform C1 of C at p1. f1 is a unit, so that νp1(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)xr1

.

f1 = 0 is a local equation at p1 for C1. If α 6= 0, then νp1(C1) ≤ r − 1, so this casecannot occur.

If α = 0, and νp1(C1) = r, then f1 has the form of (13), with n decreased by 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 (10) has infinite length.

The Transformation of (11) is called a Tschirnhausen transformation (Definition3.7). The Tschirnhausen transformation finds a (formal) curve of maximal contactH = V (y) for C at p. The Tschirnhausen transformation was introduced as a funda-mental method in resolution of singularities by Abhyankar.Definition 3.10. Suppose that C is a curve on a non-singular surface S over a fieldK, and p ∈ C. A non-singular curve H = V (g) ⊂ spec(OS,p) is a formal curve ofmaximal 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 transform Ci of C inSi contains pi with νpi(Ci) = νp(C) for i ≤ n, then the strict transform Hi of H inSi contains pi.

Exercises1. 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, ri ∈ N such that ID = Ir1C1· · · IrnCn . D has

simple normal crossings (SNCs) on S if for every p ∈ S there exist regularparameters (x, y) in OS,p such that if f = 0 is a local equation for D at p,then f = unit xayb in OS,p.

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

24 S. DALE CUTKOSKY

3. Suppose that D is an effective divisor on a non-singular surface S. Show thatthere 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-singular surface II. In this sec-tion we consider curves embedded in a non-singular surface, defined over an arbitraryfield K.Lemma 3.11. Suppose that K is a field, S is a non-singular surface over K, C is acurve on S and p ∈ C is a closed point. Suppose that π : B = B(p) → S is the blowup of p, C is the strict transform of C on B and q ∈ 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 a naturalembedding of π−1(p) in B ×S spec(R),

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[ yx ] be the ideal of q. Since R[ yx ]/(x, y) ∼= K1[ yx ], thereexists 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, r = νp(C). We have anexpansion in R,

f =∑i+j≥r

aijxiyj

with aij ∈ K1. f = xrf1 where f1 = 0 is a local equation of the strict transform C ofC in spec(R[ yx ]).

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 ].

deg(∑i+j=r

aij(y

x)j) ≤ r

in K1[ yx ] implies n ≤ r and νq(C) = r implies deg(h( yx )) = 1 so that h = yx − α with

α ∈ K1 and (by Lemma 3.4) there exist regular parameters (x1, y1) in mq ⊂ OB(p),q

such thatOS,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 3.12. Suppose that C is a curve which is a subvariety of a non-singularsurface X over a field K. Then there exists a sequence of blow ups of points λ : Y → Xsuch that the strict transform C of C on Y is non-singular.

Proof. Let r = max(νp(C) | p ∈ C. If r = 1, C is non-singular, so we may assumethat r > 1. The set p ∈ C | νp(C) = r is a subset of the singular local of C whichis 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

· · · → Xnπn→ · · · → X1

π1→ X0 = X (15)

where each πn+1 : Xn+1 → Xn is the blow up of all points on the strict transform Cnof C which have multiplicity r on Cn. If this sequence is finite, then there is an integern such that all points on the strict transform Cn of C have multiplicity ≤ r − 1. Byinduction on r, we can then repeat this process to construct the desired morphismY → X which induces a resolution of C.

We will assume that the sequence (15) is infinite, and derive a contradiction. If itis infinite, then for all n ∈ N there are closed points pn ∈ Cn which have multiplicityr on Cn and such that πn+1(pn+1) = pn. Let Rn = OXn,pn for all n. We then have aninfinite sequence of completions of quadratic transforms (blow ups of maximal ideals)of local rings

R = R0 → R1 → · · · → Rn → · · ·

We will define

δpi ∈1r!

N

such that

δpi = δpi−1 − 1 (16)

for all i ≥ 1. We can thus conclude that pi has multiplicity < r on the strict transformCi of C for some natural number i ≤ δp + 1. From this contradiction it will followthat (15) is a sequence of finite length.

Suppose that f ∈ R0 = OX,p is such that f = 0 is a local equation C and (x, y) areregular 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 field of R. There is an expansion

f =∑i+j≥r

aijxiyj

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

δ(f ;x, y) = min

i

r − j| j < r and aij 6= 0

. (17)

26 S. DALE CUTKOSKY

δ(f ;x, y) ≥ 1 since (x, y) are good parameters We thus have an expression (withδ = δ(f ;x, y))

f =∑

i+jδ≥rδ

aijxiyj = Lδ +

∑i+jδ>rδ

aijxiyj

where

Lδ =∑

i+jδ=rδ

aijxiyj = a0ry

r + Λ (18)

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

δp = supδ(f ;x, y1) | y = y1 +n∑i=1

bixi with n ∈ N and bi ∈ K ′ ∈

1r!

N ∪ ∞.(19)

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 such that

f = γyr1.

But then r = 1 since f is reduced in R, a contradiction. We see then that δp ∈ 1r!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 3.11, 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 of C1 (inspec(R1)) is f1 = 0, where

f1 =f

yr1=∑i+j=r

aijxi1 + y1Ω = a0r + x1g + y1h

for some series 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

aij(y1 + α)j + x1Ω

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 (18), we see that r = rδ and δ = 1, so that Lδ = L. But we canreplace y with y − αx to increase δ, a contradiction to the maximality of δ. Thusνp1(C1) < r, a contradiction.

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

is

f1 =f

xr1=

∑i+jδ≥rδ

aijxi+j−r1 yj1 =

∑i+jδ′=rδ′

ai−j+r,jxi1yj1 +

∑i+jδ′>rδ′

ai−j+r,jxi1yj1

where i = i+ j − r. Since

Lδ′(x1, y1) =1xr1Lδ(x1, x1y1)

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

substitution

y1 = y′1 −∑

bixi1 = y′1 − bxd1 + 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 − bxd1)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 − bxd1)j = a0r(y′1)r.

Thus ∑i+jδ=rδ

aijxi+j−r1 yj1 =

∑i+jδ′=rδ′

ai−j+r,jxi1yj1 = 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,

28 S. DALE CUTKOSKY

a contradiction, from which we conclude that δp1 = δ′ = δp − 1. We can theninductively define δpi for i ≥ 0 by this procedure so that (16) holds. The conclusionsof the Theorem now follow.

Remark 3.13. 1. This proof is a generalization of the algorithm of Section 2.1.A more general version of this, valid in arbitrary two dimensional regular localrings, can be found in [5] or [67].

2. Good parameters (x, y) for f which achieve δp = δ(f ;x, y) are such that y = 0is a (formal) curve of maximal contact for C at p (Definition 3.10).

Exercises

1. Prove that the δp defined in formula (19) 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 isan invariant of p.

2. Suppose that νp(C) > 1. Let π : B(p) → X be the blow up of p, C be thestrict transform of C. Show that there is at most one point q ∈ π−1(p) suchthat νq(C) = νp(C).

3. The Newton Polygon N(f ;x, y) is defined as follows. Let I be the ideal inR = K[[x, y]] generated by the monomials xαyβ such that the coefficient aαβof 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 such thatN(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 forf (so that ν(f(0, y)) = ν(f) = r). Then (0, r) ∈ N(f ;x, y). Let the slope ofthe steepest segment of N(f ;x, y) be s(f ;x, y). We have 0 ≥ s(f ;x, y) ≥ −1,since aαβ = 0 if α+ β < r. Show that

δ(f ;x, y) = − 1s(f ;x, y)

.

4. Resolution type theorems

4.1. Blow ups of ideals. Suppose that X is a variety, and J ⊂ OX is an ideal sheaf.The blow up of J 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 is anisomorphism over X − V (J ), and JOB is a locally principal ideal sheaf.

If U ⊂ X is an open affine subset, andR = 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) = ∪mi=1spec(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 ).Let U = X−V (J ). The strict transform W of W is the Zariski closure of π−1(W ∩

U) in B(J ).Lemma 4.1. Suppose that R is a Noetherian ring, I, J ⊂ R are ideals and I =q1 ∩ · · · ∩ qm is a primary decomposition, with primes pi =

√qi. Then

∪∞n=0(I : Jn) = ∩i|J 6⊂piqi.The proof of Lemma 4.1 follows easily from the definition of a primary ideal.

Geometrically, Lemma 4.1 says that ∪∞n=0(I : Jn) removes the primary componentsqi 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 [45].Theorem 4.2. (Universal property of blowing up) Suppose that I is an ideal sheafon a variety V and f : W → V is a morphism of varieties such that IOW is locallyprincipal. Then there is a unique morphism g : W → B(I) such that f = π g.

This is proven in Proposition II.7.14 [45].Theorem 4.3. Suppose that C is an integral curve over a field K. Consider thesequence

· · · → Cnπn→ · · · → C1

π1→ C (20)

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

Proof. Without loss of generality, C = spec(R) is affine. Let R be the integral closureof R in the function field K(C) of C. R is a regular ring. Let C = spec(R). All idealsin R are locally principal (Proposition 9.2, page 94 [12]). By Theorem 4.2, we have afactorization

C → Cn → · · · → C1 → C

for all n. Since C → C is finite, Ci+1 → Ci is finite for all i, and there exist affinerings Ri such that Ci = spec(Ri). For all n we have sequences of inclusions

R→ R1 → · · · → Rn → R.

30 S. DALE CUTKOSKY

Ri 6= Ri+1 for all i since a maximal ideal m in Ri is locally principal if and only if(Ri)m is a regular local ring. Since R is finite over R, we have that (20) is finite.

Corollary 4.4. Suppose that X is a variety over a field K and C is an integral curveon 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 strict transformCi of C on Xi. Then this sequence is finite. That is, there exists an n such that thestrict 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 Theorem 4.3.

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

This is proved in Proposition II.7.17 [45].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 exceptional divisor of themonoidal transform π (with center Y ) to be the reduced divisor with the same supportas π∗(Y ).

In general, we will define the exceptional locus of a proper birational morpismφ : V →W to be the (reduced) closed subset of V on which φ is not an isomorphism.

Remark 4.6. If Y has codimension greater than or equal to 2 in X, then the ex-ceptional divisor of the monoidal transform with center Y , π : B(IY ) → X is theexceptional locus of π. However, if X is singular and not locally factorial, it may bepossible to blow up a non-singular codimension 1 subvariety Y of X to get a monoidaltransform π such that the exceptional locus of π is a proper closed subset of the ex-ceptional divisor of π. For an example of this kind, see the exercises at the end ofthis section. If X is non-singular and Y has codimension 1 in X, then π is an iso-morphism and we have defined the exceptional divisor to be Y . This property will beuseful in our general proof of resolution in Chapter 6.

Exercises

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 transform centeredat the non-singular curve Y . Show that π is a resolution of singularities.Compute the exceptional locus of π and compute the exceptional divisor ofthe 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 is a resolutionof singularities in each of these cases, and describe the exceptional locus andexceptional divisor (of the monoidal transform).

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 the monoidaltransform of case c.

4.2. Resolution type theorems and corollaries. Resolution of singularities.Suppose that V is a variety. A resolution of singularities of V is a proper birationalmorphism φ : W → V such that W is non-singular.

Principalization of ideals Suppose that V is a non-singular variety, I ⊂ OV is anideal sheaf. A principalization of I is a proper birational morphism φ : W → V suchthat W is non-singular and IOW is locally principal.

Suppose that X is a non-singular variety, I ⊂ OX is a locally principal ideal.I has Simple Normal Crossings (SNCs) at p ∈ X if there exist regular parame-ters (x1, . . . , xn) in OX,p such that Ip = xa1

1 · · ·xann OX,p for some natural numbersa1, . . . , an.

Suppose that D is an effective divisor on X. That is, D = r1E1 + · · ·+rnEn whereEi are irreducible codimension 1 subvarities of X, and ri are natural numbers. D hasSNCs if ID = Ir1E1

· · · IrnEn has SNCs.Suppose that W ⊂ X is a subscheme, and D is a divisor on X. Say that W has

SNCs with D at p if1. 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 such thatIW,p = (x1, . . . , xr) (or IW,p = OX,p) and ID,p = xa1

1 · · ·xann OX,p for someai ∈ N.

Embedded resolution. Suppose that W is a subvariety of a non-singular variety X.An embedded resolution of W is a proper morphism π : Y → X which is a product ofmonoidal transforms, such that π is an isomorphism on an open set which intersectsevery component of W properly, the exceptional divisor of π is a SNC divisor D, andthe strict transform W of W has SNCs with D.

Resolution of indeterminacy. Suppose that φ : W → V is a rational map ofproper K-varieties such that W is non-singular. A resolution of indeterminacy of φis a proper non-singular K-variety X with a birational morphism ψ : X → W and amorphism λ : X → V such that λ = φ ψ.Lemma 4.7. Suppose that resolution of singularities is true for K-varieties of dimen-sion n. Then resolution of indeterminacy is true for rational maps from K-varietiesof dimension n.

32 S. DALE CUTKOSKY

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

Theorem 4.8. Suppose that K is a perfect field, resolution of singularities is true forprojective hypersurfaces over K of dimension n and principalization of ideals is truefor non-singular varieties of dimension n over K. Then resolution of singularities istrue for projective K-varieties of dimension n.

Proof. Suppose that V is an n dimensional projective K-variety. If V1, . . . , Vr arethe irreducible components of V , we have a projective birational morphism fromthe disjoint union of the Vi to V . Thus it suffices to assume that V is irreducible.The function field K(V ) of V is a finite separable extension of a rational functionfield K(x1, . . . , xn) (Chapter II, Theorem 30, page 104 [85]). By the theorem of theprimitive element,

K(V ) ∼= K(x1, . . . , xn)[xn+1]/(f).

We can clear the denominator of f so that

f =∑

ai1...in+1xi11 · · ·x

in+1n+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+1n+1 ,

the homogenization of f . Let V be the variety defined by F = 0 in Pn+1. K(V ) ∼=K(V ) implies there is a birational rational map from V to V . That is, there is abirational morphism φ : V → V where V is a dense open subset of V . Let Γφ bethe Zariski closure of (a, φ(a)) | a ∈ V in V × V . We have birational projectionmorphisms π1 : Γφ → V and π2 : Γφ → V . Γφ is the blow up of an ideal sheaf Jon V (Theorem 4.5). By resolution of singularities for n dimensional hypersurfaces,we have a resolution of singularities f : W ′ → V . By principalization of ideals innon-singular varieties of dimension n, we have a principalization g : W → W ′ forJOW ′ . By the universal property of blowing up (Theorem 4.2), we have a morphismh : W → Γφ. Hence π1 h : W → V is a resolution of singularities.

Corollary 4.9. Suppose that C is a projective curve over a perfect field K. Then Chas a resolution of singularities.

Proof. By Theorem 3.12 resolution of singularities is true for projective plane curvesover K. All ideal sheaves on a non-singular curve are locally principal since the localrings of points are Dedekind local rings.

Note that Theorem 4.3 and Corollary 4.4 are stronger results than Corollary 4.9.However, the ideas of the proofs of resolution of plane curves in Theorems 3.9 and3.12 extend to higher dimensions, while Theorem 4.3 does not. As a consequence ofembedded resolution of curve singularities on a surface, we will prove principalization


of ideals in non-singular varieties of dimension two. This simple proof is by Abhyankar(Proposition 6, [67]).Theorem 4.10. Suppose that S is a non-singular surface over a field K and J ⊂ OSis an ideal sheaf. Then there exists a finite sequence of blow ups of points T → S suchthat JOT is locally principal.

Proof. Without loss of generality S is affine. Let J = Γ(J ,OS) = (f1, . . . , fm). Byembedded resolution of curve singularities on a surface (Exercise 3 of Section 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 everywhere on S1.Let p1, . . . , pr be the finitely many points on S1 where JOS1 is not locally

principal (they are contained in the singular points of V (√JOS1). Suppose that

p ∈ p1, . . . , pr. By induction on the number of generators of J , we may assumethat JOS1,p = (f, g). After possibly multiplying f and g by units in OS1,p, there areregular 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 our assumption,tp < 0. Let π2 : S2 → S1 be the blow up of p, and suppose that q ∈ π−1(p). We willshow that tq > tp.

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

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

If x = x1, y = x1y1, thenf = xa+b

1 yb1, g = xc+d1 yd1 .

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.

5. Surface singularities

5.1. Resolution of surface singularities. In this section, we give a simple proofof resolution of surface singularities in characteristic zero.Theorem 5.1. Suppose that S is a projective surface over an algebraically closed fieldK of characteristic 0. Then there exists a resolution of singularities

Λ : T → S.

Theorem 5.1 is a consequence of the following Theorem 5.2, Theorem 4.8 andTheorem 4.10.Theorem 5.2. Suppose that S is a hypersurface of dimension 2 in a non-singularvariety V of dimension 3, over an algebraically closed field K of characteristic 0.Then there exists a sequence of blow ups of points and non-singular curves containedin the strict transform Si of S

Vn → Vn−1 → · · · → V1 → V

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

34 S. DALE CUTKOSKY

The remainder of this section will be devoted to the proof of Theorem 5.2. Supposethat V is a non-singular three dimensional variety over an algebraically closed fieldK of characteristic 0, and S ⊂ V is a surface.

For a natural number t, define

Singt(S) = p ∈ V | νp(S) ≥ t.By Theorem 10.19, Singt(S) is Zariski closed in V .

Letr = maxt | Singt(S) 6= ∅

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

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

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 p in V such thatmp = (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]).

π−1(p) ∼= P1, π−1(C ∩ U) ∼= (C ∩ U) × P1. Let E = π−1(C) be the exceptionaldivisor. E is a projective bundle over C. At each point q ∈ π−1(p), we have regularparameters (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 , by passing to a formalneighborhood spec(OV,p) of p and analyzing the map π : B(IW,p)→ spec(OV,p). We


have a natural isomorphism B(IW,p) ∼= B(IW )×V spec(OV,p). Observe that we havea natural identification of π−1(p) with π−1(p).

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

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

The maximal ideal mpOV is always algebraic. However, the ideal sheaf I of anon-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, is the idealsheaf of the irreducible curve

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

which we studied after Theorem 3.5. In R = K[[x, y, z]] we have regular parameters

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 formal branchesx = 0, z = 0 or y = 0, z = 0, but the resulting blown up scheme will not extend to ablow 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-singlar three dimensional variety, p ∈ V isa closed point and π : V1 = B(p) → V is the blow up of p with exceptional divisorE = π−1(p) ∼= P2. Let R = OV,p. We have a commutative diagram of morphisms ofschemes

B = B(mpR) → V1 = B(p)π ↓ ↓ πspec(R) → V

such that π−1(mp)→ π−1(p) = E is an isomorphism of schemes. Suppose that I ⊂ Ris any ideal, and I ⊂ OB is the strict transform of I. Then there exists an ideal sheafJ on V1 such that JOB = IEOB + I.

Proof. IOE is an ideal sheaf on E, so there exists an ideal sheaf J ⊂ OV1 such thatIE ⊂ 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 isa 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). Suppose that p ∈ C. Then νq(S) ≤ r forall q ∈ π−1(p), and there exists at most one point q ∈ π−1(p) such that νq(S) = r.In particular, if E = π−1(C), then either Singr(S)∩E is a non-singular curve whichmaps isomorphically onto C or Singr(S) ∩ E is a finite union of points.

36 S. DALE CUTKOSKY

Proof. By the Weierstrass preparation theorem and after a Tschirnhausen transfor-mation (Definition 3.7), a local equation f = 0 of S in OS,p = K[[x, y, z]] has theform

f = zr + a2(x, y)zr−2 + · · ·+ ar(x, y). (21)

f ∈ (I(r)C,p)= IrC,p implies ∂f

∂z ∈ Ir−1C,p , and r!z = ∂r−1f

∂zr−1 ∈ IC,p. Thus z ∈ IC,p. Aftera change of variables in x and y, we may assume that IC,p = (x, z). f ∈ IrC,p impliesxi | ai for all i. If q ∈ π−1(p), then OB(C),q has regular parameters (x1, y, z1) suchthat

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 + · · ·+ arxr1.

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 the blow upof p, S is the strict transform of S in B(p), and E = π−1(p). Then νq(S) ≤ r forall q ∈ π−1(p), and either Singr(S) ∩ E is a non-singular curve or Singr(S) ∩ E is afinite union of points.

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

f = zr + a2(x, y)zr−2 + · · ·+ ar(x, y). (22)

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

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 the line which isthe intersection of E with the strict transform of z = 0 in B(C)×V spec(OV,p).

Definition 5.6. Singr(S) has simple normal crossings (SNCs) if1. All irreducible components of Singr(S) (which could be points or curves) are

non-singular.2. If p is a singular point of Singr(S), then there exist regular parameters (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 a point or anirreducible curve in Singr(S), π : V ′ = B(W )→ V is the blow up of W and S′ is thestrict transform of S on V ′. Then Singr(S′) has simple normal crossings.

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


Definition 5.8. A closed point p ∈ S is a pregood point if in a neighborhood ofp, Singr(S) is either empty, a non-singular curve through p, or a a union of twonon-singular curves intersecting transversally at p (satisfying 2. of Definition 5.6).

Definition 5.9. p ∈ S is a good point if p is pregood, and if 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 transform ofS ∩ spec(OV,p) on Xi, then q is pregood for all closed points q ∈ Singr(Sn). Inparticular, 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 there exists asequence of blow ups

V ′ = Vn → · · · → V1 → V

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

Proof. Suppose that C is a non-singular curve in Singr(S). Let π1 : V1 = B(C)→ Vbe the blow up of C, S1 be the strict transform of S. If Singr(S1) 6= ∅, we can chooseanother non-singular curve C1 in Singr(S1) and blow up by π2 : V2 = B(C1)→ V1. LetS2 be the strict transform of S1. We either reach a surface Sn such that Singr(Sn) = ∅,or we obtain an infinite 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), where Si is thestrict transform of S on Vi. Each curve Ci which is blown up must map onto a curvein S by Lemma 5.5. Thus there exists a curve γ ⊂ S such that there are infinitelymany blow ups of curves mapping onto γ in the above sequence. Let R = OV,γ , atwo dimensional regular local ring. IS,γ is a height 1 prime ideal in this ring, andP = Iγ,γ is the maximal ideal of R.

dim R+ trdegKR/P = 3

by the dimension formula (Theorem 15.6 [61]). Thus R/P has transcendence degree 1over K. Let t ∈ R be the lift of a transcendence basis of R/P over K. 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 is a maximal ideal in A. Thus R is the local ringof a non-singular point q on the K(t) surface spec(A). q is a point of multiplicity ron the curve in spec(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 are of multi-plicity r on the strict transform of the curve spec(R/IS,γ). But this is impossible byTheorem 3.12.

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

38 S. DALE CUTKOSKY

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,

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 multiplicity r,

embedded in a non-singular surface over a field of transcendence degree 1 over K (asin 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 points over spec(OS,C).So for n >> 0, Sn×Sspec(OS,C) is non-singular. Thus there are no curves in Singr(Sn)which dominate C. For n >> 0 there are thus no curves in Singr(Sn) which dominatecurves of Singr(S), so that Singr(Sn) ∩ (Vn − Bn) is empty for large n, and all badpoints of S are contained in a finite set Tn.

Theorem 5.12. Let

· · · → Vn → Vn−1 → · · · → V1 → V (23)

be the sequence where πn : Vn → Vn−1 is the blow up of all bad points on the stricttransform Sn−1 of S. Then this sequence is finite, so that it terminates after a finitenumber 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 (23). Then there is an infinite

sequence of points pn ∈ Singr(Sn) such that πn(pn) = pn−1 for all n. We then havean 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 transformation(Definition 3.7) there exist regular parameters (x, y, z) in R0 such that there is a localequation 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 regular parameters(xi, yi, zi) in Ri for all i and a local equation fi = 0 for Si such that

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

with αi ∈ K orxi−1 = xiyi, yi−1 = yi, zi−1 = yi−1zi−1


andfi = zri + a2,i(xi, yi)zr−2

i + · · · ar,i(xi, yi)where

aji(xi, yi) =

aj,i−1(xi,xi(yi+αi))

xji

oraj,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. In K[[xi, yi]]we have relations

aj = γj,ixcj,ii y

dj,ii aj,i−1

for all i and 2 ≤ j ≤ r, where γji is a unit. By embedded resolution of curvesingularities (Exercise 3 of Section 3.4), 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

fi = zri + a2i(xi, yi)xa2ii yb2ii zr−2

i + · · ·+ ari(xi, yi)xarii ybrii (24)

where aji are units (or zero) in Ri and aji + bji ≥ j if aji 6= 0.We observe that, by the proof of Theorem 10.19 and Remark 10.21,

ISingr(Si),pi =√J

where J is the ideal in Ri generated by

∂j+k+lfi

∂xji∂yki ∂z

li

| 0 ≤ j + k + l < r.

Thus ISingr(Si),pi must 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 E at pi−1. Without loss of generality, wemay suppose that xi = 0 is a local equation of E. zi = 0 is a local equation ofthe strict transform of z = 0 in Vi ×V spec(R), so that (xi, zi) is the completion atqi of the ideal sheaf of a subscheme of E. Thus there is a curve C in Vi such thatIC,pi = (xi, zi) by Lemma 5.3. The ideal sheaf J = (ISingr(Si) : IC) in OVi is suchthat (J ∩ IC)pi ∼= ISingr(Si),pi , so that (xiyi, zi) is not the completion of an ideal ofan irreducible curve on Vi, and Singr(Si) has simple normal crossings at pi.

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

We now make the observation that we must have bji 6= 0 and aki 6= 0 for some jand k if i ≥ m0. If we did have bji = 0 for all j (with a similar analysis if aji = 0 forall j) in (24) then, since

ISingr(Si),pi =√J

where J is the ideal in Ri generated by

∂j+k+lfi

∂xji∂ykj ∂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

40 S. DALE CUTKOSKY

are local equations of C in Ri. Let λ : W → Vi be the blow up of C. If q ∈ λ−1(pi),we have regular parameters (u, v, y) in OW,q such that

xi = 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 have thatνq(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 (24), with bji = 0 for all j, but

minajij| 2 ≤ j ≤ r

has decreased by 1. We see, using Lemma 5.4, that after a finite number of blow upsof non-singular curves in Singr of the strict transform of S, the multiplicity must be< r at all points above pi. Thus pi is a good point, a contradiction to our assumption.Thus bji 6= 0 and aki 6= 0 in (24) for some j 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 (24), we must have that for 2 ≤ j ≤ r,aj,i+1 = aji + bji − j, bj,i+1 = bji oraj,i+1 = aji, bj,i+1 = aj,i + bji − j

(25)

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 such that aji 6= 0and

1.ajij≤ aki

k,bjij≤ bki

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

2. ajij

+bjij

< 1.

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

ISingr(Si),pi ⊂ (xi, zi)Ri.Since Singr(Si) has SNCs, there exists a non-singular curve C ⊂ Singr(Si) such thatIC,pi = (xi, zi). Let π : V ′ = B(C) → Vi be the blow up of C, S′ be the stricttransform of S on V ′. By Lemma 5.7, Singr(S′) has SNCs. A direct local calculation


shows that there is at most one point q ∈ π−1(pi) such that νq(S′) = r, and if such apoint 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 (24), and conditions 1. and 2. of the statementof this Lemma hold for f ′. After repeating this procedure a finite number of timeswe will construct V → Vi such that νa(S) < r for all points a on the strict transformS of S such that a maps to pi, since aji

j + bjij must drop by 1 every time we blow

up.

Lemma 5.14. Let

δj,k,i =(ajij− 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 (25) holds. Then

δj,k,i+1 = δj,k,i +(bjij− bki

k

)2

.

If δj,k,i < 0 we must have bjij −

bkik 6= 0. Thus(

bjij− bki

k

)2

≥ 1j2k2

≥ 1r4.

Corollary 5.15. There exists m1 such that i ≥ m1 implies condition 1. of Lemma5.13 holds.

Proof. There exists m1 such that i ≥ m1 implies δjki ≥ 0 for all j, k. Then the pairs(akik ,

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

(ajij ,bjij ).

Lemma 5.16. There exists an i′ ≥ m1 such that we have condition 2. of Lemma5.13,

aji′

j+ bji

′

j < 1.

Proof. A calculation shows thatajij

+bjij

≥ 1

implies aj,i+1

j

+bj,i+1

j

≤ajij

+bjij

− 1r.

42 S. DALE CUTKOSKY

With the above i′, pi′ is a good point, since pi′ satisfies the conditions of 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 multiplicity ofpoints of S. By Theorem 5.12, there exists a finite sequence of blow ups of pointsVm → V such that all points of Singr(Sm) are good points, where Sm is the stricttransform of S on Vm. By Lemma 5.10, there exists a sequence of blow ups of non-singular curves Vn → Vm such that Singr(Sn) = ∅, where Sn is the strict transform ofS on Vn. By descending induction on r we can reach the case where Sing2(Sn) = ∅,so that Sn is non-singular.Remark 5.17. The resolution algorithm of this section is the good point algorithmof Abhyankar ([7], [57], [67]).

Zariski discusses early approaches to resolution in [84]. Some early proofs of res-olution of surface singularities are by Albanese [11], Beppo Levi [56], Jung [53] andWalker [75]. Zariski gave several proofs of resolution of algebraic surfaces over fieldsof characteristic zero, including the first algebraic proof [79], which we present laterin Chapter 8.

Exercises

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

a. Show that a Tschirnhausen transformation of a suitable local equationof S at p gives a formal hypersurface of maximal contact for S at p.

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

∂zr−1 = 0 is ahypersurface of maximal contact for f = 0 in a neighborhood of theorigin.

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

5.2. Embedded resolution of singularities. In this Section we will prove thefollowing theorem, which is an extension of the main resolution theorem, Theorem5.2 of the previous section.Theorem 5.18. Suppose that S is a surface which is a subvariety of a non-singular3 dimensional variety V over an algebraically closed field K of characteristic 0. thenthere exists a sequence of monoidal transforms over the singular locus of S such thatthe strict transform of S is non-singular, and the total transform π∗(S) is a SNCdivisor.Definition 5.19. A resolution datum R = (E0, E1, S, V ) is a 4-tuple where E0, E1, Sare reduced effective divisors on the non-singular 3 dimensional variety V such thatE = E0 ∪ E1 is a SNC divisor.R is resolved at p ∈ S if S is non-singular at p and E ∪ S has SNCs at p. 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 Zariski closedsubset 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.Singr,t(R) is a Zariski closed subset of Singr(R) (and of S). For the rest of this sectionwe 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 5.20. A permissible transform of R is a monoidal transform π : V ′ → Vwhose center is either a point of Singr,t(R) or a non-singular curve C ⊂ Singr,t(R)such that C makes SNCs with E.

Let F be the exceptional divisor of π. π∗(E) ∪ F is a SNC divisor. We define thestrict transform R′ of R to be R′ = (E′0, E

′1, S′) where S′ is the strict transform of S,

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 5.21. With the notation of the previous definition,

ν(R′) ≤ ν(R)

andν(R′) = ν(R) implies η(R′) ≤ η(R).

The proof of Lemma 5.21 is a generalization of Lemmas 5.3 and 5.4.Definition 5.22. p ∈ Singr,t(R) is a pregood point if

(1) In a neighborhood of p, Singr,t(R) is either a non-singular curve through p, ortwo non-singular curves through p intersecting transversally at p (satisfying2. of Definition 5.6).

(2) Singr,t(R) makes SNCs with E at p. That is, there exist regular parametersx, y, z in OV,p such that the ideal sheaf of each component of E and eachcurve in Singr,t(R) containing p is generated by a subset of x, y, z at p.

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

π : Xn → Xn−1 → · · · → X1 → spec(OV,p)of permissible monodial transforms centered at curves in Singr,t(Ri), where Ri is thetransform of Ri 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 5.23. The number of bad points in Singr,t(R) is finite.Lemma 5.24. Suppose that all points of Singr,t(R) are good. Then there exists asequence of permissible monoidal transforms, π : V ′ → V , centered at non-singularcurves contained in Singr,t(Si), where Si is the strict transform of S on the i-thmonoidal transform such that if R′ is the strict transform of R on V ′ then either

ν(R′) < ν(R)

44 S. DALE CUTKOSKY

orν(R′) = ν(R) and η(R′) < η(R).

The proofs of the above two lemmas are similar to the proofs of Lemmas 5.11 and5.10 respectively, with the use of embedded resolution of plane curves.Theorem 5.25. Suppose that R is a resolution datum such that E0 = ∅. Let

· · · → Vnπn→ Vn−1 → · · ·

π1→ V0 = V (26)

be the sequence where πn is the monoidal transform centered at the union of bad pointsin Singr,t(Ri), where Ri is the transform of R on Vi. Then this sequence is finite, sothat it terminates in a Vm such that all points of Singr,t(Rm) are good.

Proof. Suppose that (26) is an infinite sequence. Then there exists an infinite sequenceof points pn ∈ Vn such that πn+1(pn+1) = pn for all n and pn is a bad point. LetRn = OVn,pn for n ≥ 0. Since the sequence (26) is infinite, its length is larger than 1,so we may assume that t ≤ 2.

Let f = 0 be a local equation of S at p0, gk, 0 ≤ k ≤ t be local equations of thecomponents of E1 containing p0. In R0 = OV,p there are regular parameters (x, y, z)such that ν(f(0, 0, z)) = r and ν(gk(0, 0, z)) = 1 for all k. This follows since for eachequation this is a non-trivial Zariski open condition on linear changes of variables in aset of regular parameters. By the Weierstrass preparation theorem, after multiplyingf and the gk by units and performing a Tschirnhausen transformation on f , we haveexpressions

f = zr +∑rj=2 aj(x, y)zr−j

gk = z − φk(x, y), ( or gk is a unit)(27)

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

x = x1, y = x1(y1 + α), z = x1z1

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

The strict transforms f1 of f and gk1 of gk have the form of (27) with aj replaced byaj

xj1(or aj

yj1), φk with φk

x1(or φk

y1). After a finite number of monoidal transforms centered

at points pj with ν(fj) = r, we have forms in Ri = OVi,pi for the strict transforms off and gk

fi = zri +∑rj=2 aj(xi, yi)x

ajii y

bjii zr−ji

gki = zrk + φki(xi, yi)xckii ydkii

(28)

such that aj , φki are either units or zero (as in the proof of Theorem 5.12). We furtherhave that local equations of the exceptional divisor Ei0 of Vi → V are one of xi = 0,yi = 0 or xiyi = 0. This last case can only occur if t < 2. Note that our assumptionη(pi) = t implies gki is not a unit in Ri for 1 ≤ k ≤ t.

We can further assume, by an extension of the argument in the proof of Theorem5.12, that all irreducible curves in Singr,t(Ri) are non-singular.

By Lemma 5.14, we can assume that the set

T =(

ajij,bjij

), | aj 6= 0

∪ (cki, dki) | φki 6= 0


is totally ordered. Now by Lemma 5.16 we can further assume thataj0ij0

+bj0ij0

< 1,

where (aj0ij0 ,bj0ij0

) is the minimum of the (ajij ,bjij ). Recall that, if t > 0,

∏ti=1 gi = 0

makes SNCs with the exceptional divisor Ei0 of Vi → V . These conditions imply that,after possibly interchanging the gki, interchanging xi, yi and multiplying xi, yi byunits in Ri, that the gki can be described by one of the following cases at pi:

1. t = 0.2. t = 1,

a. g1i = zi + xλi , λ ≥ 1.b. g1i = zi + xλi y

µ, 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 + εxλi y

µi , where ε is a unit in Ri, λ ≥ 1 and

λ+ µ ≥ 2. If µ > 0 we can take ε = 1.We can check explicitely by blowing up permissible curves that pi is a good point.

In this calculation, If t = 0, this follows from the proof of Theorem 5.12, recallingthat xi = 0, yi = 0 or xiyi = 0 are local equations of E0 at pi. If t = 1, we mustremember that we can only blow up a curve if it lies on g1 = 0, and if t = 2, we canonly blow up the curve g1 = g2 = 0. Although we are constructing a sequence ofpermissible monoidal transforms over spec(OVi,pi), this is equivalent to constructingsuch a sequence over spec(OVi,pi), as all curves blown up in this calculation areactually algebraic, since all irreducible curves in Singr,t(Ri) are non-singular.

We have thus found a contradiction, by which we conclude that the sequence (26)has finite length.

The proof of Theorem 5.18 now follows easily from Theorem 5.25, Lemma 5.24 anddescending induction on (r, t) ∈ N×N, with the lexicographic order.Remark 5.26. This algorithm is outlined in [67] and [57].

Exercises

Resolve (or at least make (ν(R), η(R)) drop in the lexicographic order).1. R = (∅, E1, S) where S has local equation f = z = 0, E1 is the 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, E1 is the

union of two components with respective local equations g1 = z + xy, g2 =z + y(x2 + y3).

6. Resolution of singularities of varieties in characteristic zero

In recent years several proofs of canonical resolution of singularities have appeared([73], [13], [38], [39], [15], [37]). These papers have significantly simplified Hironaka’soriginal proof in [45]. In this chapter we prove the basic theorems of resolution of

46 S. DALE CUTKOSKY

singularities in characteristic 0. The final statements of resolution are proved inSection 6.8.

Throughout this chapter, unless explicitely stated otherwise, we will assume thatall varieties are over a field K of characteristic zero. Suppose that X1 and X2 aresubschemes of a variety W . We will denote the reduced set theoretic intersection ofX1 and X2 by X1 ∩X2. However, we will denote the scheme theoretic intersection ofX1 and X2 by X1 ·X2. X1 ·X2 is the subscheme of W with ideal sheaf IX1 + IX2 .

6.1. The operator 4 and other preliminaries for resolution in arbitrarydimension.Definition 6.1. Suppose that K is a field of characteristic zero, R = K[[x1, . . . , xn]]is a ring of formal power series over K, 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 choice of generators fi of J and regular parameters xj in R.Lemma 6.2. Suppose that W is a non-singular variety and J ⊂ OW is an idealsheaf. 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 define4(J) as follows. We can coverW by affine open subsets U = spec(R)which satisfy the conclusions of Lemma 10.11, so that Ω1

R/K = dy1R⊕ · · · ⊕ dynR. IfΓ(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 parameters 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 10.11 andTheorem 10.10. Thus

| ( ∂yi∂xj

) | (q) 6= 0

from which the conclusions of the Lemma follow.

Given J ⊂ OW and q ∈ W we define νJ(q) = νq(J) (Definition 10.17). If X is asubvariety of W we denote νX = νIX . We can define 4r(J) for r ∈ N inductively, bythe 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 is an idealsheaf, 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 10.19.


If f : X → I is an upper semi-continuous function, we define max f to be thelargest value assumed by f on X, and we define a closed subset of X,

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 . We shall denote

by

R(1)(X) ⊂ X (29)

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 an ideal sheaf,b = max νJ and Y ⊂ Max νJ is a non-singular subvariety. Let π : W1 → W be themonoidal transform with center Y . Let D be the exceptional divisor of π. Then

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

J1 = I−bD J.

and ID 6 | 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 of q inW1 and B be the Zariski closure of p in W . Then A → B is proper. By uppersemi-continuity of νJ , there exists a closed point x ∈ B such that νJ(x) = νJ(p). Lety ∈ 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 10.5, there is a regular system of parameters y1, . . . , yn in OW,p suchthat y1 = y2 = · · · = yr = 0 are local equations of Y at p. By Remark 10.18, we mayassume that K = K(p) = K(q). Then we can make a linear change of variables iny1, . . . , yn to assume that there are regular parameters 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 10.18, if suffices to verify the conclusionsof 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 component of Ywhose closure contains p. Suppose that f ∈ JR. Let t = νR(f) ≥ b. Then there is anexpansion

f =∑

ai1,... ,inyi11 · · · yinn

in R, with ai1,... ,in ∈ K and ai1,... ,in = 0 if i1 + · · · + ir < t. Further, there existsi1, . . . , in with i1 + · · ·+ in = t such that ai1,... ,in 6= 0.

In S, we havef = yt1f1

48 S. DALE CUTKOSKY

wheref1 =

∑ai1,... ,iny

i1+···+ir−t1 · · · yinn

is such that y1 does not divide f1. Thus we have νS(f1) ≤ νR(f). In particular, wehave yb1 | JS and since there exists f ∈ R with νR(f) = b, yb+1

1 6 | JS. Thus we havethat

J1 = I−bD J

satisfies 1. andνJ1

(q) ≤ νJ(p) = b.

Suppose that W is a non-singular variety and J ⊂ OW is an ideal sheaf. Supposethat Y ⊂ W is a non-singular subvariety. Let Y1, . . . , Ym be the distinct irreducible(connected) components of Y . Let π : W1 → W be the monoidal transform withcenter Y and exceptional divisor D. Let D = D1 + · · · + Dm where Di are thedistinct irreducible (connected) components of D, indexed so that π(Di) = Yi. Thenby an argument as in the proof of Lemma 6.4, there exists an ideal sheaf J1 ⊂ OW1

such that

JOW1 = Ic1D1· · · Icm

DmJ1 (30)

where J1 is such that for 1 ≤ i ≤ m, IDi 6 | J1 and ci = νηi(J), with ηi the genericpoint 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 ifq ∈ Di such that

JOW1 = IcDJ1. (31)

In the situation of Lemma 6.4, J1 has the 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 of W1 with idealsheaf J1. We see that the weak transform X of X is much easier to calculate thanthe 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 andY ⊂ Max νX is a non-singular subvariety. Let π : W1 → W be the monoidaltransform of W with center Y and exceptional divisor E. In this case the stricttransform X and the weak transform X are the same scheme, and π∗(X) = X + rE.Example 6.6. Let X ⊂ W = A3 be the nodal plane curve with ideal IX = (z, y2 −x3) ⊂ k[x, y, z]. Let m = (x, y, z), π : B = B(m) → A3 be the blow up of the pointm. 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.

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 in resolution. Supposethat W is a non-singular variety, J ⊂ OW is an ideal sheaf and b ∈ N. Define

Sing(J, b) = q ∈W | νJ(q) ≥ bSing(J, b) is Zariski closed in W by Lemma 6.3. Suppose that Y ⊂ Sing(J, b) is anon-singular (but not necessarily connected) subvariety of W . Let π1 : W1 → W bethe monodial transform with center Y , D1 = π−1(Y ) be the exceptional divisor.

Recall (31) that the weak transform J1 of J is defined by

JOW1 = J cD1J1

where c is locally constant on connected components of D1. c ≥ b by upper semi-continuity of νJ .

We can now define J1 by

JOW1 = J bD1J1, (32)

so thatJ1 = J c−bD1

J1.

Suppose that

Wnπn→Wn−1 → · · · →W1

π1→W (33)

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

Ji−1OWi = IbDiJiand Di is the exceptional divisor of πi. We will say that (33) is a resolution of (W,J, b)if Sing(Jn, b) = ∅.Definition 6.7. Suppose that r = max νJ , q ∈ Sing(J, r). A non-singular codimen-sion 1 subvariety H of an affine neighborhood U of q in W is called a hypersurface ofmaximal contact for J at q if

1. Sing(J, r) ∩ U ⊂ H and2. If

Wn → · · · →W1 →W

is a sequence of monoidal transforms of the form of (33) (with b = r), then thestrict transform Hn of H on Un = Wn ×W U is such that Sing(Jn, r) ∩ Un ⊂Hn.

With the assumptions of Definition 6.7, a non-singular codimension 1 subvarietyH of U = spec(OW,q) is called a formal hypersurface of maximal contact for J at q if1. 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 ourdefinition of hypersurface of maximal contact for J = IX coincides with the definitionof a hypersurface of maximal contact for X given in Definition 10.20. In this case,the ideal sheaf Jn in (33) is the ideal sheaf of the strict transform Xn of X on Wn.

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

r = max νX > 1be the set of points of maximal multiplicity on X. Let q ∈ Sing(IX , r) be a closedpoint of maximal multiplicity r.

50 S. DALE CUTKOSKY

The basic strategy of resolution is to construct a sequence

Wn → · · · →W1 →W

of monoidal transforms centered at non-singular subvarieties of the locus of points ofmultiplicity r on the strict transform of X so that we eventually reach a situationwhere all points of the strict transform of X have multiplicity < r. We know thatunder such a sequence the multiplicity can never go up, and always remains ≤ r(Lemma 6.4). However, getting the multiplicity to drop to less that r everywhere ismuch more difficult.

Notice that the desired sequence Wn →W1 will be a resolution of the form of (33),with (Ji, b) = (IXi , r), where Xi is the strict transform of X on Wi.

Let q ∈ Sing(IX , r) be a closed point. After Weierstrass Preparation and perform-ing a Tschirnhausen transformation, there exist regular parameters (x1, . . . , xn, y) inR = OW,q such that there exists f ∈ R such that f = 0 is a local equation of X at q,and

f = yr +r∑i=2

ar(x1, . . . , xn)yr−i. (34)

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 subvarietycontaining q. Let J = IZ,q.

Write J = P1∩· · ·∩Pr where Pi are prime ideals of the same height in R. Let Ri =RPi . By semi-continuity of νX , we have f ∈ P ri RPi for all i. Since ∂

∂y : RPi → RPi , we

have (r − 1)!y = ∂r−1f∂yr−1 ∈ JRPi for all i. Thus y ∈ ∩JRPi = J and Z ∩ spec(R) ⊂ H.

Now we will verify that if Y ⊂ Sing(IX , r) is a non-singular subvariety, π : W1 →Wis the monoidal transform with center Y , X1 is the strict transform of X and q1 is aclosed point of X1 such that π(q1) = q which has maximal multiplicity r, then q1 ison the strict transform H1 of H (over the formal neighborhood of q). Suppose thatq1 is such a point. That is, q1 ∈ π−1(q) ∩ Sing(IX1 , r). We have shown above thaty ∈ IY . Without changing the form of (34), we may then assume that there is s ≤ nsuch that x1 = . . . = 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 such that

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

(35)

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

f1 = fxr1

= y(1)r +∑ri=2

aixi1y(1)r−i

= y(1)r +∑ri=2 ai(1)(x1(1), . . . , xn(1))y(1)r−i

(36)

and y1(1) = 0 is a local equation of H1 at q1, x1(1) = 0 is a local equation of theexceptional divisor of π. With our assumption that f1 also has multiplicity r, f1 has


an expression of the same form as (34), so by induction, H is a (formal) hypersurfaceof maximal contact.

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

We will now describe this resolution problem. Set

I = (ar!2

2 , . . . , ar!rr ) ⊂ S = K[[x1, . . . , xn]].

Observe that the scheme of points of multiplicity r in the formal neighborhood of qon X coincides with the scheme of points of multiplicity ≥ r! of I,

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 (35) on I. Since H is

a hypersurface of maximal contact, π certainly induces a morphism H1 → H, whereH1 is the strict transform of H. By direct verification, we see that the transform I1of I (this transform is defined by (32)) satisfies

I1S1 =1xr!1

IS1

where S1 = K[[x1(1), . . . , xn(1)]] = OH1,q1 , and this ideal is thus, by (36), 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 the exactsame way that we obtained the ideal I from our local equation f of X. We furtherobserve that νS1(I1S1) ≥ r! if and only if νq1(X1) ≥ r. We see then that to reducethe multiplicity of the strict transform of X over q, we are reduced to resolving asequence of the form (33), but over a formal scheme.

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

When X is a curve (dim W = 2) the induced resolution problem on the formalcurve H is essentially trivial, as the ideal I is in fact just the ideal generated by amonomial, and we only need blow up points to divide out powers of the terms in themonomial.

When X is a surface (dim W = 3), we were able to make use of another induction,by realizing that resolution over a general point of a curve in the locus of pointsof maximal multiplicity reduces to the problem of resolution of curves (over a non-closed field). In this way, we were able to reduce to a resolution problem over finitelymany points in the locus of points of maximal multiplicity on X. The potentiallyworrisome problem of extending a resolution of an object over a formal germ of ahypersurface to a global proper morphism over W was not a great difficulty since theblowing up of 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-singular curvescontained in the locus of maximal multiplicity. This required a little attention, but wewere able to fairly easily reduce to a situation were these formal curves were alwaysalgebraic. In this case we first blew up points to reduce to the situation where theideal I was locally a monomial ideal. Then we blew up more points to make it aprincipal monomial ideal. At this point, we finished the resolution process by blowingup non-singular curves in the locus of maximal multiplicity on the strict transform

52 S. DALE CUTKOSKY

of the surface X, which amounts to dividing out powers of the terms in the principalmonomial ideal.

In Section 5.2, we extended this algorithm to obtain embedded resolution of thesurface X. This required keeping track of the exceptional divisors which occur underthe resolution process, and requiring that the centers of all monoidal transforms makeSNCs with the previous exceptional divisors. We introduced the η invariant, whichcounts the number of components of the exceptional locus which have existed sincethe multiplicity last dropped.

In this chapter we give a proof of resolution in arbitrary dimension (over fieldsof characteristic zero) which incorporates all of these ideas into a general inductionstatement. The fact that we must consider some kind of covering by local hyper-surfaces which are not related in any obvious way is incorporated in the notion ofGeneral Basic Object.

The first proof of resolution in arbitrary dimension (over a field of characteristiczero) was by Hironaka [48]. The proof consists of a local proof, and a complex webof inductions to produce a global proof. The local proof uses sophisticated methodsin commutative algebra to reduce the problem of reducing the multiplicity of thestrict transform of an ideal after a permissible sequence of monoidal transforms tothe problem of reducing the multiplicity of the strict transform of a hypersurface. Theuse of the Tschirnhausen transformation to find a hypersurface of maximal contactlies at the heart of this proof. The order ν and the invariant τ (of Chapter 7) are theprincipal invariants considered.

The proof has been significantly simplified by Hironaka and others. Hironakaoutlined a proof of canonical resolution where he introduced the idea of general basicobjects [51] (We define this concept in Section 6.5). Since then several accessibleproofs of canonical equivariant resolution have emerged ([73], [13], [38], [39],[15],[37]).

De Jong’s theorem [34], referred to at the end of Chapter 7, can be refined to givea new proof of resolution of singularities of varieties of characteristic zero ([9], [14],[65]). The resulting theorem is not as strong as the classical results on resolutionof singularities in characteristic zero (c.f. Section 6.8), as the resulting resolutionX ′ → X may not be an isomorphism over the non-singular locus of X.

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

6.3. Pairs and basic objects.

Definition 6.8. A pair is (W0, E0) where W0 is a non-singular variety over a field Kof characteristic zero, and E0 = D1, . . . , Dr is an ordered collection of reduced non-singular divisors on W0 such that D1 + · · ·+Dr is a reduced simple normal crossingsdivisor.

Suppose that (W0, E0) is a pair, and Y0 is a non-singular closed subvariety of W0.Y0 is a permissible center for (W0, E0) if Y0 has SNCs with E0. Let π : W1 → W bethe monoidal transform of W with center Y0 and exceptional divisor D. By abuseof notation we identify Di (1 ≤ i ≤ r) with its strict transform on W1 (which couldbecome the empty set if Di is a union of components of Y0). By a local analysis, usingthe regular parameters of the proof of Lemma 6.4, we see that D1 + · · ·+Dr +D isa SNC divisor on W1.


Now set E1 = D1, . . . , Dr, Dr+1 = D. We have thus defined a new pair (W1, E1)called the transform of (W0, E0) by π.

(W1, E1)→ (W0, E0)

will be called a transformation. Observe that some of the strict transforms Di maybe the empty set.

Definition 6.9. Suppose that (W0, E0) is a pair where E0 = D1, . . . , Dr, and W0

is a non-singular subvariety of W0. Define scheme theoretic intersections Di = Di ·W0

for all i. Suppose that Di are non-singular for 1 ≤ i ≤ r and that D1 + · · ·+ Dr is aSNC divisor. Set E0 = D1, . . . , Dr. Then (W0, E0) is a pair and

(W0, E0)→ (W0, E0)

is called an immersion of pairs.Definition 6.10. A basic object is (W0, (J0, b), E0) where (W0, E0) is a pair, J0 isan ideal sheaf of W0 which is non-zero on all connected components of W0, and b isa natural number. The singular locus of (W0, (J0, b), E0) is

Sing(J0, b) = q ∈W0 | νq(J0) ≥ b = V (4b−1(J0)) ⊂W0.

Definition 6.11. A basic object (W0, (J0, b), E0) is a simple basic object if

νq(J) = b if q ∈ Sing(J, b).

Suppose that (W0, (J0, b), E0) is a basic object and Y0 is a non-singular closedsubvariety of W0. Y0 is a permissible center for (W0, (J0, b), E0) if Y0 is permis-sible for (W0, E0) and Y0 ⊂ Sing(J0, b). Suppose that Y0 is a permissible cen-ter for (W0, (J0, b), E0). Let π : W1 → W0 be the monoidal transform of W0

with center Y0 and exceptional divisor D. Thus we have a transformation of pairs(W1, E1) → (W0, E0). Let J1 be the weak transform of J0 on W1, so that there is alocally constant function c1 on D (constant on each connected component of D) suchthat the total transform of J0 is

J0OW1 = Ic1D J1.

We necessarily have c1 ≥ b, so we can define

J1 = I−bD J0OW1 = Ic1−bD J1.

We have thus defined a new basic object (W1, (J1, b), E1) called the transform of(W0, (J0, b), E0) by π.

(W1, (J1, b), E1)→ (W0, (J0, b), E0)

will be called a transformation.Observe that a transform of a basic object is a basic object, and (by Lemma 6.4)

the transform of a simple basic object is a simple basic object.Suppose that

(Wk, (Jk, b), Ek)→ · · · → (W0, (J0, b), E0) (37)

is a sequence of transformations.Definition 6.12. (37) is a resolution of (W0, (J0, b), E0) if Sing(Jk, b) = ∅.

54 S. DALE CUTKOSKY

Definition 6.13. Suppose that (37) is a sequence of transformations. Set J0 = J0,and let J i+1 be the weak transform of J i for 0 ≤ i ≤ k − 1. For 0 ≤ i ≤ k define

w-ordi : Sing(Ji, b)→1bZ

by

w-ordi(q) =νq(J i)b

and define

ordi : Sing(Ji, b)→1bZ

by

ordi(q) =νq(Ji)b

.

If we assume that Yi ⊂ Max w-ordi ⊂ Sing(Ji, b) for 1 ≤ i ≤ k in (37), then

max w-ord0 ≥ · · · ≥ max w-ordkby Lemma 6.4.Definition 6.14. Suppose that (37) is a sequence of transformations such that Yi ⊂Max w-ordi ⊂ Sing(Ji, b) for 0 ≤ i < k. Let k0 be the smallest index such thatmax w-ordk0−1 > max w-ordk0 = · · · = max w-ordk. In particular, k0 is defined to be0 if max w-ord0 = · · · = max w-ordk. Set Ek = E+

k ∪E−k where E−k is the ordered set

of divisors D in Ek which are strict transforms of divisors of Ek0 and E+k = Ek−E−k .

Definetk : Sing(Jk, b)→ Q× Z

where Q× Z has the lexicographic order by tk(q) = (w-ordk(q), η(q)),

η(q) =

CardD ∈ Ek | q ∈ D if w-ordk(q) < max w-ordk0

CardD ∈ E−k | q ∈ D if w-ordk(q) = max w-ordk0

The above definition allows us to inductively define upper semi-continuous func-tions t0, t1, . . . , tk on a sequence of transformations (37) such that Yi ⊂ Max w-ordi ⊂Sing(Ji, b) for 1 ≤ i ≤ k.Lemma 6.15. Suppose that (37) is a sequence of transformations such that Yi ⊂Max ti ⊂ Max w-ordi for 0 ≤ i ≤ k. Then

max tk ≤ · · · ≤ max tk.

Lemma 6.15 follows from Lemma 6.4, and a local analysis, using the regular pa-rameters of the proof of Lemma 6.4.Definition 6.16. Suppose that (W0, (J0, b), E0) is a basic object, and

(Wk, (Jk, b), Ek)→ · · · → (W0, (J0, b), E0) (38)

is a sequence of transformations such that max w-ordk = 0. Then we say that(Wk, (Jk, b), Ek) is a monomial basic object (relative to (38)).

Suppose that (Wk, (Jk, b), Ek) is a monomial basic object (relative to a sequence(38)). Let E0 = D1, . . . , Dr and let Di be the exceptional divisor of πi : Wi →Wi−1

where i = i+ r. Then we have an expansion (in a neighborhood of Sing(Jk, b))) of theweak transform Jk of J

Jk = Id1D1· · · IdkD

k


where di are functions on Wk which are zero on Wk −Di and di is a locally constantfunction on Di for 1 ≤ i ≤ k.

Suppose that B = (Wk, (Jk, b), Ek) is a monomial basic object (relative to (38)).We define

Γ(B) : Sing(Jk, b)→ IM = Z×Q× ZN

byΓ(B)(q) = (−Γ1(q),Γ2(q),Γ3(q)),

where

Γ1(q) = minp such that there exists i1, . . . , ip, with r < i1, . . . , ip ≤ r + k, such thatdi1(q) + · · ·+ dip(q) ≥ b for some q ∈ Di1 ∩ · · · ∩Dip

,

Γ2(q) = max

di1 (q)+···+dip (q)

b , with r < i1, . . . , ip ≤ r + k, such that p = Γ1(q),di1(q) + · · ·+ dip(q) ≥ b, q ∈ Di1 ∩ · · · ∩Dip

,

Γ3(q) = max

(i1, . . . , ip, 0, . . . ) such that Γ2(q) = di1 (q)+···+dip (q)

b ,with r < i1, . . . , ip ≤ r + k, q ∈ Di1 ∩ · · · ∩Dip

.

Observe that Γ(B) is upper semi-continuous, and Max Γ(B) is non-singular.Lemma 6.17. Suppose that B = (Wk, (Jk, b), Ek) is a monomial basic object (rel-ative to a sequence (38)). Then Yk = Max Γ(B) is a permissible center for B andif (Wk+1, (Jk+1, b), Ek+1) → (Wk, (Jk, b), Ek) is the transformation induced by themonoidal transform πk+1 : Wk+1 →Wk with center Yk, then B1 = (Wk+1, (Jk+1, b), Ek+1)is a monomial basic object and

max Γ(B) > max Γ(B1)

in the lexicographic order.

Proof. Max Γ(B) is certainly a permissible center. We will show that max Γ(B) >max Γ(B1).

We will first assume that max − Γ1 < −1. Let i = i + r. There exist locallyconstant functions di on W such that di(q) = 0 if q 6∈ Di and di is locally constanton Di such that

Jk = Id1D1· · · IdkD

k.

Let Di be the strict transform of Di on Wk+1 and let Dk+1 be the exceptional divisorof the blow up πk+1 : Wk+1 →Wk of Yk = Max Γ(B). Then

Jk+1 = Id1

D1

· · · Idk+1

Dk+1

where for 1 ≤ i ≤ k, di(q) = di(πk+1(q)) if q ∈ Di and di(q) = 0 if q 6∈ Di. We have

dk+1(q) = di1(πk+1(q)) + · · ·+ dip(πk+1(q))− b

if q ∈ Dk+1 and dk+1(q) = 0 if q 6∈ Dk+1.For q ∈ Sing (Jk+1, b), Γ(B1)(q) = Γ(B)(πk+1(q)) if πk+1(q) 6∈ Max Γ(B), so it

suffices to prove thatΓ(B1)(q) < Γ(B)(πk+1(q))

if q ∈ Sing (Jk+1, b) and πk+1(q) ∈ Max Γ(B). Suppose that

Γ(B1)(q) = (−p1, w1, (j1, . . . , jp1 , 0, . . . ))

56 S. DALE CUTKOSKY

andΓ(B)(πk+1(q)) = max Γ(B) = (−p, w, (i1, . . . , ip, 0, . . . )).

We will first verify that

p1 = Γ1(q) ≥ Γ1(πk+1(q)) = p. (39)

If k + 1 6∈ j1, . . . , jp1, we have

b ≤ dj1(q) + · · ·+ djp1(q) ≤ dj1(πk+1(q)) + · · ·+ djp1

(πk+1(q))

implies p1 ≥ p.Suppose that k + 1 ∈ j1, . . . , jp1, so that j1 = k + 1. If p1 < p there exists

ik ∈ i1, . . . , ip such that ik 6∈ j1, . . . , jp1.

dj1(q) + · · ·+ djp1(q) = dj2(q) + · · ·+ djp1

(q) + di1(πk+1(q)) + · · ·+ dip(πk+1(q))− b≤ (di1(πk+1(q)) + · · ·+ dik−1(πk+1(q)) + dik+1(πk+1(q))+ · · ·+ dip(πk+1(q))− b)+dik(πk+1(q)) + dj2(πk+1(q)) + · · ·+ djp1

(πk+1(q))< dik(πk+1(q)) + dj2(πk+1(q)) + · · ·+ djp1

(πk+1(q))< b

a contradiction. Thus p1 ≥ p.Now assume that p1 = p. We will verify that w1 ≤ w. If k + 1 6∈ j1, . . . , jp1 we

have

w1 =dj1 (q)+···+djp1

(q)

b

≤ dj1 (πk+1(q))+···+djp1(πk+1(q))

b≤ w.

If k + 1 ∈ j1, . . . , jp1, then j1 = k + 1. We must have

dj2(πk+1(q)) + · · ·+ djp1(πk+1(q)) < b

so that

w1b =(di1(πk+1(q)) + · · ·+ dip(πk+1(q))− b

)+ dj2(q) + · · ·+ djp1

(q)≤ di1(πk+1(q)) + · · ·+ dip(πk+1(q)) +

(dj2(πk+1(q)) + · · ·+ djp1

(πk+1(q))− b)

< wb.

Thus w1 < w.Assume that p = p1 and w = w1. We have shown that k + 1 6∈ j1, . . . , jp1 in this

case. SinceDi1 ∩ · · · ∩Dip = ∅,

we must have (j1, . . . , jp, 0, . . . ) < (i1, . . . , ip, 0, . . . ).It remains to verify the Lemma in the case when max − Γ1(B) = −1. This case

is not difficult, and is left to the reader.

6.4. Basic objects and hypersurfaces of maximal contact. Suppose that B =(W, (J, b), E) is a basic object and Wλλ∈Λ is a (finite) open cover of W . We thenhave associated basic objects Bλ = (Wλ, (Jλ, b), Eλ) for λ ∈ Λ obtained by restrictionto Wλ. Observe that

Sing(Jλ, b) = Sing(J, b) ∩Wλ.


If B1 = (W1, (J1, b), E1) → (W, (J, b), E) is a transformation, induced by a monoidaltransform π : W1 →W , then there are associated transformations

Bλ1 = (Wλ1 , (J

λ1 , b), E

λ1 )→ (Wλ, (Jλ, b), Eλ)

where Wλ1 = π−1(Wλ), Jλ1 = J1 |Wλ

1 . In particular, Wλ1 is an open cover of W1.

Definition 6.18. Let W1 = W × A1K with projection π : W1 → W . Suppose that

(W, (J, b), E) is a basic object. Then there is a basic object (W1, (J1, b), E1) where J1 =JOW1 . If E = D1, . . . , Dr and D′i = π−1(Di) for 1 ≤ i ≤ r, E1 = D′1, . . . , D′r.

(W1, (J1, b), E1)→ (W, (J, b), E)

is called a restriction.Remark 6.19. Suppose that (W, (J, b), E) is a basic object and Wh ⊂ W is a non-singular hypersurface. Suppose that IWh

⊂ 4b−1(J). Then Sing(J, b) ⊂ V (4b−1(J)) ⊂Wh and νq(4b−1(J) = 1 for any q ∈ Sing(J, b).

Conversely, If (W, (J, b), E) is a simple basic object, and q ∈ Sing(J, b), thenνq(4b−1(J)) = 1, so there exists an affine neighborhood U of q in W and a non-singular hypersurface Uh ⊂ U such that U ∩ Sing(J, b) ⊂ Uh.Lemma 6.20. Suppose that

(W1, (J1, b), E1)→ (W, (J, b), E)

is a transformation of basic objects. Let D be the exceptional divisor of W1 → W .Then for all integers i with 1 ≤ i ≤ b we have

4b−iW (J)OW1 ⊂ IiDand

1IiD4b−iW (J)OW1 ⊂ 4b−iW1

(J1).

Proof. Let Y ⊂ Sing(J, b) be the center of π : W1 →W .The first inclusion follows since νq(4b−iW (J)) ≥ i if q ∈ Y , so that IiD divides

4b−iW (J)OW1 .Now we will prove the second inclusion. The second inclusion is trivial if i = b,

since JOW1 = IbDJ1 by definition. We will prove the second inclusion by descendinginduction on i. Suppose that the inclusion is true for some i > 1. Let q ∈ D be aclosed point, p = π(q). Let K ′ = K(q). By Lemma 10.16, it suffices to prove theinclusion on W ×K K ′, so we may assume that K(q) = K. Then there exist regularparameters x0, x1, . . . , xn in R = OW,p such that IY,p = (x0, x1, . . . , xm), withm ≤ n,x0,

x1x0, . . . , xmx0

, xm+1, . . . , xn, is a regular system of parameters in R1 = OW1,q, and

ID,q = (x0). It suffices to show that for f ∈ 4b−(i−1)W (J)p,

f

xi−10

∈ 4b−(i−1)W1

(J1)q. (40)

If f ∈ 4b−iW (J)p ⊂ 4b−(i−1)W (J)p then the formula follows by induction, so we may

assume that f = D(g) with g ∈ 4b−iW (J)p, D an R-derivation. Set D′ = x0D. D′ isa R1-derivation on R1 = OW1,q since D′( xix0

) ∈ OW1,q for 1 ≤ i ≤ m. By induction,

g

xi0∈ 1IiD4b−iW (J)OW1,q ⊂ 4b−iW1

(J1)q ⊂ 4b−(i−1)W1

(J1)q.

58 S. DALE CUTKOSKY

ThusD′(

g

xi0) ∈ 4b−(i−1)

W1(J1)q.

D′(g

xi0) =

D(g)xi−1

0

− iD(x0)g

xi0which implies

f

xi−10

=D(g)xi−1

0

= D′(g

xi0) + iD(x0)

g

xi0∈ 4b−(i−1)

W1(J1)q.

and (40) follows.

Lemma 6.21. Suppose that (W, (J, b), E) is a simple basic object, and Wh ⊂W is anon-singular hypersurface such that IWh

⊂ 4b−1(J). Suppose that

(W1, (J1, b), E1)→ (W, (J, b), E)

is a transformation with center Y . Let (Wh)1 be the strict transform of Wh on W1.Then Y ⊂Wh, (Wh)1 is a non-singular hypersurface in W1 and I(Wh)1 ⊂ 4b−1(J1).

Proof. Let D be the exceptional divisor of π : W1 → W . Suppose that q ∈ D is aclosed point,

p = π(q) ∈ Y ⊂ Sing(J, b) = V (4b−1(J)) ⊂Wh.

Recall that Y ⊂ Sing(J, b) be definition of transformation. Let f = 0 be a localequation for Wh at p, g = 0 be a local equation of D at q. Then f

g = 0 is a localequation of (Wh)1 at q. f

g ∈ 4b−1(J1)q by Lemma 6.20. Thus

Sing(J1, b) = V (4b−1(J1)) ⊂ V (f

g).

Definition 6.22. Suppose that B = (W, (J, b), E) is a simple basic object and W isa non-singular closed subvariety of W such that IW ⊂ 4b−1(J). Then the coefficientideal of B on W is

C(J) =b−1∑i=0

4i(J)b!b−iOW .

Lemma 6.23. With the notation of Lemma 6.21 and Definition 6.22

Sing(J, b) = Sing(C(J), b!) ⊂Wh ⊂W.

Proof. It suffices to check this at closed points q ∈ Wh. By Lemma 10.16 we mayassume that K(q) = K. Then there exist regular parameters x1, . . . , xn at q in W

such that x1 = 0 is a local equation for Wh at q. Let R = OW,q = K[[x1, . . . , xn]],S = OWh,q = K[[x2, . . . , xn]] with natural projection R→ S. It suffices to show that

νR(J) ≥ b if and only if νS(4j(JS)) ≥ b− j (41)

for j = 0, 1, . . . , b− 1. Suppose that f ∈ J ⊂ R. Write

f =∑i≥0

aixi1


with ai ∈ S for all i. We have νR(f) ≥ b if and only if νS(aj) ≥ b − j for j =0, 1, . . . , b− 1.aj ∈ 4j(J)S, since aj is the projection in S of 1

j!∂jf

∂xj1. Thus we have the if direction

of (41).The ideal 4j(J) is spanned by elements of the form

ci1,... ,in =∂i1+···+inf

∂xi11 · · · ∂xinn

with f ∈ J and i1 + · · ·+ in ≤ j, and the projection of ci1,... ,in in S is

i1!∂i2+···+inai1∂xi22 · · · ∂x

inn

.

Thus we have the only if direction of (41).

Theorem 6.24. Suppose that (W, (J, b), E) is a simple basic object, and Wλλ∈Λ

is an open cover of W such that for each λ there is a non-singular hypersurfaceWλh ⊂Wλ such that IWλ

h⊂ 4b−1(Jλ) and Wλ

h has simple normal crossings with Eλ.Let Eλh be the set of non-singular reduced divisors on Wλ

h obtained by intersection ofE with Wλ

h . Recall that R(1)(Z) denotes the codimension 1 part of a closed reducedsubscheme Z, as defined in (29).

1. If R = R(1)(Sing(J, b)) 6= ∅ then R is non-singular, open and closed inSing(J, b), and has simple normal crossings with E. Furthermore, the monoidaltransform of W with center R induces a transformation

(W1, (J1, b), E1)→ (W, (J, b), E)

such that R(1)(Sing(J1, b)) = ∅.2. If R(1)(Sing(J, b)) = ∅, for each λ ∈ Λ, the basic object

(Wλh , (C(Jλ), b!), Eλh)

has the following properties:a. Sing(Jλ, b) = Sing(C(Jλ), b!)b. Suppose that

(Ws, (Js, b), Es)→ · · · → (W, (J, b), E) (42)

is a sequence of transformations and restrictions, with induced sequencesfor each λ

(Wλs , (J

λs , b), E

λs )→ · · · → (Wλ, (Jλ, b), Eλ).

Then for each λ there is an induced sequence of transformations andrestrictions

((Wh)λs , (C(Jλ)s, b!), (Eh)λs )→ (Wλh , (C(Jλ), b!), Eλh)

and a diagram where the vertical arrows are closed immersions of pairs

(Wλs , E

λs ) → · · · → (Wλ, Eλ)

↑ ↑((Wh)λs , (Eh)λs ) → · · · → (Wλ

h , Eλh)

(43)

such thatSing(C(Jλ)i, b!) = Sing(Jλi , b)

60 S. DALE CUTKOSKY

for all i.

Proof. We may assume that Wλ = W .First suppose that T = R(1)(Sing(J, b)) 6= ∅. T ⊂ Sing(J, b) ⊂ Wh implies T is

a union of connected components of Wh, so T is non-singular, open and closed inSing(J, b) and has SNCs with E.

Suppose that q ∈ T . Let f = 0 be a local equation of T at q in W . Since(W, (J, b), E) is a simple basic object, Jq = f bOW,q. If π1 : W1 → W is the monoidaltransform with center T then (J1)q1 = OW,q1 for all q1 ∈ W1 such that π1(q1) ∈ T ,and necessarily, R(1)(Sing(J1, b)) = ∅.

Now suppose that R(1)(Sing(J, b)) = ∅, and (42) is a sequence of transformationsand restrictions of simple basic objects.

Suppose that (W1, (J1, b), E1)→ (W, (J, b), E) is a transformation with center Y0.By Lemma 6.21, Y0 ⊂Wh, the strict transform (Wh)1 of Wh is a non-singular hyper-surface in W1 and

I(Wh)1 ⊂ 4b−1(J1).

Since Wh has SNCs with E, Y ⊂ Wh and Y has SNCs with E, we have that Y hasSNCs with E ∪Wh. Thus (Wh)1 has SNCs with E1, and if (Eh)1 = E1 · (Wh)1 is thescheme theoretic intersection, we have a closed immersion of pairs

((Wh)1, (Eh)1)→ (W1, E1).

Since the case of a restriction is trivial, and each (Wi, (Ji, b), Ei) is a simple basicobject, we thus can conclude by induction that we have a diagram of closed immersionsof pairs (43) such that

I(Wh)j ⊂ 4b−1(Jj) (44)

for all j.We may assume that the sequence (42) consists only of transformations. Let Dk

be the exceptional divisor of Wk →Wk−1. It remains to show that

Sing(C(J)j , b!) = Sing(Jj , b) (45)

for all j. By Lemma 10.16 and Remark 10.18 we may assume that K is algebraicallyclosed. For k > 0 and 0 ≤ i ≤ b− 1, set[

4b−i(J)]k

=1IiD

k

[4b−i(J)

]k−1OWk

so that

C(J)k =b∑i=1

[4b−i(J)

] b!i

kO(Wh)k .

We will establish the following formulas (46) and (47) for 0 ≤ k ≤ s and 0 ≤ i ≤b− 1. [

4b−i(J)]k⊂ 4b−i(Jk) (46)

Suppose that q ∈ Sing(C(J0)k, b!) is a closed point. Set Rk = OWk,q, Sk = O(Wh)k,q.Then there are regular parameters (xk,1, . . . , xk,n, zk) in Rk such that I(Wh)k,q = (zk)


and there are generators fk of JkRk such that

fk =∑α

ak,αzαk (47)

with ak,α ∈ K[[xk1, . . . , xk,n]] = Sk such that for 0 ≤ α ≤ b− 1

(ak,α)b!b−α ∈ C(J)kSk.

Assume that the formulas (46) and (47) hold. (46) implies C(J)k ⊂ C(Jk), so byLemma 6.23, Sing(Jk, b) ⊂ Sing(C(J)k, b!). Now (47) shows that Sing(C(J)k, b!) ⊂Sing(Jk, b), so that Sing(C(J)k, b!) = Sing(Jk, b) as desired.

We will now verify formulas (46) and (47). (46) is trivial if k = 0 and (47) withk = 0 follows from the proof of Lemma 6.23.

We now assume that (46) and (47) are true for k = t, and prove them for k = t+1.Since [

4b−i(J)]t⊂ 4b−i(Jt),

we have[4b−i(J)

]t+1

=1

(IDt+1

)i[4b−i(J)

]t⊂ 1

(IDt+1

)i4b−i(Jt) ⊂ 4b−i(Jt+1),

where the last inclusion follows from the second formula of Lemma 6.20. We havethus verified formula (46) for k = t+ 1.

Let qt+1 ∈ Sing(C(J0)t+1, b!) be a closed point, qt the image of qt+1 in Sing(C(J0)t, b!).By assumption, there exist regular parameters (xt1, . . . , xtn, zt) in Rt which sat-

isfy (47). Recall that we have reduced to the assumption that K is algebraicallyclosed. Let Yt be the center of Wt+1 → Wt. By (44), there exist regular parame-ters (x1, . . . , xn) in St = K[[xt1, . . . , xtn]] ⊂ Rt such that (x1, . . . , xn, zt) are regularparameters in Rt, there exists r with r ≤ n such that x1 = · · · = xr = zt = 0are local equations of Yt in OWt,qt and Rt+1 = OWt+1,qt+1 has regular parameters(xt+1,1, . . . , xt+1,n, zt+1) such that

x1 = xt+1,1

xi = xt+1,1xt+1,i for 2 ≤ i ≤ rzt = xt+1,1zt+1

xi = xt+1,i for r < i ≤ n.In particular, xt+1,1 = 0 is a local equation of the exceptional divisor Dt+1 andzt+1 = 0 is a local equation of (Wh)t+1. We have generators

ft+1 =ft

xbt+1,1

=∑α

at+1,αzαt+1

of Jt+1Rt+1 = 1xbt+1,1

JtRt+1 such that

at+1,α =at,α

xb−αt,1

if α < b, and

(at+1,α)b!b−α =

1(xt,1)b!

(at,α)b!b−α ∈ 1

(IDt+1

)b!C(J)tSt+1 = C(J)t+1St+1.

62 S. DALE CUTKOSKY

6.5. General basic objects.Definition 6.25. Suppose that (W0, E0) is a pair. We define a general basic ob-ject (GBO) (F0,W0, E0) on (W0, E0) to be a collection of pairs (Wi, Ei) with closedsets Fi ⊂ Wi which have been constructed inductively to satisfy the following threeproperties:

1. F0 ⊂W0

2. Suppose that Fi ⊂ Wi has been defined for the pair (Wi, Ei). If πi+1 :(Wi+1, Ei+1)→ (Wi, Ei) is a restriction then Fi+1 = π−1

i+1(Fi)3. Suppose that Fi ⊂ Wi has been defined for the pair (Wi, Ei). If πi+1 :

(Wi+1, Ei+1)→ (Wi, Ei) is a transformation centered at Yi ⊂ Fi then Fi+1 ⊂Wi+1 is a closed set such that Fi − Yi ∼= Fi+1 − πi+1(Yi) by the map πi+1.

The closed sets Fi will be called the closed sets associated to (F0,W0, E0).If

(W1, E1)→ (W0, E0)is a transformation with center Y0 ⊂ F0 then we can use the pairs (Wi, Ei) and closedsubsets Fi ⊂Wi in Definition 6.25 which map to (W1, E1) by a sequence of restrictionsand transformations satisfying 2. and 3. to define a general basic object (F1,W1, E1)over (W1, E1). We will say that Y0 is a permissible center for (F0,W0, E0) and call

(F1,W1, E1)→ (F0,W0, E0)

a transformation of general basic objects.If

(W1, E1)→ (W0, E0)is a restriction then we can use the pairs (Wi, Ei) and closed subsets Fi definedin Definition 6.25 above which map to (W1, E1) by a sequence of restrictions andtransformations satisfying (2) and (3) to define a general basic object (F1,W1, E1)over (W1, E1). We will call

(F1,W1, E1)→ (F0,W0, E0)

a restriction of general basic objects.A composition of restrictions and transformations of general basic objects

(Fr,Wr, Er)→ · · · → (F0,W0, E0) (48)

will be called a permissible sequence for the GBO (F0,W0, E0). (48) will be called aresolution if Fr = ∅.Definition 6.26. Suppose that (W0, E0) is a pair, (F0,W0, E0) is a GBO over(W0, E0), with associated closed sets Fi. Further, suppose that Wλ

0 λ∈Λ is a finiteopen covering of W0 and d ∈ N.

Suppose that we have a collection of basic objects Bλ0 λ∈Λ with

Bλ0 = (Wλ0 , (a

λ0 , b

λ), Eλ0 ).

We will say that Bλ0 λ∈Λ is a d-dimensional structure on the GBO (F0,W0, E0) iffor each λ ∈ Λ

1. we have an immersion of pairs

jλ0 : (Wλ0 , E

λ0 )→ (Wλ

0 , Eλ0 )

where Wλ0 has dimension d.


2. If(Fr,Wr, Er)→ · · · → (F0,W0, E0)

is a permissible sequence for the GBO (F0,W0, E0) (with associated closedsets Fi ⊂Wi for 0 ≤ i ≤ r), then the sequence of morphisms of pairs

(Wλr , E

λr )→ · · · → (Wλ

0 , Eλ0 )

induce diagrams of immersions of pairs

(Wλr , E

λr ) → · · · → (Wλ

0 , Eλ0 )

↑ ↑(Wλ

r , Eλr ) → · · · → (Wλ

0 , Eλ0 )

such that

(Wλr , (a

λr , b

λ), Eλr )→ · · · → (Wλ0 , (a

λ0 , b

λ), Eλ0 )

is a sequence of transformations and restrictions of basic objects, and

Fλi = Fi ∩Wλi = Sing(aλi , b

λ)

for 0 ≤ i ≤ r.

6.6. Functions on a general basic object.

Proposition 6.27. Let (F0,W0, E0) be a GBO which admits a d-dimensional struc-ture, and fix notation as in Definition 6.26. Let Fλ0 = Sing(aλ0 , b

λ).There is a function ordd : F0 → Q such that for all λ ∈ Λ, the composition

Fλ0jλ0→ F0

ordd→ Q

is the function ordλ : Fλ0 → Q defined by Definition 6.13 for Fλ0 = Sing(aλ0 , bλ).

Proof. Let Bλ0 λ∈Λ where Bλ0 = (Wλ0 , (a

λ0 , b

λ), Eλ0 ) be the given d-dimensional struc-ture on (F0,W0, E0). Let E0 = D1, . . . , Dr. We have closed immersions jλ0 : Wλ

0 →Wλ

0 for all λ ∈ Λ.Suppose that q0 ∈ F0 is a closed point, and λ, β ∈ Λ are such that there are

qλ0 ∈ Wλ0 and qβ0 ∈ W

β0 such that jλ0 (qλ0 ) = jβ0 (qβ0 ) = q0. We must show that

νqλ0 (aλ0 )

bλ=νqβ0

(aβ0 )

bβ.

We will prove this by expressing these numbers intrinsicly, in terms of the GBO(F0,W0, E0), so that they are independent of λ. We will carry out the calculation forBλ0 .

Let (F1,W1, E1) π1→ (F0,W0, E0) be the restriction where W1 = W0 × A1K . Let

q1 = (q0, 0) ∈W1. Let L1 = q0 ×A1K . L1 ⊂ F1 = π−1

1 (F0).For any integer N > 0 we can construct a permissible sequence of pairs

(WN , EN ) πN→ · · · π2→ (W1, E1) π1→ (W0, E0) (49)

where for i ≥ 1, Wi+1 →Wi is the blow up of a point qi. qi is defined as follows. LetLi be the strict transform of L1 on Wi, and let Di be the exceptional divisor of πi,where i = i+ r. Then qi = Li ∩Di.

64 S. DALE CUTKOSKY

We have q1 ∈ L1 ⊂ F1. We must have L2 ⊂ F2 by 3. of Definition 6.25. Thusπ2 induces a transformation of GBOs (F2,W2, E2) → (F1,W1, E1) and q2 ∈ F2. Byinduction, we have for any N > 0 a permissible sequence

(FN ,WN , EN ) πN→ · · · π2→ (F1,W1, E1) π1→ (F0,W0, E0) (50)

(50) induces a permissible sequence of basic objects with qλ1 = (qλ0 , 0) and centersqλi for i ≥ 1

(WλN , (a

λN , b

λ), EλN ) πN→ · · · π1→ (Wλ0 , (a

λ0 , b

λ), Eλ0 ). (51)

Observe that Li ⊂ Wλi for all i ≥ 1 so that qλi ∈ Wλ

i for all i ≥ 1. Set b′ = νqλ0 (aλ0 ).Suppose that (x1, . . . , xd) are regular parameters at qλ0 in Wλ

0 . Then (x1, . . . , xd, t)are regular parameters at qλ1 in Wλ

1 (where A1K = spec(K[t])), and thus there are

regular parameters (x1(N), x2(N), . . . , xd(N), t) in WλN at qλN which are defined by

xi = xi(N)tN−1 for 1 ≤ i ≤ d. t = 0 is a local equation of the exceptional divisor DλN

of WλN → Wλ

N−1.Suppose that f ∈ OWλ

0 ,qλ0

and νqλ0 (f) = r. Let K ′ = K(qλ0 ). We have an expression

f =∑

i1+···id≥r

ai1,... ,idxi11 · · ·x

idd ∈ K

′[[x1, . . . , xd]] = OWλ0 ,q

λ0

where ai1,... ,id ∈ K ′. Thus in K ′[[x1(N), . . . , xd(N), t]] = OWλN,qλN

we have an expan-sion

f =∑i1+···+id≥r ai1,... ,idx1(N)i1 · · ·xd(N)idt(i1+···+id)(N−1)

= t(N−1)r(∑

i1+···+id=r ai1,... ,idx1(N)i1 · · ·xd(N)id + tN−1ΩN)

Thus we have an expression f = t(N−1)rfN in OWλN,qN

, where t 6 | fN .Since νqλ0 (aλ0 ) = b′,

aλ0OWλN

= I(N−1)b′

DλN

KλN

where IDλN

6 | KλN . We thus have an expression

aλN = I(N−1)(b′−bλ)

DλN

KλN (52)

and νqλN

(aλN ) ≥ bλ since qN,λ ∈ FN ∩ WλN = Sing(aλN , b

λ).

Under the closed immersion WλN ⊂ Wλ

N we have (for N ≥ 1) DλN

= DN · WλN ,

where DλN

has dimension d and is irreducible. Since FλN ⊂ WλN it follows that

dim (FN ·DN ) = dim (FλN ·DN ) ≤ dim DλN

= d

and the following three conditions are equivalent

1. dim (FN ·DN ) = dim (FλN ·DN ) = d

2. DλN ⊂ FλN

3. (N − 1)(b′ − bλ) ≥ bλ.


Thus b′

bλ= 1 holds if and only if b′ − bλ = 0, which holds if and only if for any N

dim (FN ·DN ) < d. Thus the condition

νqλ0 (aλ0 )

bλ=b′

bλ= 1

is independent of λ.Now assume that b′ − bλ > 0. Then for N sufficently large, (N − 1)(b− bλ) ≥ bλ,

and FN ·DN = DλN

is a permissible center of dimension d for (FN ,Wn, EN ).We now define an extension of (50) by a sequence of transformations

(FN+S ,WN+S , EN+S)πN+S→ · · · πN+1→ (FN ,WN , EN ) πN→ · · · π1→ (F0,W0, E0)

(53)

where πN+i is the transformation with center YN+i = FN+i ·DN+i, and we assumethat dim YN+i = d for i = 0, 1, . . . , S − 1. Observe that

dim YN+i = d if and only if DλN+i

= DN+i · WλN+i ⊂ FN+i. (54)

We thus have YN+i = DλN+i

. (54) induces an extension of (51) by a sequence oftransformations

(WλN+S , (a

λN+S , b

λ), EN+S)πN+S→ · · ·

πN+1→ (WλN , (a

λN , b

λ), EN ) πN→ · · · π1→ (Wλ0 , (a

λ0 , b

λ), E0)(55)

The centers of πi in (54) are Dλj

for j = N, . . . , N + S − 1. Thus WN+i+1 → WN+i

is the identity for i = 0, . . . , S − 1.We see then that

aλN+i = I(N−1)(b′−bλ)−ibλ

DλN+i

KλN

for i = 1, . . . , S. The statement that the extension (55) is permissible is preciselythe statment that (N − 1)(b′ − bλ) ≥ Sbλ, which is equivalent to the statement thatdim FN+i ·DN+i = d for i = 0, 1, . . . , S − 1. This last statement is equivalent to

S ≤ `N = b (N − 1)(b′ − bλ)bλ

c

where bxc denotes the greatest integer in x. Thus `N is determined both by N andby the GBO (F0,W0, E0).

limN→∞

`NN − 1

=b′

bλ− 1

is defined in terms of the GBO, hence is independent of λ.

Theorem 6.28. Let (F0,W0, E0) be a GBO which admits a d-dimensional structure,and fix notation as in Definition 6.26. Let Fi be the closed sets associated to the GBO.Let Fλi = Sing(aλi , b

λ), and w-ordλi , tλi be the functions of Definition 6.13 defined forFλi .

Consider any permissible sequence of transformations

(Fr,Wr, Er)πr→ · · · π1→ (F0,W0, E0).

66 S. DALE CUTKOSKY

1. Then for 1 ≤ i ≤ r, there are functions w-orddi : Fi → Q such that thecomposition

Fλijλi→ Fi

w-orddi→ Qis the function w-ordλi : Fλi → Q.

2. Suppose that the centers Yi of πi+1 are such that Yi ⊂ Max w-orddi . Thenthere are functions tdi : Fi → Q× Z such that the composition

Fλijλi→ Fi

tdi→ Q× Z

is the function tλi : Fλi → Q× Z.

Proof. Let E0 = D1, . . . , Ds. The functions ti are completely determined byw-orddi , with the constraint that Yi ⊂ Max w-orddi for all i, so we need only showthat w-orddi can be defined for GBOs with d-dimensional structure.

The functions w-ordλ0 are equal to ordλ0 , which extend to ord0 on F0 by Proposition6.27. We thus have defined w-ord0 = ord0. Let Di be the exceptional divisor of πi,where i = i+ s, Dλ

i= Wλ

i ·Di.For each λ and i we have expressions

aλi = Icλ1

Dλ1

· · · Icλi

Dλi

aλi (56)

where aλi is the weak transform of aλ0 on Wλi and cj are locally constant functions on

Dλj

We further have that

aλi =1IbλDλi

aλi−1OWλi.

Suppose that ηi−1 is a generic point of an irreducible component Y ′i of Yi such thatηi−1 is contained in Wλ

i (and thus necessarily in Wλi ), and that qi ∈ Di is a point

which we do not require to be closed, but is contained in Dλi

= Wλi ·Di, and πi(qi) is

contained in the Zariski closure of ηi−1.From (56), we deduce

(aλi )qi =

1IbλDλi

Icλ1 (qi)

Dλ1

· · · Icλi−1(qi)

Dλi−1

I

(∑i−1

j=1cλj (ηi−1)νηi−1 (I

Dλ

j

)

)+νηi−1 (aλi−1)

Dλi

aλi

qi

(57)

Thus

cλi (qi)bλ

=

w-ordλi−1(ηi−1) +i−1∑j=1

cλj (ηi−1)bλ

νηi−1(IDλj

)

− 1. (58)

We further have

ordλi (qi) = w-ordλi (qi) +1bλ

i∑j=1

cλj (qj).

Suppose that q′i ∈ Fi, and define q′j for 0 ≤ j ≤ i − 1 by q′j = πj+1(q′j+1). Tosimplify notation, we can assume that q′j ∈ Yj for all j.

Let Y ′j be the irreducible component of Yj which contains q′j , and let η′j be thegeneric point of Y ′j for 0 ≤ j ≤ i− 1.


Suppose that λ ∈ Λ is such that q′i ∈ Wλi . We have associated points qj and ηj in

Wλj for 0 ≤ j ≤ i.We thus have sequences of points

q0, q1, . . . , qi and η0, η1, . . . , ηi−1 with qj , ηj ∈ Wλj for 0 ≤ j ≤ i

and

q′0, q′1, . . . , q′i and η′0, η′1, . . . , η′i−1 with q′j , η′j ∈Wλ

j for 0 ≤ j ≤ i.

From equation (58) and induction we see that the values

cλj (qj)bλ

for j = 1, . . . , i (59)

and w-ordλi (qi) are determined by

1. The rational numbers

ordd0(q′0), ordd1(q′1), . . . , orddi (q′i), (60)

ordd0(η′0), ordd1(η′1), . . . , orddi−1(η′i−1) (61)

2. The functions

Y H(q′s, j) =

1 if q′s ∈ Ys and Ys ⊂ Dj locally at qs0 if q′s 6∈ Ys or Ys 6⊂ Dj locally at qs

(62)

Our theorem now follows from Proposition 6.27, since w-ordλi (qi) is completelydetermined by the data (60), (61) and (62).

Proposition 6.29. Let B0 = (F0,W0, E0) be a GBO which admits a d-dimensionalstructure. Fix notation as in Definition 6.26. Let Fλi = Sing(aλi , b

λ).Consider a permissible sequence of transformations

Br = (Fr,Wr, Er)πr→ · · · π1→ (F0,W0, E0)

such that max w-ordr = 0. Then there are functions

Γr = Γr(Br) : Fr → IM = Z×Q× ZN

such that the composition

Fλrjλr→ Fr

Γr→ IM

is the function

Γ = Γ(Wλr , (a

λr , b

λ), Eλr ) : Fλr → IM

of Definition 6.16.

Proof. Suppose that E0 = D1, . . . , Ds. The function Γ on the basic objects(Wλ

r , (aλr , b

λ), Eλr ) depends only on the values of cibλ

for 1 ≤ i ≤ r (with the nota-tion of the proof of Theorem 6.28). Thus the conclusions follow from (59) of the proofof Theorem 6.28.

68 S. DALE CUTKOSKY

6.7. Resolution theorems for a general basic object.

Theorem 6.30. Let (Fd0 ,W0, Ed0 ) be a GBO with associated closed sets F di which

admits a d-dimensional structure. Fix notation as in Definition 6.26. Let Fλi =Sing(aλi , b

λ) = F di ∩ Wλi .

Consider a permissible sequence of transformations

(Fdr ,Wr, Edr ) πr→ · · · π1→ (Fd0 ,W0, E

d0 ) (63)

such that the center Yi of πi+1 satisfies

Yi ⊂ Max ti ⊂ Max w-ordi ⊂ Fifor i = 0, 1 . . . , r − 1 and max w-ordr > 0.

This condition implies w-ordi(q) ≤ w-ordi−1(πi(q)), ti(q) ≤ ti−1(πi(q)) for q ∈ Fiand 1 ≤ i ≤ r. Thus

max w-ord0 ≥ max w-ord1 ≥ · · · ≥ max w-ordr,

max t0 ≥ max t1 ≥ · · · ≥ max tr.

Let R(1)(Max tr) be the set of points where Max tr has dimension d− 1. Then R(1)is open and closed in Max tr and non-singular in Wr.

1. If R(1)(Max tr) 6= ∅ then R(1) is a permissible center for (Fdr ,Wr, Edr ) with

associated transformation

(Fdr+1,Wr+1, Edr+1)

πr+1→ (Fdr ,Wr, Edr )

and either R(1)(Max tr+1) = ∅ or max tr > max tr+1.2. If R(1)(Max tr) = ∅, there is a GBO (Fd−1

r ,Wr, Ed−1r ) with (d−1)-dimensional

structure and associated closed sets F d−1i such that if

(Fd−1N ,WN , E

d−1N )→ · · · → (Fd−1

r ,Wr, Ed−1r ) (64)

is a resolution (F d−1N = ∅) then (64) induces a permissible sequence of trans-

formations(FdN ,WN , E

dN )→ · · · → (Fdr ,Wr, E

dr )

such that the center Yi of πi+1 satisfies Yi ⊂ Max ti ⊂ Max w-ordi ⊂ F di forr ≤ i ≤ N−1, max tr = · · · = max tN−1, Max ti = F d−1

i for i = r, . . . , N−1and one of the following holds

a. F dN = ∅b. F dN 6= ∅ and max w-ordN = 0c. F dN 6= ∅, max w-ordN > 0 and max tN−1 > max tN .

We will now prove Theorem 6.30. Let E0 = D1, . . . , Ds LetDi be the exceptionaldivisor of πi where i = s+ i. Set bλr = bλ(max w-ordr) > 0. Let

r0 be the smallest integer such that max w-ordr0 = max w-ordr in (63).

There exists an open cover Wλ0 of W0 and a d-dimensional structure Bλ0 λ∈Λ of

(F0,W0, E0) where Bλ0 = (Wλ0 , (a

λ0 , b

λ), Eλ0 ) for λ ∈ Λ. Thus for λ ∈ Λ, we havesequences of transformations

(Wλr , (a

λr , b

λ), Eλr ) πr→ · · · π1→ (Wλ0 , (a

λ0 , b

λ), Eλ0 )


where the center Y λi of Wλi+1 → Wλ

i is contained in Max ti. There is an expression

aλr = Icλ1Dλ

1

· · · Icλr

Dλr

aλr .

We now define a basic object

(B′r)λ = (Wλ

r , ((a′r)λ, (b′)λ), Eλr )

on Wλr by

(a′r)λ =

aλr if bλr ≥ bλ

(aλr )bλ−bλr + (Ic

λ1Dλ

1

· · · Icλr

Dλr

)bλr if bλr < bλ,

(65)

(b′)λ =bλr if bλr ≥ bλbλr (bλ − bλr ) if bλr < bλ

Lemma 6.31. 1. Sing((a′r)λ, (b′)λ) = (Max w-ordr) ∩Wλ

r .2. (Wλ

r , ((a′r)λ, (b′)λ), Eλr ) is a simple basic object.

3. A transformation

(Wλr+1, ((a

′r+1)λ, (b′)λ), Eλr+1)→ (Wλ

r , ((a′r)λ, (b′)λ), Eλr )

with center Y λr induces a transformation

(Wλr+1, (a

λr+1, b

λ), Eλr+1)→ (Wλr , (a

λr , b

λ), Eλr )

such that the center Y λr ⊂ Max w-ordr ∩ Wλr .

4. Sing((a′r+1)λ, (b′)λ) = ∅ if and only if one of the following holds:a. max w-ordr > max w-ordr+1 orb. max w-ordr = max w-ordr+1 and (Max w-ordr) ∩Wλ

r = ∅.5. If max w-ordr = max w-ordr+1 then

Sing((a′r+1)λ, (b′)λ) = Max w-ordr+1 ∩ Wλr+1

and (a′r+1)λ is defined in terms of aλr+1 by (65).

Proof. 1. For q ∈ Wλr , q ∈ Max w-ordr if and only if νq(aλr ) ≥ bλr and νq(aλr ) ≥ bλ.

These conditions hold if and only if νq((a′r)λ) ≥ (b′)λ.

2. follows since νq(aλr ) ≤ bλr for q ∈ Wλr .

3. is immediate from 1.If bλr ≥ bλ, we have

aλr+1 = I−bλr

Dλr+1

ar = I−(b′)λ

Dλr+1

(a′r)λ = (a′r+1)λ. (66)

Suppose that bλr < bλ. Then

(aλr+1)bλ−bλr +

(Ic

λ1Dλ

1

· · · Icλr+1

Dλr+1

)bλr= (I−b

λr

Dλr+1

ar)bλ−bλr +

(Ic

λ1Dλ

1

· · · Icλr+1

Dλr+1

)bλr (67)

where

cλr+1(q) =r∑j=1

cj(η)νη(IDλj

) + bλr − bλ

70 S. DALE CUTKOSKY

if η is the generic point of the component of Y λr containing πr+1(q).

If we set K = Icλ1Dλ

1

· · · Icλr+1

Dλr+1

(the ideal sheaf on Wλr ) we see that (67) is equal to

(ar +K)I(bλ−bλr )bλrDλr+1

= (a′r+1)λ. (68)

Now 4. and 5. follow from 1. and equations(66), (67) and (68).

At q ∈ Max tr ∩ Wλr there exist N = η(q) hypersurfaces Dλ

i1, . . . , Dλ

iN∈ (E−r )λ

such thatq ∈ Dλ

i1 ∩ · · · ∩Dλin ∩Max w-ordr 6= ∅.

Thus there exists an affine neighborhood Uλ,φr of q such that

Max tr ∩ Uλ,φr = Dλ,φi1∩ · · · ∩Dλ,φ

in∩Max w-ordλ,φr (69)

(The superscript λ, φ denotes restriction to Uλ,φr ). Now define a basic object

(B′′r )λ,φ = (Uλ,φr , ((a′′r )λ,φ, (b′′)λ,φ), (E+r )λ,φ)

by

(a′′r )λ,φ = (a′r)λ,φ + (IDλ,φ

i1)(b′′)λ,φ + · · ·+ (IDλ,φ

iN

)(b′′)λ,φ

(b′′)λ,φ = (b′)λ(70)

Lemma 6.32. 1. Sing((a′′r )λ,φ, (b′′)λ,φ) = Max tr ∩ Uλ,φr .2. (Uλ,φr , ((a′′r )λ,φ, (b′′)λ,φ), (Er+)λ,φ) is a simple basic object.3. A transformation

(Uλ,φr+1, ((a′′r+1)λ,φ, (b′′)λ,φ), (E+

r+1)λ,φ)→ (Uλ,φr , ((a′′r )λ,φ, (b′′)λ,φ), (E+r )λ,φ)

with center Y λ,φr induces a transformation

(Uλ,φr+1, (aλ,φr+1, b

λ), Eλ,φr+1)→ (Uλ,φr , (aλ,φr , bλ), Eλ,φr )

such that Y λ,φr ⊂ Max tr.4. Sing((a′′r+1)λ,φ, (b′′)λ,φ) = ∅ if and only if one of the following conditions hold:

a. max tr > max tr+1 orb. max tr = max tr+1 and Max tr+1 ∩ Uλ,φr+1 = ∅.

5. If max tr = max tr+1 and Max tr+1∩Wλr+1 6= ∅, then Sing((a′′r+1)λ,φ, (b′′)λ,φ) =

Max w-ordr+1 ∩ Uλ,φr+1 and (a′′r+1)λ,φ is defined in terms of aλr+1 and (a′r+1)λ

by (63) and (65).

Proof. 1. For q ∈ Uλ,φ1 , q ∈ Max tr if and only if q ∈ Max w-ordr and η(q) = N ,which hold if and only if νq((a′r)

λ) ≥ (b′)λ (by 1. of Lemma 6.31) and νq(Iλ,φDij ) ≥ 1

for 1 ≤ j ≤ N which holds if and only if q ∈ Sing((a′′r )λ,φ, (b′′)λ,φ).2. follows since νq((a′′r )λ,φ) ≤ (b′′)λ,φ for q ∈ Uλ,φr .3. is immediate from 1.4. and 5. follow from 4. and 5. of Lemma 6.31, and the observation that if

max tr+1 = max tr and q ∈ Max tr+1∩Uλ,φr+1 then q must be on the strict transformsof the hypersurfaces Dλ,φ

i1, . . . , Dλ,φ

iN.


We now observe that the conclusions of Lemmas 6.31 and 6.32, which are formu-lated for transformations, can be naturally formulated for restrictions also. Then itfollows that the (B′′r )λ,φ define a GBO with d-dimensional structure on (Wr, (Edr )+)

(F ′′r ,Wr, (Edr )+) (71)

with associated closed sets F ′′i = Max ti.If Y is a permissible center for (71) then Y makes simple normal crossings with

E+r . The local description (69) and (70) then implies that Y makes simple normal

crossings with Er. Thus Y ⊂ Max tr is a permissible center for (Fdr ,Wr, Edr ). If the

induced transformations are

(F ′′r+1,Wr+1, (Edr+1)+)→ (F ′′r ,Wr, (Edr )+)

and(Fdr+1,Wr+1, E

dr+1)→ (Fdr ,Wr, E

dr )

then either max tr > max tr+1 in which case F ′′r+1 = ∅, or max tr = max tr+1 andF ′′r+1 = Max tr+1.

We will now verify that the assumptions of Theorem 6.24 hold for the simple basicobject (Uλ,φr , ((a′′r )λ,φ, (b′′)λ,φ), (E+

r )λ,φ).Define (by (65)) (B′r0)λ in the same way that we defined (B′r)

λ. Now 5. of Lemma6.31 implies (B′r)

λ is obtained by a sequence of transformations of the simple basicobject (B′r0)λ. (Eλr )+ is the exceptional divisor of the product of these transforma-tions.

Since (B′r0)λ is a simple basic object, we can find an open cover V λ,Ψr0 of Wλr0

with non-singular hypersurfaces (Vh)λ,Ψr0 in V λ,Ψr0 such that

I(Vh)λ,Ψr0⊂ 4(b′)λ−1((a′r0)λ,Ψ).

Let (Vh)λ,Ψr be the strict transform of (Vh)λ,ψr0 in V λ,ψr . By Lemma 6.21, (Vh)λ,ψr isnon-singular,

I(Vh)λ,ψr⊂ 4(b′)λ−1((a′r)

λ,ψ),

and we have that (Vh)λ,ψr makes SNCs with (E+r )λ,ψ. Let Uλ,φ,ψr = V λ,ψr ∩Uλ,φr . For

fixed λ, φ, Uλ,φ,ψr is an open cover of Uλ,φr . For q ∈ Uλ,φ,ψr ,

(I(Vh)λ,ψr)q ⊂

(4(b′)λ−1((a′r)

λ,ψ))q⊂ 4(b′′)λ,φ−1

((a′′r )λ,φ

)q

by (70). Thus the assumptions of Theorem 6.24 hold for

(B′′r )λ,φ = (Uλ,φr , ((a′′r )λ,φ, (b′′)λ,φ), (E+r )λ,ψ)

and Uλ,φ,ψr . 1. of Theorem 6.30 now follows from Theorem 6.24 applied to ourGBO of (71), with d-dimensional structure (B′′r )λ,φ.

If R(1)(Max tr) is empty, then by Theorem 6.24, applied to the GBO of (71), andits d-dimensional structure, (71) has a (d− 1)-dimensional structure, and we set

(Fd−1r ,Wr, E

d−1r ) = (F ′′r ,Wr, (Edr )+).

Given a resolution (64), by Lemma 6.32, we have an induced sequence

(FdN ,WN , EdN )→ · · · → (Fdr ,Wr, E

dr )

such that the conclusions of 2. of Theorem 6.30 hold.

72 S. DALE CUTKOSKY

Lemma 6.33. Let I1 and I2 be totally ordered sets. Suppose that

(WN , EN ) πN→ · · · π1→ (W0, E0) (72)

is a sequence of transformations of pairs together with closed sets Fi ⊂ Wi such thatthe center Yi of πi+1 is contained in Fi for all i, πi+1(Fi+1) ⊂ Fi for all i and FN = ∅.Assume that for 0 ≤ i ≤ N the following conditions hold:

1. There is an upper semi-continuous function gi : Fi → I1.2. There is an upper semi-continuous function g′i : Max gi → I2.3. Yi = Max g′i in (72)4. For any q ∈ Fi+1, with 0 ≤ i ≤ N − 2,

gi(πi+1(q)) ≥ gi+1(q) if πi+1(q) ∈ Yigi(πi+1(q)) = gi+1(q) if πi+1(q) 6∈ Yi

This condition implies that

max g0 ≥ max g1 ≥ · · · ≥ max gN−1

and if max gi = max gi+1, then πi+1(Max gi+1) ⊂ Max gi.5. If max gi = max gi+1 and q ∈ Max gi+1, with 0 ≤ i ≤ N − 2, then

g′i(πi+1(q)) > g′i+1(q) if πi+1(q) ∈ Yig′i(πi+1(q)) = g′i+1(q) if πi+1(q) 6∈ Yi

Then there are upper semi-continuous functions

fi : Fi → I1 × I2defined by

fi(q) =

(max gi,max g′i) if q ∈ Yifi+1(q) if q 6∈ Yi

(73)

where in the second case we can view q as a point on Wi+1 as Wi+1 → Wi is anisomorphism in a neighborhood of q. For q ∈ Fi+1,

fi(πi+1(q)) > fi(q) if πi+1(q) ∈ Yifi(πi+1(q)) = fi+1(q) if πi+1(q) 6∈ Yimax fi = (max gi,max g′i) and Max fi = Max g′i

(74)

Proof. We first show, by induction on i, that the fi of (73) are well defined functions.Since FN = ∅, YN−1 = FN−1. By assumption 3.,

Max g′N−1 = Max gN−1 = FN−1,

so we can define

fN−1(q) = (gN−1(q), g′N−1(q)) = (max gN−1,max g′N−1)

for q ∈ FN−1.Suppose that q ∈ Fr. If q ∈ Yr, we define

fr(q) = (gr(q), g′r(q)) = (max gr,max g′r).

Suppose that q 6∈ Yr. As πi+1(Fi+1) ⊂ Fi for all i, we have an isomorphism Fr−Yr ∼=Fr+1 − Dr+1, where Dr+1 is the reduced exceptional divisor of Wr+1 → Wr. SinceFN = ∅, we can define fr(q) = fr+1(q) for q ∈ Fr − Yr.

The properties (74) are immediate from the assumptions and (73).


It remains to prove that fr is upper semi-continuous. We prove this by descendinginduction on r. Fix α = (α1, α2) ∈ I1 × I2. We must prove that q ∈ Fr | fr(q) ≥ αis closed. If max fr < α the set is empty. If max fr ≥ α,

q ∈ Fr | fr(q) ≥ α = Max g′r ∪ πr+1(q′ ∈ Fr+1 | fr+1(q′) ≥ α)which is closed by upper semi-continuity of fr+1 and properness of πr+1.

Theorem 6.34. Fix an integer d ≥ 0. There is a totally ordered set Id and functionsfdi with the following properties:

1. For each GBO (Fd0 ,W0, Ed0 ) with a d-dimensional structure and associated

closed subsets F di there is a function

fd0 : F d0 → Id

with the property that Max fd0 is a permissible center for (W0, Ed0 ).

2. If a sequence of transformations with permissible centers Yi

(Fdr ,Wr, Edr )→ · · · → (Fd0 ,W0, E

d0 ) (75)

and functions fdi : F di → Id, i = 0, . . . , r − 1, have been defined with theproperty that Yi = Max fdi , then there is a function fdr : F dr → Id such thatMax fdr is permissible for (Wr, E

dr ).

3. For each GBO (Fd0 ,W0, Ed0 ) with a d-dimensional structure, there is an index

N so that the sequence of transformations

(FdN ,WN , EdN )→ · · · → (Fd0 ,W0, E

d0 ). (76)

constructed by 1. and 2. is a resolution (FN = ∅).4. The functions fdi in 2. have the following properties:

a. If q ∈ Fi, with 0 ≤ i ≤ N − 1 and if q 6∈ Yi, then fdi (q) = fdi+1(q).b. Yi = Max fdi for 0 ≤ i ≤ N − 1 and

max fd0 > max fd1 > · · · > max fdN−1

c. For 0 ≤ i ≤ N − 1 the closed set Max fdi is smooth, equidimensional andits dimension is determined by the value of max fdi .

Proof. (of Theorem 6.34) The proof is by induction on d.First assume that d = 1. Set I1 to be the disjoint union I1 = Q×Zt ∞, where

Q× Z is ordered lexicographically, and ∞ is the maximum of I1.Suppose that (F1

0 ,W0, E10) is a GBO with one dimensional structure, and with

associated closed sets F 1i . Define f1

0 = t10 (t10 is defined by Theorem 6.28). Theclosed set F 1

0 is (locally) a codimension greater than or equal to one subset of a onedimensional subvariety of W0, so dim F0 = 0. Thus R(1)(Max t10) 6= ∅, and we arein the situation of 1. of Theorem 6.30. If we perform the permissible transformationwith center Max t10,

(F11 ,W1, E

11)→ (F1

0 ,W0, E10)

we have max t10 > max t11. We can thus define a sequence of transformations

(F1r ,Wr, E

1r )→ · · · → (F1

0 ,W0, E10)

where each transformation has center Max t1i , f1i = t1i : F 1

i → I1 and

max t10 > · · · > max t1r.

74 S. DALE CUTKOSKY

Since there is a natural number b such that max t1i ∈ 1b!Z× Z for all i, there is an r

such that the above sequence is a resolution.Now assume that d > 1 and that the conclusions of Theorem 6.34 hold for GBOs

of dimension d − 1. Thus there is a totally ordered set Id−1, and functions fd−1i

satisfying the conclusions of the Theorem.Let I ′d be the disjoint union

I ′d = Q× Z t IM t ∞,

where IM is the ordered set of Definition 6.16. We order I ′d as follows: Q × Z hasthe lexicographic order. If α ∈ Q×Z and β ∈ IM then β < α and ∞ is the maximalelement of I ′d. Define Id = I ′d × Id−1 with the lexicographic ordering.

Suppose that (Fd0 ,W0, Ed0 ) is a GBO of dimension d, with associated closed sets

F di . Define g0 : F d0 → I ′d by

g0(q) = td0(q),

and g′0 : Max g0 → Id−1 by

g′0(q) =∞ if q ∈ R(1)(Max td0)fd−1

0 (q) if q 6∈ R(1)(Max td0)

where fd−10 is the function defined by the GBO of dimension d− 1 (Fd−1

0 ,W0, Ed−10 )

of 2. of Theorem 6.30.Assume that we have now inductively defined a sequence of transformations

(Fdr ,Wr, Ed1 ) πr→ · · · π1→ (Fd0 ,W0, E

d0 ) (77)

and we have defined functions

gi : F di → I ′d and g′i : Max gi → Id−1

satisfying the assumptions 1. - 5. of Lemma 6.33 (with I1 = I ′d and I2 = Id−1) fori = 0, . . . , r − 1. In particular, the center of πi+1 is Yi = Max g′i.

If F dr 6= ∅, define gr : F dr → I ′d by

gr(q) =

Γ(Br)(q) if w-orddr(q) = 0tdr(q) if w-orddr(q) > 0.

where Γ(Br) is the function defined in Proposition 6.29 for the GBOBr = (Fdr ,Wr, Edr ).

Define g′r : Max gr → Id−1 by

g′r(q) =

∞ if w-orddr(q) = 0∞ if q ∈ R(1)(Max tdr) and w-ordr(q) > 0fd−1r (q) if q 6∈ R(1)(Max tdr) and w-ordr(q) > 0

where fd−1r is the function defined by the GBO of dimension d− 1 (Fd−1

r ,Wr, Ed−1r )

in 2. of Theorem 6.30. Observe that if max w-ordr > 0, then the center Yi of πi+1 in(77) is defined by Yi = Max g′i ⊂ Max tdi for 0 ≤ i ≤ r− 1. Thus (if max w-ordr > 0)we have that (77) is a sequence of the form of (63) of Theorem 6.30.

Now Theorem 6.30, our induction assumption, and Lemma 6.17 show that thesequence (77) extends uniquely to a resolution

(FN ,WN , EN )→ · · · → (F0,W0, E0)


with functions gi and g′i satisfying the assumptions 1. - 5. of Lemma 6.33. By Lemma6.33, the functions (gi, g′i) extend to functions

F di → Id = I ′d × Id−1

satisfying the conclusions of Theorem 6.34.Consider the values of the functions fdi which we constructed at a point q ∈ Fi.

fdi (q) has d coordinates, and has one of the following three types:1. fdi (q) = (td(q), td−1(q), . . . , td−r(q),∞, · · · ,∞).2. fdi (q) = (td(q), td−1(q), . . . , td−r(q),Γ(q),∞, · · · ,∞).3. fdi (q) = (td(q), td−1(q), . . . , t1(q)).

Each coordinate is a function defined for a GBO of the corresponding dimension.ti(q) = (w-ordi(q), ηi(q)) with w-ordi(q) > 0.

In case 1. td−r is such that q ∈ R(1)(Max td−r), that is Max td−r has codimension1 in a d− r dimensional GBO, and dim(Max fdi ) = d− r − 1.

In case 2. w-ordd−(r+1)(q) = 0 and Γ is the function defined in Proposition 6.29 fora monomial GBO and dim(Max fdi ) = (d−r−Γ1−1, where Γ1 is the first coordinateof max Γ.

In case 3. dim(Max fdi ) = 0.Thus Max fdi is non-singular and equidimensional, and 4.c. follows.

Theorem 6.35. Suppose that d is a positive integer. Then there are a totally orderedset I ′d and functions pdi with the following properties:

1. For each pair (W0, E0) with d = dim W0 and ideal sheaf J0 ⊂ OW0 there is afunction

pd0 : V (J0)→ I ′dwith the property that Max pd0 ⊂ V (J0) is permissible for (W0, E0).

2. If a sequence of transformations of pairs with centers Yi ⊂ V (J i)

(Wr, Er)→ · · · → (W0, E0)

and functions pdi : V (J i)→ I ′d, 0 ≤ i ≤ r−1 has been defined with the propertythat Yi = Max pdi and V (Jk) 6= ∅, then there is a function pdr : V (Jr) → I ′dsuch that Max pdr is permissible for (Wr, Er).

3. For each pair (W0, E0) and J0 ⊂ OW0 there is an index N such that thesequence of transformations

(WN , EN )→ · · · → (W0, E0)

constructed inductively by 1. and 2. is such that V (JN ) = ∅. The corre-sponding sequence will be called a “Strong Principalization” of J0.

4. Property 4. of Theorem 6.34 holds.

Proof. Let J0 = J0, and set b0 = max νJ0. By Theorem 6.34, there exists a resolution

(WN1 , (JN1 , b0), EN1)→ · · · → (W0, (J0, b0), E0)

of the simple basic object (W0, (J0, b0), E0). Thus we have that JN1 (which is theweak transform of J0 on OWN1

) satisfies b1 = max νJN1< b0. We now can construct

by Theorem 6.34 a resolution

(WN2 , (JN2 , b1), EN2)→ · · · → (WN1 , (JN1 , b1), EN1)

76 S. DALE CUTKOSKY

of the simple basic object (WN1 , (JN1 , b1), EN1).After constructing a finite number of resolutions of simple basic objects in this way,

we have a sequence of transformations of pairs

(WN , EN )→ · · · → (W0, E0) (78)

such that the strict transform JN of J0 on WN is JN = OWN.

We have upper semi-continuous functions

fdi : Max νJi → Id

satisfying the conclusions of Theorem 6.34 on the resolution sequences

(WNi+1 , (JNi+1 , bi), ENi+1)→ · · · → (WNi , (JNi , bi), ENi)

Now apply Lemma 6.33 to the sequence (78), with Fi = V (J i), gi = νJi , g′i = fdi .

We conclude that there exist functions pdi : V (J i) → I ′d = Z × Id with the desiredproperties.

6.8. Resolutions of singularities of algebraic varieties in characteristic zero.Recall our conventions on varieties in section 1.1 on Notations.Theorem 6.36. (principalization of ideals) Suppose that I is an ideal sheaf on a non-singular variety W over a field of characteristic zero. Then there exists a sequence ofmonodial transforms

π : W1 →W

which is an isomorphism away from the closed locus of points where I is not locallyprincipal, such that IOW1 is locally principal.

Proof. We can factor I = I1I2 where I1 is an invertible sheaf, and V (I2) is the set ofpoints where I is not locally principal. Then we apply Theorem 6.35 to J0 = I2.

Theorem 6.37. (embedded resolution of singularities) Suppose that X is an algebraicvariety over a field of characteristic zero which is embedded in a non-singular varietyW . Then there exists a birational projective morphism

π : W1 →W

such that π is a sequence of monodial transforms, π is an embedded resolution 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 that the weaktransform of J0 on WN is JN = OWN

.With the notation of the proofs of Theorem 6.34 and Theorem 6.35, if X0 has

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

pd0(q) = (1, (1, 0), . . . , (1, 0), 0, . . . , 0) ∈ I ′dwhere there are r copies of (1, 0) followed by d − r zeros. The function pd0 is thusconstant on the non-empty open set Reg(X0) of non-singular points of X0. Let thisconstant be c ∈ I ′d. By property 4. of Theorem 6.34 there exists a unique indexr ≤ N − 1 such that max pdr = c. Since Wr → W0 is an isomorphism over the denseopen set Reg(X0), the strict transform Xr of X0 must be the union of the irreducible


components of the closed set Max pdr of Wr. Since Max pdr is a permissible center,Xr is non-singular and makes simple normal crossings with the exceptional divisorEr.

Theorem 6.38. (resolution of singularities) Suppose that X is an algebraic varietyover a field of characteristic zero. Then there is a resolution of singularities

π : X1 → X

such that π is a projective morphism, which is an isomorphism away from the singularlocus of X.

Proof. This is immediate from Theorem 6.37, after choosing an embedding of X intoa projective space W .

Theorem 6.39. (resolution of indeterminacy) Suppose that K is a field of charac-teristic zero and φ : W → V is a rational map of projective K-varieties. Then thereexists a projective birational morphism π : W1 → W such that W1 is non-singularand a morphism λ : W1 → V such that λ = φ π. If W is non-singular, then π is aproduct of monoidal transforms.

Proof. By Theorem 6.38, there exists a resolution of singularities ψ : W1 →W . Afterreplacing W with W1 and φ with ψ φ, we may assume that W is non-singular. LetΓφ be the graph of φ, with projections π1 : Γφ → W and π2 : Γφ → V . As π1 isbirational and projective, there exists an ideal sheaf I ⊂ OW such that Γφ ∼= B(I)is the blow up of I (Theorem 4.5). By Theorem 6.36, there exists a principalizationπ : W1 → W of I. The Universal property of blowing up (Theorem 4.2) now showsthat there is a morphism λ : W1 → V such that λ = φ π.

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

Hi(Y,OY ) ∼= Hi(X,OX) ∀ i.

Proof. By Theorem 6.39, there exists a commutative diagram of projective morphisms

Zf→ Y

g ↓ πX

such that g is a product of blowups of nonsingular subvarieties,

g : Z = Zngn→ Zn−1

gn−1→ . . .g2→ Z1

g1→ Z0 = X.

We have ([60] or Lemma 2.1 [29])

Rigj∗OZj =

0, i > 0OZj−1 , i = 0

Thus, by the Leray spectral sequence,

Rig∗OZ =

0, if i > 0OX , if i = 0 (79)

andg∗ : Hi(X,OX) ∼= Hi(Z,OZ)

78 S. DALE CUTKOSKY

for all i. Now, by considering the commutative diagram

Hi(Z,OZ)f∗← Hi(Y,OY )

g∗ ↑ π∗Hi(X,OX)

we conclude that π∗ is one-to-one. To show that π∗ is an isomorphism we now onlyneed to show that f∗ is also one-to-one.

Resolution of indeterminancy also gives a new diagram

Wγ→ Z

β ↓ fY

where β is a product of blowups of nonsingular subvarieties, so we have

β∗ : Hi(Y,OY ) ∼= Hi(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 give a completeproof only the case of a hypersurface. In the embedded resolution constructed inTheorem 6.37, which induces the resolution of 6.38, if X is not a hypersurface, theremay be monodial transforms Wi+1 →Wi whose centers are not contained in the stricttransform Xi of X, so that the induced morphism Xi+1 → Xi of strict transforms ofX will in general not be a monodial transform.Theorem 6.41. (A stronger theorem of resolution of singularities) Suppose that Xis an algebraic variety over a field of characteristic zero. Then there is a resolutionof singularities

π : X1 → X.

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

Proof. In the case of a hypersurface X = X0 embedded in a non-singular varietyW = W0, the proof given in Theorem 6.37 actually produces a resolution of the kindasserted by this theorem, since the resolution sequence is constructed by patchingtogether resolutions of the simple basic objects (Wi, (J i, b), Ei), where J i is the weaktransform of the ideal sheaf J0 of X0 in W0. Since J0 is a locally principal ideal,and each transformation in these sequences is centered at a non-singular subvarietyof Max J i, the weak transform J i is actually the strict transform of J0.

For the case of a general variety X, we choose an embedding of X in a projectivespace W . We then modify the above proof by considering the Hilbert Samuel functionof X. It can be shown that there is a simple basic object (W0, (K0, b), E0) such thatMax νK0 is the maximal locus of the Hilbert Samuel function. In the constructionof pdi of Theorem 6.35 we use νK0 in place of νJi . Then the resolution of this basicobject induces a sequence of monodial transformations of X, such that the maximumof the Hilbert Samuel function has dropped.

More details about this process are given, for instance, in [51], [13] and [73].

Exercises:


1. Follow the algorithm of this chapter to construct an embedded resolution ofsingularities 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 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 typeK-scheme. Prove that there exists a proper birational morphism π : Y → Xsuch that Y is non-singular.

4. Suppose that K is a field of characteristic zero and X is a reduced finite typeK-scheme. Prove that there exists a proper birational morphism π : Y → Xsuch 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 thefunction pd0 of Theorem 6.35. Show that pd0(q) depends only on the localbasic object (U,E) and the ideal (J0)q ⊂ OW0,q. In particular, pd0(q) isindependent of K-algebra isomorphism of OX,q which takes J0OX,q toJ0OX,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)) = pd0(q) for all q ∈ V (J0).

c. Suppose that Θ : W0 → W0 is a K-automorphism, and Y ⊂ W0 is anonsingular subvariety such that Θ(Y ) = Y . Let π1 : W1 → W bethe monoidal transform centered at Y . Show that Θ extends to a K-automorphism Θ : W1 → W1 such that Θ(π1(q)) = π1(Θ(q)) for allq ∈W1 and Θ(D) = D if D is the exceptional divisor of π1.

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

πi(Θ(q)) = Θ(πi(q))

for all q ∈ Wi. Furthermore, the functions pdi of Theorem 6.35 satisfypdi (Θ(q)) = pdi (q) for all q ∈Wi.

7. (Equivariant resolution of singularities) Suppose thatX is a variety over a fieldK of characteristic zero. Suppose that G is a group of K-automorphisms of X.Show that there exists an equivariant resolution of singularities π : X1 → Xof X. That is, G extends to a group of K-automorphisms of X so thatπ(Θ(q)) = Θ(π(q)) for all q ∈ X1. More generally, deduce equivariant versionsof all the other theorems of Section 6.8 and of exercises 3 and 4.

7. Resolution of surface singularities in positive characteristic

7.1. Resolution and some invariants. In this chapter we prove resolution of sin-gularities for surfaces over an algebraically closed field of characteristic p ≥ 0. Thecharacteristic 0 proof in Section 5.1 depended on the existence of hypersurfaces of

80 S. DALE CUTKOSKY

maximal contact, which is false in positive characteristic (see the exercises at the endof this chapter). We prove the following theorem in this chapter.

Theorem 7.1. Suppose that S is a surface over an algebraically closed field K ofcharacteristic p ≥ 0. Then there exists a resolution of singularities

Λ : T → S.

Theorem 7.1 in the case when S is a hypersurface follows from induction on r =max t | Singt(S) 6= 0 in Theorems 7.6, 7.7 and 7.8. We then obtain Theorem 7.1 ingeneral from Theorem 4.8 and Theorem 4.10.

For the remainder of this chapter we will assume that K is an algebraically closedfield of characteristic p ≥ 0, V is a non-singular 3 dimensional variety over K, and Sis a surface in V .

It follows from Theorem 10.19 that for t ∈ N,

Singt(S) = p ∈ S | νp(S) ≥ t

is Zariski closed.Let

r = maxt | Singt(S) 6= ∅.Suppose that p ∈ Singr(S) is a closed point, f = 0 is a local equation of S at p,

and (x, y, z) are regular parameters at p in V (or in OV,p). There is an expansion

f =∑

i+j+k≥r

aijkxiyjzk

with aijk ∈ K in OV,p = K[[x, y, z]]. The leading form of f is defined to be

L(x, y, z) =∑

i+j+k=r

aijkxiyjzk.

We define a new invariant, τ(p), to be the dimension of the smallest linear subspaceM of the K-subspace spanned by x, y and z in K[x, y, z] such that L ∈ k[M ]. Thissubspace is uniquely determined. This dimension is in fact independent of choice ofregular parameters (x, y, z) at p (or in OV,p). If x, y, z are regular parameters in OV,p,we will call the subvariety N = V (M) of spec(OV,p) to be an “approximate manifold”to S at p. If (x, y, z) are regular parameters in OV,p, we call N = V (M) ⊂ spec(OV,p)a (formal) approximate manifold to S at p. M is dependent of our choice of regularparameters at p. Observe that

1 ≤ τ(q) ≤ 3.

If there is a non-singular curve C ⊂ Singr(S) such that q ∈ C, then τ(q) ≤ 2.

Example 7.2. Let f = x2 + y2 + z2 + x5. If char(K) 6= 2, then τ = 3 and x = y =z = 0 are local equations of the approximate manifold at the origin.

However, if char(K) = 2, then τ = 1 and x + y + z = 0 is a local equation of anapproximate manifold at the origin.

Example 7.3. Let f = y2 + 2xy + x2 + z2 + z5, char(K) > 2. L = (y + x)2 + z2 sothat τ = 2 and x + y = z = 0 are local equations of an approximate manifold at theorigin.


Lemma 7.4. Suppose that q ∈ S is a closed point such that νq(S) = r, τ(q) = n(with 1 ≤ n ≤ 3), L1 = · · · = Ln = 0 are local equations at q of a (possibly formal)approximate manifold N of S at q. Let π : V1 → V be the blow up of q, E = π−1(q)be the exceptional divisor of π, S1 be the strict transform of S on V1, N1 be the stricttransform of N on V1. Suppose that q1 ∈ π−1(q). Then νq1(S1) ≤ νq(S), and ifνq1(S1) = r, then τ(q) ≤ τ(q1) and q1 ∈ N1 ∩ E.

We further have that N1 ∩ E = ∅ if τ(q) = 3, N1 ∩ E is a point if τ(q) = 2 andN1 ∩ E is a line if τ(q) = 1.

Proof. Let f = 0 be a local equation of S at q, (x, y, z) be regular parameters at psuch that

1. x = y = z = 0 are local equations of N if τ(q) = 3,2. y = z = 0 are local equations of N if τ(q) = 2,3. z = 0 is a local equation of N if τ(q) = 1.

In OV,p = K[[x, y, z]], write

f =∑

aijkxiyjzk = L+G

where L is the leading form of degree r, andG has order> r. q1 has regular parameters(x1, y1, z1) such that one of the following cases hold:

1. x = x1, y = x1(y1 + α), z = x1(z1 + β) with α, β ∈ K2. x = x1y1, y = y1, z = y1(z1 + β) with β ∈ K3. x = x1z1, y = y1z1, z = z1

Suppose that the first case holds. S1 has a local equation f1 = 0 at q1 such that

f1 = L(1, y1 + α, z1 + β) + x1Ω.

Thus νq1(f1) ≤ r, and νq1(f1) = r implies

L(1, y1 + α, z1 + β) =∑

i+j+k=r

aijk(y1 + α)j(z1 + β)k =∑j+k=r

bjkyj1zk1

for some bjk ∈ k. Thus

L(1,y

x,z

x) =

∑j+k=r

(y

x− α)j(

z

x− β)k

implies V (y − αx, z − βx) ⊂ N . We conclude that τ(q) ≤ 2 if νq1(S1) = νq(S).Suppose that τ(q) = 2 and νq1(S1) = νq(S). Then we must have that α = β = 0,

so that q1 ∈ N1. We further have

f1 = L(y1, z1) + x1Ω

so that τ(q1) ≥ τ(q).If τ(q) = 1, then L = czr for some (non-zero) c ∈ K, and we must have β = 0, so

that q1 ∈ N1. We further have

f1 = L(z1) + x1Ω

so that τ(q1) ≥ τ(q).There is a similar analysis in cases 2 and 3.

82 S. DALE CUTKOSKY

Lemma 7.5. Suppose that C ⊂ Singr(S) is a non-singular curve and q ∈ C is a closedpoint. Let π : V1 → V be the blow up of C, E = π−1(C) be the exceptional divisor,S1 be the strict transform of S on V1. Suppose that (x, y, z) are regular parametersin OV,q (or OV,q) such that x = y = 0 are local equations of C at q, and x = 0 orx = y = 0 are local equations of a (possible formal) approximate manifold N of Sat q. Let N1 be the strict transform of N on V1. Suppose that q1 ∈ π−1(q). Thenνq1(S1) ≤ r, and if νq1(S1) = r then τ(q) ≤ τ(q1) and q1 ∈ N1 ∩ E.

Furthermore, N1 ∩ E = ∅ if τ(q) = 2 and N1 ∩ E is a curve which maps isomor-phically onto spec(OC,q) (or spec(OC,q) in the case where N is formal) if τ(q) = 1.

We omit the proof of Lemma 7.5 as it is similar to Lemma 7.4.

Theorem 7.6. There exists a sequence of blow ups of points in Singr(Si)

Vn → Vn−1 → · · · → V,

where Si is the strict transform of S on Vi, so that all curves in Singr(Sn) are non-singular.

Proof. This is possible by Corollary 4.4 and since (by Lemma 7.4) the blow up of apoint in Singr(Si) can introduce at most one new curve into Singr(Si+1), which mustbe non-singular.

Theorem 7.7. Suppose that all curves in Singr(S) are non-singular, and

· · · → Vn → Vn−1 → · · · → V

is a sequence of blow ups of non-singular curves in Singr(Si) where Si is the stricttransform of S on Vi. Then Singr(Si) is a union of non-singular curves and a finitenumber of points for all i. Further, this sequence in finite. That is, there exists an nsuch that Singr(Sn) is a finite set.

Proof. If π : V1 → V is the blow up of the non-singular curve C in Singr(S), andC1 ⊂ Singr(S1)∩π−1(C) is a curve, then by Lemma 7.5, C1 is non-singular and mapsisomorphically onto C. Furthermore, the strict transform of a non-singular curvemust be non-singular, so Singr(Si) must be a union of non-singular curves, and afinite number of points for all i.

Suppose that we can construct an infinite sequence

· · · → Vn → Vn−1 → · · · → V

by blowing up non-singular curves in Singr(Si). Then there exist curves Ci ⊂Singr(Vi) for all i such that Ci+1 → Ci are dominant morphisms for all i, and

νOVi,Ci (ISi,Ci) = r

for all i. We have an infinite sequence

OV,C → OV1,C1 → · · · → OVn,Cn → · · · (80)

of local rings, where, after possibly reindexing to remove the maps which are theidentity, each homomorphism is a localization of the blow up of the maximal idealof a 2 dimensional regular local ring. As explained in the characteristic zero proofof resolution of surface singularities (Lemma 5.10), we can view the OSi,Ci as local


rings of a closed point of a curve on a non-singular surface defined over a field oftranscendence degree 1 over K. By Theorem 3.12 (or Corollary 4.4).

νOVn,Cn (ISn,Cn) < r

for large n, a contradiction.

We can now state the main resolution theorem.Theorem 7.8. Suppose that Singr(S) is a finite set. Consider a sequence of blowups

· · · → Vn → · · · → V2 → V1 → V (81)

where Vi+1 → Vi is the blow up of a curve in Singr(Si) if such a curve exists, andVi+1 → Vi is the blow up of a point in Singr(Si) otherwise. Let Si be the stricttransform of S on Vi. Then Singr(Si) is a union of non-singular curves and a finitenumber of points for all i. Further, this sequence is finite. That is, there exists Vnsuch that Singr(Sn) = ∅.

Theorem 7.8 is an immediate consequence of Theorem 7.9 below, and Theorems7.6 and 7.7.Theorem 7.9. Let the assumptions be as in the statement of Theorem 7.8, andsuppose that q ∈ Singr(S) is a closed point. Then there exists an n such that allclosed points qn ∈ Sn such that qn maps to q and νqn(Sn) = r satisfy τ(qn) > τ(q).

The proof of this Theorem is very simple if τ(q) = 3. By Lemma 7.4 the multiplicitymust drop at all points above q in the blow up of q. We will prove Theorem 7.9 in thecase τ(q) = 2 in Section 7.2 and prove Theorem 7.9 in the case τ(q) = 1 in Section7.3.

7.2. τ(q) = 2. In this section we prove Theorem 7.9 in the case that τ(q) = 2.Suppose that τ(q) = 2, and that there exists a sequence (81) of infinite length. We

can then choose an infinite sequence of closed points qi ∈ Singr(Si) such that qi+1

maps to qi for all i. By Lemma 7.4, we have τ(qi) = 2 for all i. and that each qi isisolated in Singr(Si).

Thus each map Vi+1 → Vi is either the blow up of the unique point qi ∈ Singr(Si)which maps to q or Vi+1 → Vi is an isomorphism over q. After reindexing, we mayassume that each map is the blow up of a point qi such that qi+1 maps to qi for all i.Let Ri = OVi,qi . We have a sequence

R0 → R1 → · · · → Ri → · · · . (82)

of infinite length, where each Ri+1 is the completion of a local ring of a closed pointof the blow up of the maximal ideal of spec(Ri), and νqi(Si) = r and τ(qi) = 2 forall i. We will derive a contradiction. Since τ(q) = 2, there exist regular parameters(x, y, z) in R0 such that a local equation f = 0 of S in R0 has the form

f =∑

i+j+k≥r

aijkxiyjzk

and the leading form of f is

L(x, y) =∑i+j=r

aij0xiyj .

84 S. DALE CUTKOSKY

For an arbitrary seriesg =

∑bijkx

iyjzk ∈ R0,

define

γxyz(g) = min k

r − (i+ j)| bijk 6= 0 and i+ j < r ∈ 1

r!N ∪ ∞

with the convention that γxyz(g) =∞ if and only if bijk = 0 whenever i+ j < r. Wehave γxyz(g) < 1 if and only if νR0(g) < r. Furthermore, γxyz(g) = ∞ if and only ifg ∈ (x, y)r. Set γ = γxyz(g). Define

[g]xyz =∑

(i+j)γ+k=rγ

bijkxiyjzk. (83)

There is an expansion

g = [g]xyz +∑

(i+j)γ+k>rγ

bijkxiyjzk. (84)

Returning to our series f which is a local equation of S, set γ = γxyz(f),

Tγ = (i, j) | k

r − (i+ j)= γ, i+ j < r, for some k such that aijk 6= 0.

Then

[f ]xyz = L+∑

(i,j)∈Tγ

ai,j,γ(r−i−j)xiyjzγ(r−i−j). (85)

Definition 7.10. [f ]xyz is solvable if γ ∈ N and there exists α, β ∈ K such that

[f ]xyz = L(x− αzγ , y − βzγ).

Lemma 7.11. There exists a change of variables

x1 = x−n∑i=1

αizi, y1 = y −

n∑i=1

βizi

such that [f ]xyz is not solvable.Remark 7.12. We can always choose (x, y, z) that are regular parameters in OV0,q0 .Then (x, y, z) are also regular parameters in OV0,q0 . However, since we are onlyblowing up maximal ideals in (81), we may work with any formal system of regularparameters (x, y, z) in R0 (see Lemma 5.3 ).

Proof. (of Lemma 7.11) If there doesn’t exist such a change of variables then thereexists an infinite sequence of changes of variables for n ∈ N

xn = x−n∑i=1

αizγi , yn −

n∑i=1

βizγi

such that γn ∈ N for all n, γn+1 = γxnynz(f), and γn+1 > γn for all n. Let

x∞ = x−∞∑i=1

αizγi , y∞ = y −

∞∑i=1

βizγi .

Thusγ∞ = γx∞,y∞,z(f) =∞


and

f ∈ (x−∞∑i=1

αizi, y −

∞∑i−1

βizi)r ⊂ R0.

But by construction of V0, p is isolated in Singr(S). If I ⊂ OV,q is the reduced idealdefining Singr(S) at q, then by Remark 10.21, I = IR0 is the reduced ideal whosesupport is the locus where f has multiplicity r in spec(R0). Thus (x, y, z) = I ⊂(x−

∑αiz

i, y −∑βiz

i), which is impossible.

We may thus assume that [f ]xyz is not solvable. Since the leading form of f isL(x, y) and τ(p) = 2, R1 has regular parameters (x1, y1, z1) defined by

x = x1z1, y = y1z1, z = z1. (86)

Let f1 = fzr = 0 be a local equation of the strict transform of f in R1.

Lemma 7.13. 1. γx1y1z(f1) = γxyz(f)− 1.2. [f1]x1y1z = 1

zr [f ]xyz so [f ]xyz not solvable implies [f1]xyz is not solvable.3. If γx1y1z(f1) > 1, Then the leading form of f1 is L(x1, y1) = 1

zrL(x, y).

Proof. The Lemma follows from substitution of (86) in (83), (84) and (85).

We claim that the case γx1,y1,z(f1) = 1 cannot occur (with our assumption thatνq1(S1) = r and τ(q1) = 2). Suppose that it does. Then the leading form of f1 is[f1]x1y1z and there exists a form Ψ of degree r and a, b, c, d, e, f ∈ K such that

[f1]x1y1z = Ψ(ax1 + by1 + cz, dx1 + ey1 + fz).

We have an expression

L(x1, y1) + zΩ = Ψ(ax1 + by1 + cz, dx1 + ey1 + fz)

so thatL(x, y) = Ψ(ax+ by, dx+ ey).

We have ae− bd 6= 0 since τ(q) = 2. By Lemma 7.14 below, [f1]x1y1z is thus solvable,a contradiction to 2. of Lemma 7.13.

Since γx1y1z(f1) < 1 implies νR1(f1) < r, it now follows from Lemma 7.13 that aftera finite sequence of blow ups R0 → Ri, where Ri has regular parameters (xi, yi, zi)with

x = xizii , y = yiz

ii , z = zi

by 3. of Lemma 7.13, we reach a reduction νqi(Si) < r or νqi(Si) = r, τ(qi) = 3, acontradiction to our assumption that (82) has infinite length. Thus Theorem 7.9 hasbeen proven when τ(q) = 2.Lemma 7.14. Let Φ = Φ(x, y), Ψ = Ψ(u, v) be forms of degree r over K and supposethat

Φ(x, y) = Ψ(ax+ by, dx+ ey)

for some a, b, d, e ∈ K with ae − bd 6=0. Then for all c, f ∈ K there exist α, β ∈ Ksuch that

Φ(x+ αz, y + βz) = Ψ(ax+ by + cz, dx+ ey + fz).

86 S. DALE CUTKOSKY

Proof.

Φ(x+ αz, y + βz) = Ψ(a(x+ αz) + b(y + βz), d(x+ αz) + e(y + βz))= Ψ(ax+ by + (aα+ bβ)z, dx+ ey + (dα+ eβ)z)

so we need only solveaα+ bβ = c, dα+ eβ = f

for α, β.

7.3. τ(q) = 1. In this section we prove Theorem 7.9 in the case that τ(q) = 1.We now assume that τ(q) = 1. Suppose that there is a sequence (81) which does

not terminate. We can then choose a sequence of points qn ∈ Vn with q0 = q, suchthat qn+1 maps to qn for all n and νqn(Sn) = r, τ(qn) = 1 for all n. After possiblyreindexing, we may assume that for all n, qn is on the subvariety of Vn which isblownup under Vn+1 → Vn. Let Rn = OVn,qn for n ≥ 0. We then have an infinitesequence

R = R0 → R1 → · · · → Rn → · · · (87)

Let Ji ⊂ OVi,qi be the reduced ideal defining Singr(Si) at qi. Then by Lemmas7.4 and 7.5, Ji is either the maximal ideal mi of OVi,qi or a regular height 2 primeideal pi in OVi,qi . Then the multiplicity r locus of OSi,qi (which is a quotient of Riby a principal ideal) is defined by the reduced ideal Ji = JiRi by Remark 10.21.Ji = mi = miRi or Ji = pi = piRi, a regular prime in Ri. By the construction of thesequence (81), we see that OVi+1,qi+1 is a local ring of the blow up of mi if Ji = mi

and a local ring of the blow up of pi otherwise. Thus Ri+1 is the completion of a localring of the blow up of mi or pi.

We are thus free to work with formal parameters and equations (which define theideal ISi,qi = ISi,qiRi) in the Ri, since the ideals mi and pi are determined in Ri bythe multiplicy r locus of Singr(OSi,qi).

Suppose that T = K[[x, y, z]] is a power series ring, r ∈ N and

g =∑

bijkxiyjzk ∈ T.

We can construct a polygon in the following way.Define

∆ = ∆(g;x, y, z) = ( i

r − k,

j

r − k) ∈ Q2 | k < r and bijk 6= 0.

Let | ∆ | be the smallest convex set in R2 such that ∆ ⊂| ∆ | and (a, b) ∈| ∆ | implies(a + c, b + d) ∈| ∆ | for all c, d ≥ 0. For a ∈ R, let S(a) be the line through (a, 0)with slope -1, V (a) be the vertical line through (a, 0).

Suppose that | ∆ |6= ∅. We define αxyz(g) to be the smallest a appearing in any(a, b) ∈| ∆ |, βxyz(g) to be that smallest b such that (αxyz(g), b) ∈| ∆ |. Let γxyz(g)be the smallest number γ such that S(γ)∩ | ∆ |6= ∅ and let δxyz(g) be such that(γxyz(g)−δxyz(g), δxyz(g)) is the lowest point on S(γxyz(g))∩ | ∆ |. (αxyz(g), βxyz(g))and (γxyz(g)− δxyz(g), δxyz(g)) are vertices of | ∆ |. Define εxyz(g) to be the absolutevalue of the largest slope of a line through (αxyz(g), βxyz(g)) such that no points of| ∆(g;x, y, z) | lie below it.Lemma 7.15. 1. The vertices of | ∆ | are points of ∆, which lie on the lattice

1r!Z×

1r!Z.


2. νR(g) < r holds if and only if | ∆ | contains a point on S(c) with c < 1 whichholds if and only if there is a vertex (a, b) with a+ b < 1.

3. αxyz(g) < 1 if and only if g 6∈ (x, z)r.4. A vertex of | ∆ | lies below the line b = 1 if and only if g 6∈ (y, z)r.

Proof. The proof of Lemma 7.15 follows directly from the definitions.

Suppose that g ∈ T is such that νT (g) = r. Let

L =∑

i+j+k=r

bijkxiyjzk

be the leading form of g. (x, y, z) will be called good parameters for g if b00r = 1.We now suppose that νT (g) = r, and (x, y, z) are good parameters for g. Let

∆ = ∆(g;x, y, z) and suppose that (a, b) is a vertex of | ∆ |. Let

S(a,b) = k | ( i

r − k,

j

r − k) = (a, b) and bijk 6= 0.

Definegabxyz = zr +

∑k∈S(a,b)

ba(r−k),b(r−k),kxa(r−k)yb(r−k)zk.

We will say that gabxyz is solvable (or that the vertex (a, b) is not prepared on| ∆(g;x, y, z) |) if a, b are integers and

gabxyz = (z − ηxayb)r

for some η ∈ K. We will say that the vertex (a, b) is prepared on | ∆(g;x, y, z) |if gabx,y,z is not solvable. If gabxyz is solvable, we can make an (a, b) preparation,which is the change of parameters

z1 = z − ηxayb.If all vertices (a, b) of | ∆ | are prepared, then we say that (g;x, y, z) is well prepared.Lemma 7.16. Suppose that T is a power series ring over K, g ∈ T is such thatνT (g) = r, τ(g) = 1, (x, y, z) are good parameters of g, and (g;x, y, z) is well prepared.Then z = 0 is an approximate manifold of g = 0.

Proof. νT (g) = r implies all vertices (a, b) of | ∆(g;x, y, z) | satisfy a + b ≥ 1. Theleading form of g is

L =∑

i+j+k=r

bijkxiyjzk

where b00r = 1. τ(g) = 1 implies there exists a linear form ax+ by + cz such that

(ax+ by + cz)r = L.

As b00r = 1, we must have c = 1. If a 6= 0, then (1, 0) is a vertex of | ∆(g;x, y, z) |,which is not prepared since

g(1,0)xyz = (z + ax)r.

Thus a = 0. If b 6= 0, then (0, 1) is a vertex of | ∆(g;x, y, z) | which is not preparedsince

g(0,1)xyz = (z + by)r.

Thus L = zr and z = 0 is an approximate manifold of g = 0.

88 S. DALE CUTKOSKY

Lemma 7.17. Consider the terms in the expansion

h =k∑λ=0

ηk−λ(k

λ

)xi+(k−λ)ayj+(k−λ)bzλ1 (88)

obtained by substituting z1 = z−ηxayb into the monomial xiyjzk. Define a projectionfor (a, b, c) ∈ N3 such that c < r,

π(a, b, c) = (a

r − c,

b

r − c).

Then1. Suppose that k < r. Then the exponents of monomials in (88) with non-zero

coefficients project into the line segment joining (a, b) to ( ir−k ,

jr−k ).

a. If (a, b) = ( ir−k ,

jr−k ), then all these monomials project to (a, b).

b. If (a, b) 6= ( ir−k ,

jr−k ), then xiyjzk1 is the unique monomial in (88) which

projects onto ( ir−k ,

jr−k ). No monomial in (88) projects to (a, b).

2. Suppose that r ≤ k, and (i, j, k) 6= (0, 0, r). Then all exponents in (88) withnon-zero coefficients and z1 exponent less that r project into(

(a, b) + Q2≥0

)− (a, b).

3. Suppose that (i, j, k) = (0, 0, r). Then all exponents in (88) with non-zerocoefficients and z1 exponent less than r project to (a, b).

Proof. If k 6= r, we havei+(k−λ)ar−λ = a+ (r−k)

(r−λ)

(i

r−k − a)

j+(k−λ)br−λ = b+ (r−k)

(r−λ)

(j

r−k − b).

Suppose that k < r. Since 0 ≤ λ ≤ k, we have 0 < r−kr−λ ≤ 1. Thus(

i+ (k − λ)ar − λ

,j + (k − λ)b

r − λ

)is on the line segment joining (a, b) to ( i

r−k ,j

r−k ) and 1. follows.If k > r and 0 ≤ λ < r ≤ k, then r−k

r−λ < 0, ir−k − a < 0 and j

r−k − b < 0. Thus 2.follows if r 6= k.

Suppose that k = r and 0 ≤ λ < r. Then(i+ (k − λ)a

r − λ,j + (k − λ)b

r − λ

)=(

i

r − λ+ a,

j

r − λ+ b

)and the last case of 2. and 3. follow.

We deduce from this Lemma thatLemma 7.18. Suppose that z1 = z − ηxayb is an (a, b) preparation. Then

(1) | ∆(g;x, y, z1) |⊂| ∆(g;x, y, z) | −(a, b).(2) If (a′, b′) is another vertex of | ∆(g;x, y, z) |, then (a′, b′) is a vertex of

| ∆(g;x, y, z1) |

and ga′,b′x,y,z1 is obtained from ga′,b′x,y,z by substituting z1 for z.


Example 7.19. It is not always possible to well prepare after a finite number ofvertex preparations. Consider over a field of characteristic 0 (or p > r)

gm = y(y − x)(y − 2x) · · · (y − rx) + (z − xm + xm+1 − · · · )r

with m ≥ 2. The vertices of | ∆(g;x, y, z) | are (0, 1 + 1r ), (1, 1

r ) and (m, 0). We canremove the vertex (m, 0) by the (m, 0) preparation z1 = z − xm. Thus

g = y(y−x)(y−2x) · · · (y−rx)+(z− x2

1 + x)r = y(y−x)(y−2x) · · · (y−rx)+(z−x2+x3−· · · )r

can only be well prepared by the formal substitution

z1 = z −∞∑i=2

(−1)ixi.

Lemma 7.20. Suppose that g is reduced, νT (g) = r, τ(g) = 1 and (x, y, z) are goodparameters for g. Then there is a formal series φ(x, y) ∈ K[[x, y]] such that underthe substitution z = z1 + φ(x, y), (x, y, z1) are good parameters for g and (g;x, y, z1)is well prepared.

Proof. Let vL be the lowest vertex of | ∆(g;x, y, z) |. Let h(vL) be the second coordi-nate of vL. Set b = h(vL). If vL is not prepared, make a vL preparation z1 = z−ηxayb(where (a, b) = vL) to remove vL in | ∆(g;x, y, z1) |.

Let vL1 be the lowest vertex of | ∆(g;x, y, z1) |. If vL1 is not prepared and h(vL1) =h(vL) we can again prepare vL1 by a vL1 preparation z2 = z1 − η1x

a1yb. We caniterate this procedure to either achieve | ∆(g;x, y, zn) | such that the lowest vertexvLn is solvable or h(vLn) > h(vL), or we can construct an infinite sequence of vLnpreparations (with h(vLn) = b for all n)

zn+1 = zn − ηnxanyb

where (an, b) = vLn such that the lowest vertex vLn of | ∆(g;x, y, zn) | is not preparedand h(vLn) = h(vL) for all n. Since an+1 > an for all n, we can then make the formalsubstitution z′ = z −

∑∞i=0 ηix

aiyb, to get | ∆(g;x, y, z′) | whose lowest vertex vL′satisfies h(vL′) > h(vL).

In summary, there exists a series Φ′(x) such that if we set z′ = z−yh(vL)Φ′(x), andvL′ is the lowest vertex of | ∆(g;x, y, z′) | then either vL′ is prepared or h(vL′) > h(vL).

By iterating this procedure, we construct a series Φ(x, y) such that if z = z−Φ(x, y),then either the lowest vertex vL of | ∆(g;x, y, z) | is prepared, or | ∆(g;x, y, z) |= ∅.This last case only occurs if g = unit zr, and thus cannot occur since g is assumed tobe reduced.

We can thus assume that vL is prepared. We can now apply the same procedureto vT , the highest vertex of | ∆(g;x, y, z) |, to reduce to the case where vT and vLare both prepared. Then after a finite number of preparations we find a change ofvariables z′ = z − Φ(x, y) such that (g;x, y, z′) is well prepared.

We will also consider change of variables of the form

y1 = y − ηxn

for η ∈ K, n a positive integer, which we will call translations.

90 S. DALE CUTKOSKY

Lemma 7.21. Consider the expansion

h =j∑

λ=0

ηj−λ(j

λ

)xi+(j−λ)nyλ1 z

k (89)

obtained by substituting y1 = y − ηxn into the monomial xiyjzk. Consider the pro-jection for (a, b, c) ∈ N3 such that c < r defined by

π(a, b, c) = (a

r − c,

b

r − c).

Suppose that k < r. Set (a, b) = ( ir−k ,

jr−k ). Then xiyj1z

k is the unique monomial in(89) whose coefficients projects onto (a, b). All other monomials in (89) with non-zerocoefficient project to points below (a, b) on the line through (a, b) with slope − 1

n .

Proof. The slope of the line through the point (a, b) and(i+ (j − λ)n

r − k,

λ

r − k

)is

j − λi− (i+ (j − λ)n)

= − 1n.

Since λr−k <

jr−k for λ < j, the Lemma follows.

Definition 7.22. Suppose that g ∈ T is reduced, νT (g) = r, τ(g) = 1 and (x, y, z)are good parameters for g. Let α = αxyz(g), β = βxyz(g), γ = γxyz(g), δ = δxyz(g),ε = εxyz(g). (g;x, y, z) will be said to be very well prepared if it is well prepared andone of the following conditions holds.

1. (γ − δ, δ) 6= (α, β) and if we make a translation y1 = y − ηx, with subsequentwell preparation z1 = z − Φ(x, y), then

αxy1z1(g) = α, βxy1z1(g) = β, γxy1z1(g) = γ

andδxy1z1(g) ≤ δ.

2. (γ − δ, δ) = (α, β) and one of the following cases hold:a. ε = 0b. ε 6= 0 and 1

ε is not an integerc. ε 6= 0 and n = 1

ε is a (positive) integer and for any η ∈ K, if y1 = y−ηxnis a translation, with subsequent well preparation z1 = z − Φ(x, y), thenεxy1z(g) = ε. Further, if (c, d) is the lowest point on the line through(α, β) with slope −ε in | ∆(g;x, y, z1) | and (c1, d1) is the lowest point onthis line in | ∆(g;x, y1, z1) |, then d1 ≤ d.

Lemma 7.23. Suppose that g ∈ T is reduced, νT (g) = r, τ(g) = 1 and (x, y, z) aregood parameters for g. Then there are formal substitutions

z1 = z − φ(x, y), y1 = y − ψ(x)

where φ(x, y), ψ(x) are series such that (g;x, y1, z1) is very well prepared.


Proof. By Lemma 7.20, we may suppose that (g;x, y, z) is well prepared. Let α =αxyz(g), β = βxyz(g), γ = γxyz(g), δ = δxyz(g), ε = εxyz(g). Let (γ − t, t) be thehighest vertex on | ∆(g;x, y, z) | on the line a + b = γ. Let η ∈ K and consider thetranslation y1 = y − ηx. By Lemma 7.21, (γ − t, t) and (α, β) are prepared verticesof | ∆(g;x, y1, z) |. Thus if z1 = z − Φ(x, y) is a well preparation of (g;x, y1, z), then(γ − t, t) and (α, β) are vertices of | ∆(g;x, y1, z1) |.

We can thus choose η ∈ K so that after the translation y1 = y − ηx, and asubsequent well preparation z1 = z − Φ(x, y), we have αxy1z1 = α, βxy1z1 = β,γxy1z1 = γ and δxy1z1 ≥ δ is a maximum for substitutions y1 = y−ηx, z1 = z−Φ(x, y)as above. If (γ − δxy1z1 , δxy1z1) 6= (α, β), then after replacing (x, y, z) with (x, y1, z1),we have achieved the condition 1. of the Lemma, so that (g;x, y, z) is very wellprepared.

We now assume that (g;x, y, z) is well prepared, and (α, β) = (γ − δ, δ). If ε = 0or 1

ε is not an integer then (g;x, y, z) is very well prepared.Suppose that n = 1

ε is an integer. We then choose η ∈ K such that with thetranslation y1 = y − ηxn, and subsequent well preparation, we maximize d for points(c, d) of the line through (α, β) with slope −ε on the boundary of | ∆(f ;x, y1, z1) |.By Lemma 7.21 α, β and γ are not changed. If we now have that d 6= β, we are verywell prepared.

If this process does not end after a finite number of iterations, then there exists aninfinite sequence of changes of variables (translations followed by well preparations)

yi = yi+1 + ηi+1xni+1 , zi = zi+1 + φi+1(x, yi+1)

such that ni+1 > ni for all i. Given n ∈ N, there exists σ(n) ∈ N such thatσ(n) > σ(n − 1) for all n, and i ≥ σ(n) implies all vertices (a, b) of | ∆(g;x, yi, zi) |below (α, β) have a ≥ n. This last condition follows since all vertices must lie in thelattice 1

r!Z×1r!Z, and there are only finitely many points common to this lattice and

the region 0 ≤ a ≤ n, 0 ≤ b ≤ β. Thus xn | φi(x, yi) if i ≥ σ(n). Let

Ψi(x, y) =i∑

j=1

φj(x, y −j∑

k=1

ηkxnk),

so that zi = z −Ψi(x, y) for all i. Ψi(x, y) is a Cauchy sequence in K[[x, y]]. Thusz′ = z−

∑∞i=1 φi(x, yi) is a well defined series in K[[x, y, z]]. Set y′ = y−

∑∞i=1 ηix

ni .| ∆(g;x, y′, z′) | then has the single vertex (α, β), and (g;x, y′, z′) is thus very wellprepared.

Definition 7.24. Suppose that g ∈ T , g is reduced, νT (g) = r, τ(g) = 1, and (x, y, z)are good parameters for g. We consider 4 types of monodial transforms T → T1,where T1 is the completion of the local ring of a monoidal transform of T , and T1 hasregular parameters (x1, y1, z1) related to the regular parameters (x, y, z) of T by oneof the following rules.

T1 Singr(g) = V (x, y, z),

x = x1, y = x1(y1 + η), z = x1z1,

with η ∈ K. Then g1 = gxr1

is the strict transform of g in T1, and if νT1(g1) = r

and τ(g1) = 1, then (x1, y1, z1) are good parameters for g1.

92 S. DALE CUTKOSKY

T2 Singr(g) = V (x, y, z),

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

Then g1 = gyr1

is the strict transform of g in T1, and if νT1(g1) = r andτ(g1) = 1, then (x1, y1, z1) are good parameters for g1.

T3 Singr(g) = V (x, z),

x = x1, y = y1, z = x1z1.

Then g1 = gxr1


T4 Singr(g) = V (y, z),

x = x1, y = y1, z = y1z1.

Then g1 = gyr1


Lemma 7.25. With the assumptions of Definition 7.24, suppose that g ∈ T is re-duced, νT (g) = r, τ(g) = 1, (x, y, z) are good parameters for g, (x, y, z) and (x1, y1, z1)are related by a transformation of one of the above types T1 - T4, and νT1(g) = r1,τ(g1) = 1. Then there is a 1-1 correspondence

σ : ∆(g, x, y, z)→ ∆(g1, x1, y1, z1)

defined by1. σ(a, b) = (a+ b− 1, b) if the transformation is a T1 with η = 0.2. σ(a, b) = (a, a+ b− 1) if the transformation is a T2.3. σ(a, b) = (a− 1, b) if the transformation is a T3.4. σ(a, b) = (a, b− 1) if the transformation is a T4.

The proof of Lemma 7.25 is straightforward, and is left to the reader.Lemma 7.26. In each of the four cases of the preceeding lemma, if σ(a, b) = (a1, b1)is a vertex of | ∆(g1;x1, y1, z1) |, then (a, b) is a vertex of | ∆(g;x, y, z) |, and if(g;x, y, z) is (a, b) prepared then (g1;x1, y1, z1) is (a1, b1) prepared. In particular,(g1;x1, y1, z1) is well prepared if (g;x, y, z) is well prepared.

Proof. In all cases, σ (when extended to R2) takes line segments to line segmentsand interior points of | ∆(g;x, y, z) | to interior points of σ(| ∆(g;x, y, z) |). Thus theboundary of σ(| ∆(g;x, y, z) |) is the union of the image by σ of the line segments onthe boundary of σ(| ∆(g;x, y, z) |). If (a1, b1) is a vertex of σ(| ∆(g;x, y, z) |), it mustthen necessarily be σ(a, b) with (a, b) a vertex of σ(| ∆(g;x, y, z) |).

To see that (g1;x1, y1, z1) is (a1, b1) prepared if (g;x, y, z) is (a, b) prepared, weobserve that

g1a1,b1x1,y1,z1 =

1xr g

a,bx,y,z if a T3 transformation or a T1 with η = 0

1yr g

a,bx,y,z if a T2 or T4 transformation

Lemma 7.27. Suppose that assumptions are as in Definition 7.24 and (x, y, z) and(x1, y1, z1) are related by a T3 transformation. Suppose that νT1(g1) = r and τ(g1) =1. If (g;x, y, z) is very well prepared, then (g1;x1, y1, z1) is very well prepared.

βx1y1z1(g1) = βxyz(g), δx1y1z1(g1) = δxyz(g), εx1y1z1(g1) = εxyz(g)


andαx1y1z1(g1) = αxyz(g)− 1, γx1y1z1(g1) = γxyz(g)− 1.

We further have Singr(g1) ⊂ V (x1, z1).

Proof. Well preparation is preserved by Lemma 7.26. Suppose that (g;x, y, z) is verywell prepared and

(αxyz, βxyz) 6= (γxyz − δxyz, δxyz).Let γ = γxyz(g), γ1 = γxyz(g1) = γ − 1. The terms in g which contribute to the lineS(γ) are ∑

i+ j + γk = rγk < r

bijkxiyjzk. (90)

They are transformed to ∑(i+ k − r) + j + γ1k = rγ1

k < r

bijkxi+k−r1 yj1z

k1 (91)

which are the terms in g1 which contribute to the line S(γ1). We have

δxyz(g) = δx1,y1,z1(g1) = min j

r − k| bijk 6= 0 in (90).

If a translation y = y + ηx transforms (90) into∑i+ j + γk = rγ

k < r

cijkxiyjzk (92)

then the translation y1 = y1 + ηx transforms (91) into∑(i+ k − r) + j + γ1k = rγ1

k < r

cijkxiyj1z

k1 . (93)

Since (92) and (93) are the terms which contribute to δxyz(g) and δx1y1z1(g1), wehave δxyz(g) = δx1y1z1(g1) and (g;x, y, z) well prepared implies (g1;x1, y1, z1) wellprepared.

We can make a similiar argument if

(αxyz, βxyz) = (γxyz − δxyz, δxyz).

The statement Singr(g1) ⊂ V (x1, z1) follows since Singr(g) = V (x, y, z), so thatSingr(g1) ⊂ V (x1), the exceptional divisor of T3, and Singr(g1) ⊂ V (z1) since z = 0is an approximate manifold of g = 0 (Lemmas 7.16 and 7.5).

Lemma 7.28. Suppose that assumptions are as in Definition 7.24, (x, y, z) and(x1, y1, z1) are related by a T1 transformation with η = 0, (g;x, y, z) is very well

94 S. DALE CUTKOSKY

prepared and νT1(g1) = r, τ(g1) = 1. Let (g1;x1, y′, z′) be a very well preparation of

(g1;x1, y1, z1). Let

α = αxyz(g), β = βxyz(g), γ = γxyz(g), δ = δxyz(g), ε = εxyz(g),

α1 = αx1y′z′(g1), β1 = βx1y′z′(g1), γ1 = γx1y′z′(g1), δ1 = δx1y′z′(g1), ε1 = εx1y′z′(g1).

Then

(β1, δ1,1ε 1, α1) < (β, δ,

1ε, α)

in the Lexicographical order, with the convention that if ε = 0, then 1ε =∞. If β1 = β,

δ1 = δ and ε 6= 0, then 1ε1

= 1ε − 1.

We further have Singr(g1) ⊂ V (x1, z′).

Proof. Let σ : ∆(g;x, y, z) → ∆(g1;x1, y1, z1) be the 1-1 correspondence of Lemma7.25 1., which is defined by σ(a, b) = (a+b−1, b). σ transforms lines of slope m 6= −1to lines of slope m

m+1 , and transforms lines of slope -1 to vertical lines.First suppose that −ε ≤ −1. Then (α, β) 6= (γ − δ, δ), and we are in case 1.

of Definition 7.22. Then the line segment through (α, β) and (γ − δ, δ) (which hasslope ≤ −1) is transformed to a segment with positive slope, or a vertical line. Thusβx1,y1,z1 = δ < β. Since (g;x1, y1, z1) is well prepared, after very well preparation wehave β1 < β.

Suppose that −1 < −ε ≤ − 12 . We have (α, β) = (γ − δ, δ). Then the line segment

through (α, β) and (c, d) (with the notation of Definition 7.22) is transformed to theline segment through (α+ β − 1, β) and (c+ d− 1, d) which has slope

−εx1y1z1 =−ε−ε+ 1

< −1.

Thus(αx1y1z1 , βx1y1z1) = (α+ β − 1, β)

and(γx1y1z1(g1)− δx1y1z1(g1), δx1y1z1(g1)) 6= (αx1y1z1(g1), βx1y1z1(g1)).

Let (γx1y1z1(g1)−t, t) be the highest point on the line S(γx1y1z1(g1)) in | ∆(g1;x1, y1, z1) |.By Lemma 7.26, (g1;x1, y1, z1) is well prepared. Thus (by Lemma 7.21) very wellpreparation does not affect the vertices (αx1y1z1(g1), βx1y1z1(g1)) and (γx1y1z1(g1) −t, t) of | ∆(g1;x1, y1, z1) | which are below the horizontal line of points whose secondcoordinate is d. Thus β1 = β and δ1 ≤ t ≤ d < δ.

Suppose that − 12 < −ε < 0. In this case we have (α, β) = (γ − δ, δ). Then the line

segment through (α, β) and (c, d) (with the notation of Definition 7.22) is transformedto the line segment through (α+ β − 1, β) and (c+ d− 1, d) which has slope

−εx1y1z1 =−ε−ε+ 1

,

so that −1 < −εx1y1z1 < 0. Thus

(αx1y1z1 , βx1y1z1) = (γx1y1z1 − δx1y1z1 , δx1y1z1) = (α+ β − 1, β),

and (c1, d) = (c+d−1, d) is the lowest point on the line through (αx1y1z1 , βx1y1z1) withslope −εx1y1z1 which is on | ∆(g1;x1, y1, z1) |. By Lemma 7.26, (g1;x1, y1, z1) is well


prepared. We will now show that (g1;x1, y1, z1) is very well prepared. If 1εx1,y1,z1

6∈ N,then this is immediate, so suppose that

n =1

εx1y1z1

∈ N.

We necessarily have n > 1, and 1ε = n + 1. If we can remove the vertex (c1, d) from

| ∆(g1;x1, y1, z1) | by a translation y1 = y1− ηxn1 , then the translation y = y− ηxn+1

will remove the vertex (c, d) from | ∆(g;x, y, z) |, a contradiction to the assumptionthat (g;x, y, z) is very well prepared.

Thus (g1;x1, y1, z1) is very well prepared, with β1 = β, δ1 = δ and 1ε1

= 1ε − 1.

The final case is when ε = 0. Then β = δ, (g1;x1, y1, z1) is very well prepared,

(α1, β1) = (γ1 − δ1, δ1) = (α+ β − 1, β)

and ε1 = 0. Thus β1 = β, δ1 = δ and 1ε1

= 1ε =∞. α1 < α since Singr(g) = V (x, y, z)

(by assumption) so ε = 0 implies β < 1.By Lemmas 7.4 and 7.16, Singr(g1) ⊂ V (x1, z1). If Singr(g1) = V (x1, z1), then

all vertices of | ∆(g1;x1, y1, z1) | have the first cordinate ≥ 1, and thus Singr(g1) =V (x1, z

′).

Lemma 7.29. Suppose that assumptions are as in Definition 7.24, (x, y, z) and(x1, y1, z1) are related by a T2 or a T4 transformation and (g;x, y, z) is well pre-pared. Suppose that νT1(g1) = r and τ(g1) = 1. We have that (g1;x1, y1, z1) is wellprepared and Singr(g1) ⊂ V (y1, z1). Let (g1;x1, y

′, z′) be a very well preparation.Then

βx1,y1,z1 = βx1,y′,z′(g1) < βxyz(g).

Proof. First suppose that T → T1 is a T2 transformation. The correspondence σ :∆(g;x, y, z)→ ∆(g1;x1, y1, z1) of Lemma 7.25 2. is defined by σ(a, b) = (a, a+ b−1).σ takes lines of slope m to lines of slope m+ 1 Thus

(αx1y1z1(g1), βx1y1z1(g1) = (αxyz(g), αxyz(g) + βxyz(g)− 1).

Since Singr(g) = V (x, y, z) by assumption, we have αxyz(g) < 1, and thus

βx1y1z1(g1) < βxyz(g).

(g1;x1, y1, z1) is well prepared by Lemma 7.26. Thus the vertex (αx1y1z1(g1), βx1y1z1(g1))is not affected by very well preparation. We thus have βx1y′z′(g1) < βxyz(g).

Now suppose that T → T1 is a T4 transformation. The 1-1 correspondence σ :∆(g;x, y, z) → ∆(g1;x1, y1, z1) of Lemma 7.25 4. is defined by σ(a, b) = (a, b − 1).Thus βx1y1z1(g1) < βxyz(g). (g1;x1, y1, z1) is well prepared by Lemma 7.26, andthe vertex (αx1y1z1(g1), βx1y1z1(g1)) is not affected by very well preparation. Thusβx1y′z′(g1) = βx1y1z1 < βxyz(g).

Lemma 7.30. Suppose that p is a prime, s is a non–negative integer and r0 is apositive integer such that p 6 |r0. Let r = r0p

s. Then

1.(rλ)≡ 0 mod p, if ps 6 | λ, 0 ≤ λ ≤ r is an integer.

2.(

r

λps)≡(r0λ)

mod p, if 0 ≤ λ ≤ r0 is an integer.

Proof. Compare the expansions over Zp of (x+ y)r = (xps

+ yps

)r0 .

96 S. DALE CUTKOSKY

Theorem 7.31. Suppose that assumptions are as in Definition 7.24, (x, y, z) and(x1, y1, z1) are related by a T1 transformation with η 6= 0 and (g;x, y, z) is very wellprepared. Suppose that νT1(g1) = r, τ(g1) = 1 and βxyz(g) > 0. Then there existgood parameters (x, y′1, z

′1) in T1 such that (g1;x1, y

′1, z′1) is very well prepared, with

βx1,y′1,z′1(g1) < βx,y,z(g) and Singr(g1) ⊂ V (x1, z

′1).

Proof. Let α = αxyz(g), β = βxyz(g), γ = γxyz(g), δ = δxyz(g). Lemma 7.16 impliesz = 0 is an approximate manifold of g = 0. Thus

α+ β > 1. (94)

Apply the translation y′ = y − ηx and well prepare by some substitution z′ = z −Φ(x, y′). This does not change (α, β) or γ. Set δ′ = δx,y′,z′(g).

First assume that δ′ < β. We have regular parameters (x1, y1, z1) in T1 such thatx = x1, y

′ = x1y1, z′ = x1z1 by Lemma 7.16 so we have a 1-1 correspondence (by

Lemma 7.25 1.)σ : ∆(g;x, y′, z′)→ ∆(g1;x1, y1, z1)

defined by σ(a, b) = (a+b−1, b). (g1;x1, y1, z1) is well prepared (by Lemma 7.26). Weare essentially in case 1 (−ε ≤ −1) of the proof of Lemma 7.28 (this case only requireswell preparation) so that (α1, β1) = σ(γ − δ′, δ′) and β1 = δ′ < β. After very wellpreparation, we thus have regular parameter x1, y

′1, z′1 in T1 such that (g1;x1, y

′1, z′1)

is very well prepared and βx1,y′1,z′1(g1) < βxyz(g).

We have thus reduced the proof to showing that δ′ < β. If (α, β) 6= (γ − δ, δ), wehave δ′ ≤ δ < β since (g;x, y, z) is very well prepared.

Let g =∑aijkx

iyjzk. Suppose that (α, β) = (γ − δ, δ). Set

W = (i, j, k) ∈ N3 | k < r and (i

r − k,

j

r − k) = (α, β),

F =∑

(i,j,z)∈W

aijkxiyjzk.

By assumption, zr + F is not solvable.

F (x, y, z) = F (x, y′ + ηx, z) =∑

(i,j,k)∈W

j∑λ=0

aijkηλ

(j

λ

)xi+λ(y′)j−λzk.

(95)

By Lemma 7.21, the terms in the expansion of g(x, y′ + ηx, z) contributing to (γ, 0)in | ∆(g;x, y′, z) |, where γ = α+ β, are

F(γ,0) =∑

(i,j,k)∈W

aijkηjxi+jzk 6= 0.

If (g;x, y′, z) is (γ, 0) prepared, then δ′ = 0 < β. Suppose that

gγ0x,y′,z = zr + F(γ,0)

is solvable. Then γ ∈ N and there exists ψ ∈ K such that

(z − ψxγ)r = zr + F(γ,0) (96)

so that, with ω = −ψηβ∈ K, for 0 ≤ k < r, we have


1. If(

rr − k

)6= 0 (in K) then i = α(r − k), j = β(r − k) ∈ N and

aijk =(

r

r − k)ωr−k (97)

2. If i = α(r − k), j = β(r − k) ∈ N and(

rr − k

)= 0 (in K) then aijk = 0.

Thus (by Lemma 7.30) aijk = 0 if ps 6 | k.Suppose that K has characteristic 0. By Remark 7.32, we obtain a contradiction

to our assumption that g(γ,0)xy′z is solvable. Thus 0 = δ′ < β if K has characteristic

0.Now we consider the case where K has characteristic p > 0, and r = psr0 with

p 6 | r0, r0 ≥ 1. By (97) and Lemma 7.30, we have i = αps, j = βps ∈ N andai,j,(r0−1)ps 6= 0.

We have an expression βps = ept, where p 6 | e. Suppose that t ≥ s. Then β ∈ Nwhich implies α = γ − β ∈ N, so that

(z + ωxαyβ)r = zr + F

a contradiction, since zr + F is by assumption not solvable. Thus t < s. Supposethat e = 1. Then β = pt−s < 1 and α < 1 (since we must have Singr(g) = V (x, y, z))implies γ = α+ β = 1, a contradiction to (94). Thus e > 1.

zr + F = (zps

+ ωps

xαps

yβps

)r0

=(zps

+ ωps

xαps

(y′ + ηx)βps)r0 =

(zps

+ ωps

xαps[(y′)p

t

+ ηpt

xpt]e)r0

Now make the (γ, 0) preparation z = z′ − ηβωxγ (from (96)) so that (g;x, y′, z′) is(γ, 0) prepared.

zr + F =(

(z′)ps

+ eωps

ηpt(e−1)(y′)p

t

xαps+pt(e−1) + (y′)2ptΩ)

)r0= (z′)p

sr0 + r0

[eωp

s

ηpt(e−1)(y′)p

t

xαps+pt(e−1) + (y′)2ptΩ

](z′)p

s(r0−1)

+Λ2(x, y′)(z′)ps(r0−2) + · · ·+ Λr0(x, y′)

for some polynomials Ω(x, y′),Λ2, . . . ,Λr0 where (y′)ipt | Λi for all i. Since all

contributions to S(γ)∩ | ∆(g;x, y′, z′) | must come from this polynomial, (recallthat we are assuming (α, β) = (γ − δ, δ)) the term of lowest second coordinate onS(γ)∩ | ∆(g;x, y′, z′) | is

(a, b) =(αps + pt(e− 1)

ps,pt

ps

)which is not in N2 since t < s, and is not (α, β) since e > 1. Thus (g;x, y′, z′) is not(a, b) solvable, and

δ′ =pt

ps<ept

ps= β.

Remark 7.32. In Theorem 7.31 suppose that K has characteristic 0. From (97)we see that if gγ0

xy′z′ is solvable, then for 0 ≤ k ≤ r − 1 there exists i, j ∈ N with

98 S. DALE CUTKOSKY

i = α(r − k), j = β(r − k) so that α, β ∈ N. Comparing (97) with the binomialexpansion

(z + ωxαyβ)r =r∑i=0

(r

r − i)

(ωxαyβ)r−izi = zr + F

we see that zr + F is solvable, a contradiction. Thus (γ, 0) is not solvable on| ∆(g;x, y′, z) | if K has characteristic 0, and after well preparation, (γ, 0) is a vertexof | (g;x, y′, z′) |.Example 7.33. With the notation of Theorem 7.31, it is possible to have (γ, 0)solvable in | ∆(g;x, y′, z) | if K has positive characteristic, as is shown by the followingsimple example. Suppose that the characteristic p of K is greater than 5.

Let g = zp + xyp2−1 + xp

2−1y2.The origin is isolated in Singp(g = 0), and (g;x, y, z) is very well prepared. The

vertices of | ∆(g;x, y, z) | are (α, β) = (γ − δ, δ) = ( 1p , p −

1p ) and

(p− 1

p ,2p

). Set

y = y′ + ηx with 0 6= η ∈ K.

g = zp + x(y′ + ηx)p2−1 + xp

2−1(y′ + ηx)2

= zp +(∑p2−1

i=0

(p2−1

i

)ηp

2−1−ixp2−i(y′)i

)+ η2xp

2+1 + 2ηy′xp2

+ xp2−1(y′)2

The vertices of | ∆(g;x, y′, z) | are (α, β) =(

1p , p−

1p

)and (γ, 0) = (p, 0). However,

(γ, 0) is solvable. Set z′ = z + ηp−1pxp.

g = (z′)p +

p2−1∑i=1

(p2−1

i

)ηp

2−1−ixp2−i(y′)i

+ η2xp2+1 + 2ηy′xp

2+ xp

2−1(y′)2

Since(p2−1

1)≡ −1 mod p, δ1 = δxy′z′(g) = 1

p and the vertices of | ∆(g;x, y′, z′) | are

(α, β) = ( 1p , p−

1p ), (γ1 − δ1, δ1) = (p− 1

p ,1p ) and (p+ 1

p , 0).

Theorem 7.34. For all n ∈ N, there are fn ∈ Rn such that fn is a generator ofISn,qnRn, and good parameters (xn, yn, zn) for fn in Rn such that if

βn = βxnynzn(fn), εn = εxnynzn(fn), αn = αxnynzn(fn), δn = δxnynzn(fn)

σ(n) = (βn, δn,1εn, αn)

then σ(n+1) < σ(n) for all n in the lexicographical order in 1r!N×

1r!N×(Q+ ∪ ∞)×

1r!N. If βn+1 = βn, δn+1 = δn and 1

εn6=∞, then

1εn+1

=1εn− 1.

As an immediate corollary, we find a contradiction to the assumption that (87) hasinfinite length. We have now verified Theorem 7.9 for the final case τ(q) = 1.

Proof. We inductively construct regular parameters (xn, yn, zn) in Rn, a generator fnof ISn,qnRn such that (xn, yn, zn) are good parameters for fn, Singr(fn) ⊂ V (xn, zn)or Singr(fn) ⊂ V (yn, zn) and (fn;xn, yn, zn) is well prepared. If Singr(fn) ⊂ V (xn, zn),we will further have that (fn;xn, yn, zn) is very well prepared.


Let f be a generator of IS,qR0. We first choose (possibly formal) regular parameters(x, y, z) in R which are good for f = 0, such that | ∆(f ;x, y, z) | is very well prepared.By Lemma 7.16, z = 0 is an approximate mainfold for f = 0. Let α = αx,y,z(f),β = βx,y,z(f).

By assumption, q is isolated in Singr(f).Suppose that fi and regular parameters (xi, yi, zi) in Ri have been defined for i ≤ n

as specified above.If Singr(fn) = V (yn, zn), then Rn → Rn+1 must be a T4 transformation,

xn = x′n+1, yn = y′n+1, zn = y′n+1z′n+1.

Thus (fn+1;x′n+1, y′n+1, z

′n+1) is prepared by Lemma 7.29. Further Singr(fn+1) ⊂

V (y′n+1, z′n+1). If Singr(fn+1) = V (xn+1, y

′n+1, z

′n+1), make a change of variables to

very well prepare. Otherwise, set yn+1 = y′n+1, zn+1 = z′n+1. We have βn+1 < βn byLemma 7.29.

If Singr(fn) = V (xn, zn) then Rn → Rn+1 must be a T3 transformation

xn = xn+1, yn = yn+1, zn = xn+1zn+1,

and (fn;xn, yn, zn) is very well prepared. Thus (fn+1;xn+1, yn+1, zn+1) is also verywell prepared.

(βn+1, δn+1,1

εn+1, αn+1) < (βn, δn,

1εn, αn)

by Lemma 7.27. Further Singr(fn+1) ⊂ V (xn+1, zn+1).Now suppose that Singr(fn) = V (xn, yn, zn). Then (f ;xn, yn, zn) is very well

prepared, and Rn+1 must be a T1 or T2 transformation of Rn.If Rn+1 is a T2 transformation of Rn,

xn = xn+1y′n+1, yn = y′n+1, zn = yn+1z

′n+1,

then (fn+1;xn+1, y′n+1, z

′n+1) is well prepared with βn+1 < βn by Lemma 7.29. Fur-

ther, Singr(fn+1) ⊂ V (y′n+1, z′n+1). If Singr(fn+1) = V (xn+1, y

′n+1, z

′n+1), then make

a further change of variables to very well prepare. Otherwise, set yn+1 = y′n+1, zn+1 =z′n+1.

If Rn+1 is a T1 transformation of Rn,

xn = xn+1, yn = xn+1(y′n+1 + η), zn = xn+1z′n+1,

then Singr(fn+1) ⊂ V (xn+1, z′n+1) by Lemma 7.5.

If βn 6= 0 and η 6= 0 in the T1 transformation relating Rn and Rn+1, we can changevariables to yn+1, zn+1 to very well prepare, with βn+1 < βn and Singr(fn+1) ⊂V (xn+1, zn+1) by Theorem 7.31.

If η = 0 in the T1 transformation of Rn, and (xn+1, yn+1, zn+1) are regular pa-rameters of Rn+1 obtained from (xn+1, y

′n+1, z

′n+1) by very well preparation, then

(fn+1;xn+1, yn+1, zn+1) is very well prepared,

(βn+1, δn+1,1

εn+1, αn+1) < (βn, δn,

1εn, αn)

and Singr(fn+1) ⊂ V (xn+1, zn+1) by Lemma 7.28.If βn = 0 we can make a translation y′n = yn + ηxn, and have that (g′nxn, y

′n, zn)

is very well prepared, and βn, γn, δn, εn, αn are unchanged. Thus we are in the caseof η = 0.

100 S. DALE CUTKOSKY

7.4. Remarks and further discussion. The first proof of resolution of singularitiesof surfaces in positive characteristic was given by Abhyankar ([1], [3]). The proof usedvaluation theoretic methods to prove local uniformization (defined in Section 2.5).

The most general proof of resolution of surfaces was given by Lipman [58]. Thislucid article gives a self contained geometric proof.

The algorithm we give in this chapter is sketched in lectures by Hironaka [49].These notes are very rough, and require correction at some points.

This proof of resolution of surfaces which are hypersurfaces over an algebraicallyclosed field (Theorem 7.1) extends to give a proof of resolution of excellent surfaces.Some general remarks about the proof are given in the final 3 pages of Hironaka’snotes [49]. The first part of our proof of Theorem 7.1 generalizes to the case of anexcellent surface S, and we reduce to proving a generalization of Theorem 7.9, whenV = spec(R) where R is a regular local ring with maximal ideal m and S = spec(R/I)is a two dimensional equidimensional local ring with Singr(R/I) = V (m).

There are two main new points which must be considered. The first point is tofind a generalization of the polygon | ∆(f ;x, y, z) | to the case where OS,q ∼= R/I isas above. This is accomplished by the general theory of characteristic polyhedra [50].The second point is to extend the theory to the case of non-closed residue field. Thisis accomplished in [21].

We will now restrict to the case when our surface S is a hypersurface over anarbitrary field K. The proof of Theorem 7.9 in the case when τ(q) = 3 is the same,and if τ(q) = 2, the proof is really the same, since we must then have equality ofresidue fields K(qi) = K(qi+1) for all i in the sequence (82), under the assumptionthat τ(qi) = 2, νqi(Si) = r for all i. This is the analogue of our proof of resolution ofcurves embedded in a non-singular surface, over an arbitrary field.

In the case when τ(q) = 1, some extra care is required. It is possible to havenon-trivial extensions of residue fields k(qi)→ k(qi+1) in the sequence (87). However,if k(qi+1) 6= k(qi), we must have βi+1 < βi (This is proven in the theorem of page 26[21]).

Thus (βi, δi, 1εi, αi) drops after every transformation in (87) and we see that we

must eventually have ν or τ drop.The essential difficulty in resolution in positive characteristic, as opposed to char-

acteristic zero, is the lack of existence of a hypersurface of maximal contact (see theexercises at the end of this section).

Abhyankar is able to finesse this difficulty to prove resolution for 3 dimensionalvarieties X, over an algebraically closed field K of characteristic p > 5 (13.1, [4]). Wereduce to proving local uniformization, by a patching argument (9.1.7, [4]).

The starting point of the proof of local uniformization is to use a projection ar-gument (12.4.3, 12.4.4 [4]) to reduce the problem to the case of a point on a normalvariety of multiplicity ≤ 3! = 6. This projection method can be traced to Albanese.A simple exposition of this method is given on page 200 of [57]. We will sketch theproof of resolution, in the case of a hypersurface singularity of multiplicity ≤ 6. Wethen have that a local equation of this singularity is

wr + ar−1(x, y, z)wr−1 + · · ·+ a0(x, y, z) = 0 (98)


Since r ≤ 6 < Char(K), 1r ∈ K, and we can perform a Tschirnhausen transformation

to reduce to the case

wr + ar−2(x, y, z)wr−2 + · · ·+ a0(x, y, z) = 0.

Now w = 0 is a hypersurface of maximal contact, and we have a good theorem forembedded resolution of 2 dimensional hypersurfaces, so we reduce to the case whereeach ai is a monomial in x, y, z times a unit. Now we have reduced to a combinatorialproblem, which can be solved in a characteristic free way.

A resolution in the case where (98) has degree p = Char K is found by Cossart[22]. The proof is extraordinarily difficult.

Suppose that X is a variety over a field K. In [34], de Jong has shown that thereis a dominant proper morphism of K-varieties X ′ → X such that dim X ′ = dim Xand X ′ is non-singular. If K is perfect, then the finite extension of function fieldsK(X) → K(X ′) is separable. This is weaker than a resolution of singularities sincethe map is in general not birational (the extension of function fields K(X ′)→ K(X)is finite). This proof relies on sophisticated methods in the theory of moduli spacesof curves.

Exercises

1. (Narasimhan [64]) Let K be an algebraically closed field of characterisitc 2,and let X be the 3-fold in A4

K with equation

f = w2 + xy3 + yz3 + zx7 = 0.

a. Show that the maximal multiplicity of a singular point on X is 2, andthe locus of singular points on X is the monomial curve C with localequations

y3 + zx6 = 0, xy2 + z3 = 0, yz2 + x7 = 0, w2 + zx7 = 0

which has the parameterization t → (t7, t19, t15, t32). Thus C has em-bedding dimension 4 at the origin, so there cannot exist a hypersurface(or formal hypersurface) of maximal contact for X at the origin.

b. Resolve the singularities of X.2. (Hauser [46]) Let K be an algebraically closed field of characteristic 2, and

let S be the surface in A2K with equation f = x2 + y4z + y2z4 + z7 = 0.

a. Show that the maximal multiplicity of points on S is 2, and the singularlocus of S is defined by the singular curve y2 + z3 = 0, x+ yz2 = 0.

b. Suppose that X is a hypersurface on a non-singular variety W , and p ∈ Xis a point in the locus of maximal multiplicity r of X. A hypersurfaceH through p is said to have permanent contact with X if under anysequence of blow ups π : W1 → W of W , with non-singular centers,contained in the locus of points of multiplicity r on the strict transformof X, the strict transform of H contains all points of the intersectionof the strict transform of X with multiplicity r which are in the fiberπ−1(p). Show that there does not exist a non-singular hypersurface ofpermanent contact at the origin for the above surface S. Conclude thata hypersurface of maximal contact does not exist for S.

c. Resolve the singularities of S.


8. Local uniformization and resolution of surface singularities

In this chapter we present a modern proof of resolution of surface singularitiesthrough local uniformization of valuations in the spririt of Zariski [79] and [81]. Anintroduction to this approach was given earlier in Section 2.5

8.1. Classification of valuations in algebraic function fields of dimension 2.Let L be an algebraic function field of dimension two over an algebraically closed fieldK of characteristic zero. That is, L is a finite extension of a rational function field intwo variables over K. A valuation of the function field L is a valuation ν of the fieldL which is trivial on K. ν is a homomorphism ν : L∗ → Γ from the multiplicativegroup of L onto an ordered abelian group Γ such that

(1) ν(ab) = ν(a) + ν(b) for a, b ∈ L∗(2) ν(a+ b) ≥ min ν(a), ν(b) for a, b ∈ L∗(3) ν(c) = 0 for 0 6= c ∈ K

We formally extend ν to L by setting ν(0) = ∞. We will assume throughout thissection that ν is non-trivial, that is, Γ 6= 0.

Some basic references to the valuation theory of algebraic function fields are [85]and [3].

A basic invariant of a valuation is its rank. A subgroup Γ′ of Γ is said to be isolatedif it has the property that if 0 ≤ α ∈ Γ′ and β ∈ Γ is such that 0 ≤ β < α, thenβ ∈ Γ′. The isolated subroups of Γ form a single chain of subgroups. The lengthof this chain is the rank of the valuation. Since L is a two-dimensional algebraicfunction field, we have that rank (ν) ≤ 2 (c.f. Corollary to the Lemma of Section 10,Chapter VI of [85]). If Γ is of rank 1 then Γ is embeddable as a subgroup of R (c.f.page 45 of Section 10, Chapter VI [85]). If Γ is of rank 2, then there is a non-trivialisolated subgroup Γ1 ⊂ Γ such that Γ1 is embeddable as a subgroup of R and Γ/Γ1 isembeddable as a subgroup of R. If these subgroups of R are discrete, then ν is calleda discrete valuation. In particular, a discrete valuation (in an algebraic function fieldof dimension two) can be of rank 1 or of rank 2. The ordered chain of prime ideals pof V correspond to the isolated subgroups Γ′ of Γ by

P = f ∈ V | ν(f) ∈ Γ− Γ′ ∪ 0

(c.f. Theorem 15, Section 10, Chapter VI [85]).In our analysis, we will find a rational function field of two variables L′ over which

L is finite, and consider the restricted valuation ν′ = ν | L′. We will consider L′ =K(x, y) where x, y ∈ L are algebraically independent elements such that ν(x), ν(y)are non-negative. Let [L′ : L] = r. Suppose that w ∈ L and suppose that

F (x, y, w) = A0(x, y)wr + · · ·+Ar(x, y) = 0 (99)

is the irreducible relation satisfied by w, with Ai ∈ K[x, y] for all i. Since ν(F ) =∞,there must be two distinct terms Aiwr−i, Ajwr−j of the same value, where we canassume that i < j. Thus (j − i)ν(w) = ν(AjAi ), and we have that r!Γ′ ⊂ Γ. Thus Γand Γ′ have the same rank, and Γ is discrete if and only if Γ′ is discrete.

Associated to the valuation ν is the valuation ring

V = f ∈ L | ν(f) ≥ 0.


V has a unique maximal ideal

mV = f ∈ V | ν(f) > 0.The residue field K(V ) = V/mV of V contains K and has transcendence degree

r∗ ≤ r over K. ν is called an r∗-dimensional valuation. In our case (ν non-trivial) wehave r∗ = 0 or 1.

If R is local ring contained in L, we will say that ν dominates R if R ⊂ V andmV ∩R is the maximal ideal of R. If S is a subring of V we will say that the centerof ν on S is the prime ideal mV ∩S. We will say that R is an algebraic local ring of Lif R is essentially of finite type over K (R is a localization of a finite type K-algebra)and R has quotient field L).

We again consider our algebraic extension L′ = K(x, y) → L. The valuation ringof ν′ is V ′ = V ∩ L, with maximal ideal mV ′ = mV ∩ L. The residue field K(V ′) isthus a subfield of K(V ). Suppose that w∗ ∈ K(V ), and let w be a lift of w∗ to V .We have an irreducible equation of algebraic dependence (99) of w over K[x, y]. LetAh be such that

ν(Ah) = min ν(A1), . . . , ν(Ar).We may assume that h 6= r, because if ν(Ar) < ν(Ai) for i < r, then we would haveν(Ar) < ν(Aiwr−i) for 1 ≤ i ≤ r − 1 since ν(w) ≥ 0, so that ν(F ) = ν(Ar) < ∞ ,which is impossible. ν( AiAh ) ≥ 0 for 0 ≤ i ≤ r, so that Ai

Ah∈ V ′ for all i. Let a∗i be the

corresponding residues in K(V ′). We have a relation

a∗0(w∗)r + · · ·+ a∗h−1(w∗)r−h+1 + (w∗)r−h + a∗h+1(w∗)r−h−1 + · · ·+ a∗r = 0.

Since r − h > 0, w∗ is algebraic over K(V ′). Thus ν and ν′ = ν | L′ have the samedimension.

We will now give a classification of the various types of valuations which occur onL.

one-dimensional valuations

Choose x, y ∈ L such that x, y are algebraically independent over K, ν(x) = 0,ν(y) > 0 and the residue of x in K(V ) is transcendental over K. Let L′ = K(x, y),and let ν′ = ν | L′ with valuation ring V ′ = V ∩ L′. By construction, if mV ′ is themaximal ideal of V ′, there exists c ∈ K such that mV ′ ∩K[x, y] ⊂ (x− c, y). Suppose(if possible) that p = mV ′ ∩K[x, y] = (x− c, y). Then the residue of x in K(V ) is c, acontradiction. Thus there exists an irreducible f ∈ R = K[x, y] such that p = (f). Itfollows that the valuation ring V ′ is Rp (c.f. Theorem 9, Section 5, Chapter VI [85])and the value group of ν′ is Z. The valuation ring V is a localization of the integralclosure of the local Dedekind domain V ′ in L. Thus V is itself a local Dedekinddomain which is a localization of a ring of finite type over K. In particular, V is itselfa regular algebraic local ring of L.

zero-dimensional valuations of rank 2

Let Γ1 be the non-trivial isolated subgroup of Γ, with corresponding prime idealp ⊂ V . The composition of ν with the order perserving homomorphism Γ → Γ/Γ1

defines a valuation ν1 of L of rank 1. Let V1 be the valuation ring of ν1. V1 consists of


the elements of K whose value by ν is in Γ1 or is positive. Hence V1 = Vp. ν inducesa valuation ν2 of K(Vp) with value group Γ1, where ν2(f) = ν(f∗) if f∗ ∈ Vp is alift of f to Vp. Since K(Vp) has a non-trivial valuation, which vanishes on K, K(Vp)cannot be an algebraic extension of K. Thus K(Vp) has positive transcendence degreeover K, so it must have transcendence degree 1. Let x∗ ∈ K(Vp) be such that x∗ istranscendental over K. Let x be a lift of x∗ to Vp. Then K(x) ⊂ Vp, so that ν1 istrivial on K(x), and ν1 is a valuation of the one-dimensional algebraic function fieldL over K(x).

Let V2 ⊂ K(VP ) be the valuation ring of ν2, and let π : Vp → K(Vp) be the residuemap. Then V = π−1(V2). ν1 is a 1 dimensional valuation of L, so Vp is an algebraiclocal ring which is a local Dedkind domain. The residue field K(Vp) is an algebraicfunction field of dimension 1 over K, so V2 is also a local Dedekind domain. Thus Γis a discrete group. Let w be a regular parameter in Vp.

Let w be a regular parameter in Vp. Given any non-zero element f ∈ L, we havea unique expression f = wmg where m = ν1(f) ∈ Z, and g is a unit in Vp. Let g∗

be the residue of g in K(Vp). Let n = ν2(g∗) ∈ Z. Then, up to an order preservingisomorphism of Γ, ν(f) = (m,n) ∈ Γ = Z2, where Z2 has the lexicographic order.

discrete zero-dimensional valuations of rank 1.

Let x ∈ L be such that ν(x) = 1. If y ∈ L and ν(y) = n1, then there exists aunique c ∈ K such that n2 = ν(y − c1xn1) > ν(y). If we iterate this constructioninfinitely many times, we have a uniquely determined Laurent series expansion

c1xn1 + c2x

n2 + . . .

with ν(y − c1xn1 − · · · − cmxnm) = nm+1 > nm for all m. This construction definesan embedding of K algebras L ⊂ K x, where K x is the field of Laurentseries in x. The order valuation on K x restricts to the valuation ν on L.

non-discrete zero-dimensional valuations of rank 1

We subdivide this case by the rational rank of ν. The rational rank is the dimensionof Γ ⊗ Q as a rational vector space. We may normalize ν so that Γ is an orderedsubgroup of R and 1 ∈ Γ.

Suppose that ν has rank larger that 1. Choose x, y ∈ L such that ν(x) = 1and ν(y) = τ is a positive irrational number. Suppose that F (x, y) =

∑aijx

iyj

is a polynomial in x and y with coefficients in K. ν(aijxiyj) = i + jτ for anyterm with aij 6= 0, so all the monomials in the expansion of F have distinct values.It follows that ν(F ) = min i + τj | aij 6= 0. We conclude that x and y arealgebraically independent, since a relation F (x, y) = 0 has infinite value. Furthermore,if L1 = K(x, y) and ν1 = ν | L1, we have that the value group Γ1 of ν1 is Z + Zτ .Since Γ/Γ1 is a finite group, we have that Γ is a group of the same type. In particular,Γ has rational rank 2.

Now suppose that ν has rational rank 1. We can normalize Γ so that it is anordered subgroup of Q, whose denominators are not bounded, as Γ is not discrete.In Example 3, Section 15, Chapter VI [85], examples are given of two dimensionalalgebraic function fields with value group equal to any given subgroup of the rationals.


There is a nice interpretation of this kind of valuation in terms of certain “formal”Puiseux series (c.f. MacLane and Shilling [59]) where the denominators of the rationalexponents in the Puiseux series expansion are not bounded.

Exercise

Let K be a field, R = K[x, y] be a polynomial ring and L be the quotient field ofR.

a. Show that the rule ν(x) = 1, ν(y) = π defines a K-valuation on L whichsatisfies

ν(f) = mini+ jπ | aij 6= 0if f =

∑aijx

iyj ∈ K[x, y].b. Compute the rank, rational rank and value group Γ of ν.c. Let V be the valuation ring of ν. Suppose that τ ∈ Γ. Show that

Iτ = f ∈ V | ν(f) ≥ τis an ideal of V .

d. Show that V is not a Noetherian ring.

8.2. Local uniformization of algebraic function fields of surfaces. In this sec-tion, we prove the following local uniformization theorem.Theorem 8.1. Suppose that K is an algebraically closed field of characteristic zero,L is a two dimensional algebraic function field over K and ν is a valuation of L.Then there exists a regular local ring A which is essentially of finite type over K withquotient field L such that ν dominates A.

Our proof is by a case by case analysis of the different types of valuations of Lwhich were classified in the previous section. If ν is 1 dimensional, we have thatR = V satisfies the conclusions of the theorem. We must prove the theorem in thecase when ν has dimension zero, and one of the following three cases hold: ν hasrational rank 1, ν has rank 1 and rational rank 1, ν has rank 2. The case when ν hasrank 1 includes the cases when ν is discrete and ν is not discrete. We will follow thenotation of the preceeding section.

valuations of rational rank 1 and dimension 0Lemma 8.2. Suppose that S = K[x1, . . . , xn] is a polynomial ring, ν is a rank 1valuation with valuation ring V and maximal ideal mV which contains S such thatK(V ) = K and mV ∩ S = (x1, . . . , xn). Suppose that there exist fi ∈ S for i ∈ Nsuch that

ν(f1) < ν(f2) < · · · < ν(fn) < · · ·is a strictly increasing sequence. Then Limi→∞ν(fi) =∞.

Proof. Suppose that ρ > 0 is a positive real number. We will show that there areonly finitely many real numbers less than ρ which are the values of elements of S.Let τ = min ν(f) | f ∈ (x1, . . . , xn). τ is well defined since S is Noetherian. Infact, if there where not a minimum, we would have an infinite descending sequence ofpositive real numbers

τ1 > τ2 > · · · τi > · · ·


and elements fi ∈ (x1, . . . , xn) such that ν(fi) = τi. Let Iτi = f ∈ S | ν(f) ≥ τi.Iτi are distinct ideals of S which form a strictly ascending chain

Iτ1 ⊂ Iτ2 ⊂ · · · ⊂ Iτi ⊂ · · ·which is impossible.

Choose a positive integer r such that r > ρτ . If f ∈ S is such that ν(f) < ρ

then write f = g + h with h ∈ (x1, . . . , xn)r and deg (g) < r. ν(h) > ρ impliesν(g) = ν(f). Thus there exists a finite number of elements f1, . . . , fm ∈ S (themonomials of degree < r) such that if 0 < λ < ρ is the value of an element of S, thenthere exist c1, . . . , cm ∈ K such that ν(c1f1 + · · ·+ cmfm) = λ.

We will replace the fi with appropriate linear combinations of the fi so that

ν(f1) < ν(f2) < · · · < ν(fm).

By induction, it suffices to make a linear change in the fi so that

ν(f1) < ν(fi) if i > 1. (100)

After reindexing the fi, we can suppose that there exists an integer l > 1 such that

ν(f1) = ν(f2) = · · · = ν(fl)

andν(fi) > ν(f1) if l < i.

ν( fif1) = 0 for 2 ≤ i ≤ l implies fi

f1∈ V and there exist ci ∈ K(V ) = K such

that fif1

has residue ci in K(V ). Then fif1− ci ∈ mV implies ν( fif1

− ci) > 0, so thatν(fi − cif1) > ν(f1) for 2 ≤ i ≤ l. After replacing fi with fi − cif1 for 2 ≤ i ≤ l, wehave that (100) is satisfied.

Theorem 8.3. Suppose that S = K[x, y] is a polynomial ring over an algebraicallyclosed field K of characteristic zero, ν is a rational rank 1 valuation of K(x, y) withvaluation ring V and maximal ideal mV , K(V ) = V/mV = K such that S ⊂ V andthe center of V on S is the maximal ideal (x, y). Further suppose that f ∈ K[x, y]is given. Then there exists a birational extension K[x, y] → K[x′, y′] (K[x, y] andK[x′, y′] have a common quotient field) such that K[x′, y′] ⊂ V , the center of ν onK[x′, y′] is (x′, y′) and f = (x′)lδ where l is a non-negative integer and δ ∈ K[x′, y′]is not in (x′, y′).

Proof. Set r = ord f(0, y). We have 0 ≤ r ≤ ∞. If r = 0 we are done, so supposethat r > 0. We have an expansion

f =d∑i=1

xαiγi(x)yβi +∑i>d

xαiγi(x)yβi (101)

where the first sum is over terms with minimal value ρ = ν(xαiγi(x)yβi) for 1 ≤ i ≤ d,γi(x) ∈ K[x] are polynomials with non-zero constant term, β1 < · · · < βd, βd+1 <βd+2 < · · · and ν(xαiγi(x)yβi) > ρ if i > d.

Set ν(x)ν(y) = a

b with a, b ∈ N, (a, b) = 1. There exist non-negative integers a′, b′ such

that ab′ − ba′ = 1. ν(ya

xb) = 0 implies there exists 0 6= c ∈ K(V ) = K such that

ν(ya

xb− c) > 0. Set

x = xa1(y1 + c)a′, y = xb1(y1 + c)b

′.


We have thatx1 = xb

′y−a

′, y1 + c = x−bya

so that ν(y1) > 0 and

ν(x1) =ν(y)b

[ab′ − a′b] =ν(y)b

> 0.

Set α = α1a+β1b. We have that αia+βib = α for 1 ≤ i ≤ d. Thus there exist ei ∈ Zsuch that (

a ba′ b′

)(αi − α1

βi − β1

)=(

0ei

)for 1 ≤ i ≤ d. By Cramer’s rule, we have

βi − β1 = Det(a 0a′ ei

)= aei.

Thus ei = βi−β1a .

We have a factorization f = xα1 f1(x1, y1) where

f1(x1, y1) = (y1 + c)a′α1+b′β1(

d∑i=1

γi(y1 + c)βi−β1a ) + x1Ω

is the strict transform of f in the birational extensionK[x1, y1] ofK[x, y]. ord(f, 0, y) =r implies there exists an i such that αi = 0, βi = r. Thus ν(zr) ≥ ρ and βi ≤ r for1 ≤ i ≤ d. We have

r1 = ord f1(0, y1) ≤ βd − β1

a≤ r.

If r1 < r then we have a reduction, and the theorem follows from induction on r,since the conclusions of the theorem hold in K[x, y] if r = 0. We thus assume thatr1 = r. Then βd = r, β1 = 0, a = 1 and αd = 0. Further, there is an expression

d∑i=1

γi(0)(y1 + c)βi = γd(0)yr1.

Let u be an indeterminate, and set

g(u) =d∑i=1

γi(0)uβi = γd(0)(u− c)r.

The binomial theorem implies that βd−1 = r − 1 and thus

ν(y) = ν(xαd−1).

Let 0 6= λ ∈ k(V ) = k be the residue of yxαd−1 . Then ν(y − λxαd−1) > ν(y). Set

y′ = y − λxαd−1 and f ′(x, y′) = f(x, y). We have ord f ′(0, y′) = ord f(0, y) = r.yr is a minimal value term of f(x, y) in the expansion (101), so ν(y) ≤ ν(f).Replacing y with y′ and f with f ′ and iterating the above procedure after (101), we

either achieve a reduction ord f1(0, y1) < r or we find that there exists a′(x) ∈ K[x]such that ν(y′ − a′(x)) > ν(y′). We further have that ν(y′) ≤ ν(f). Set y′′ =y′ − a′(x) and repeat the algorithm following (101) until we either reach a reductionord f1(0, y1) < r or show that there exist polynomials a(i)(x) ∈ K[x] for i ∈ N such

that we have an infinite increasing bounded sequence

ν(a(x)) < ν(a′(x)) < · · ·


By Lemma 8.2, this is impossible. Thus there is a polynomial q(x) ∈ K[x] such thatafter replacing y with y − q(x), and following the algorithm after (101),we achieve areduction r1 < r in K[x1, y1]. By induction on r, we can achieve the conclusions ofthe theorem.

We now give the proof of Theorem 8.1 in the case of this subsection, that is, ν hasdimension 0 and rational rank 1.

Let x, y ∈ L be algebraically independent over K and of positive value. Let w bea primitive element of L over K(x, y) of positive value. Let R = K[x, y, w]. R ⊂ V ,the center of ν on R is the maximal ideal (x, y, w), and R has quotient field L. Letz be algebraically independent over K(x, y), and let f(x, y, z) in the polynomial ringK[x, y, z] be the irreducible polynomial such that R ∼= K[x, y, z]/(f). After possiblymaking a change of variables, replacing x with x+ βz and y with y + γz for suitableβ, γ ∈ K, we may assume that

r = ord f(0, 0, z) <∞.

If r = 1, thenRmV ∩R is a regular local ring which is dominated by ν, so the conclusionsof the theorem are satisfied. We thus suppose that r > 1. Let π : K[x, y, z] →K[x, y, w] be the natural surjection. For g ∈ K[x, y, z], we will denote ν(π(g)) byν(g). Write

f(x, y, z) =d∑i=1

ai(x, y)zσi +∑i>d

ai(x, y)zσi (102)

where ai(x, y) ∈ K[x, y] for all i, ai(x, y)zσi are the terms of minimal value ρ =ν(ai(x, y)zσi) for 1 ≤ i ≤ d, ν(ai(x, y)zσi) > ρ for i > d and σ1 < σ2 < · · · < σd.

By Theorem 8.3, there exists a birational extension K[x, y] → K[x1, y1] withK[x1, y1] ⊂ V and mV ∩K[x1, y1] = (x1, y1) such that

f =d∑i=1

xλi1 ai(x1, y1)zσi +∑i>d

xλi1 ai(x1, y1)zσi

where ai(0, 0) 6= 0 for all i. R1 = K[x1, y1, z1]/(f) is a domain with quotient field L,R → R1 is birational, R1 ⊂ V and mV ∩ R1 = (x1, y1, z). Write ν(z)

ν(x1) = st with s, t

relatively prime positive integers. Let s′, t′ be positive integers such that s′t−st′ = 1.Set

x1 = xt2(z2 + c2)t′, y1 = y2, z = xs2(z2 + c2)s

′

where 0 6= c2 ∈ K is determined by the condition ν(z2) > 0. Let α = λ1t+ σ1s. Wehave α = λit + σis for 1 ≤ i ≤ d. We have (as in the proof of Theorem 8.3) thatσi−σ1t is a non-negative integer for 1 ≤ i ≤ d and f = xα2 f2(x2, y2, z2) where

f2(x2, y2, z2) = (x2 + c2)t′λ1+s′σ1(

d∑i=1

ai(z2 + c2)σi−σ1t ) + x2Ω

is the strict transform of f in k[x2, y2, z2]. Set R2 = K[x2, y2, z2]/(f2). R1 → R2 isbirational, R2 ⊂ V and mV ∩R2 = (x2, y2, z2).

Setr1 = ord f2(0, 0, z2) ≤ σd − σ1

t≤ r.


If r1 < r we have a reduction, so we may assume that r1 = r. Then we have σd = r,σ1 = 0, t = 1 and ad(0, 0) 6= 0. We further have a relation

d∑i=1

ai(0, 0)(z2 + c2)σi = ad(0, 0)zr2 .

Set

g(u) =d∑i=1

ai(0, 0)uσi = ad(0, 0)(u− c2)r.

The binomial theorem shows that σd−1 = r − 1 and ν(z) = ν(ad−1(x, y)). Thereexists c ∈ k such that ν(z − cad−1(x, y)) > ν(z). radzr−1 is a minimal value term of∂f∂z so that ν(∂f∂z ) ≥ ν(z). Set z′ = z − cad−1(x, y), f ′(x, y, z′) = f(x, y, z). We haveord (f ′(0, 0, z′)) = r and ∂f ′

∂z′ = ∂f∂z .

We repeat the analysis following (102), with f replaced by f ′ and z replaced with z′.We either get a reduction r1 = ord (f2(0, 0, z2)) < r, or there exists a′(x, y) ∈ K[x, y]such that ν(z′ − a′(x, y)) > ν(z′) and

ν(z′) ≤ ν(∂f ′

∂z′) = ν(

∂f

∂z).

Set z′′ = z′ − a′(x, y) and repeat the algorithm following (102). If we do reach areduction r1 < r after a finite number of iterations, there exist polynomials a(i)(x, y) ∈K[x, y] for i ∈ N such that

ν(a(x, y)) < ν(a′(x, y)) < · · ·is an increasing bounded sequence of real numbers. By Lemma 8.2 this is impossible.

Thus there exists a polynomial a(x, y) ∈ K[x, y] such that after replacing z withz − a(x, y), the algorithm following (102) results in a reduction r1 < r. We thenrepeat the algorithm, so that by induction on r, we construct a birational extensionof local rings R→ R1 so that R1 is a regular local ring which is dominated by V .

valuations of rank 1, rational rank 2 and dimension 0

After normalizing ν, we have that its value group is Z + Zτ where τ is a positiveirrational number. Choose x, y ∈ L such that ν(x) = 1 and ν(y) = τ . By the analysisof the preceeding case (rational rank 1 and dimension 0), x and y are algebraicallyindependent over K. Let w be a primitive element of L over K(x, y) which haspositive value. There exists an irreducible polynomial f(x, y, z) in the polynomialring K[x, y, z] such that T = K[x, y, w] ∼= K[x, y, z]/(f) and mV ∩ T = (x, y, w).After possibly replacing x with x + βzm and y with y + γzm where m is sufficientlylarge that mν(w) > max ν(x), ν(y), and β, γ ∈ K are suitably chosen, we mayassume that

r = ord (f(0, 0, z) <∞.If r = 1, TmV ∩T is a regular local ring, so we assume that r > 1. Let π : K[x, y, z]→K[x, y, w] be the natural surjection. For g ∈ K[x, y, z], we will denote ν(π(g)) byν(g). Write

f =d∑i=1

aixαiyβizγi +

∑i>d

aixαiyβizγi


where all ai ∈ K are non-zero, aixαiyβizγi for 1 ≤ i ≤ d are the minimal value terms,and γ1 < γ2 < · · · < γd.

There exist integers s, t such that ν(z) = s + τt. There exist integers M,N suchthat

ν(xαiyβizγi) = M +Nτ

for 1 ≤ i ≤ d. We thus have equalities

M = α1 + sγ1 = · · · = αd + sγdN = β1 + tγ1 = · · · = βd + tγd.

(103)

We further have(αi + sγi) + (βi + tγi)τ > M +Nτ

if i > d. Since ord (f(0, 0, z)) = r and adxαdyβdzγd is a minimal value term, we haveγd ≤ r. We expand τ into a continued fraction:

τ = h1 +1

h2 + 1h3+···

.

Let fjgj

be the convergent fractions of τ (c.f. Section 10.2 [44]). Since Lim fjgj

= τ , wehave

(αi + sγi) + (βi + tγi)fjgj

> M +Nfjgj

for j = p − 1 and j = p whenever i > d and p is sufficiently large. We further havethat sgp+ tfp > 0 and sgp−1 + tfp−1 > 0 for p sufficiently large. Since ν(xsyt) = ν(z),there exists c ∈ K = K(V ) such that ν( z

xsyt − c) > 0. Consider the extension

K[x, y, z]→ K[x1, y1, z1] (104)

where

x = xgp1 y

gp−11 , y = x

fp1 y

fp−11 ,

z = xsyt(z1 + c) = xsgp+tfp1 y

sgp−1+tfp−11 (z1 + c).

(105)

We have fp−1gp − fpgp−1 = ε = ±1, (c.f. Theorem 150 [44]) and ε,−τ + fp−1gp−1

and

τ − fpgp

have the same signs (c.f. Theorem 154 [44]). From (105), we see that

xε1 =xfp−1

ygp−1, yε1 =

ygp

xfp

from which we deduce that (104) is birational, and

εν(x1) = gp−1(fp−1

gp−1− τ), εν(y1) = gp(τ −

fpgp

).

Hence ν(x1), ν(y1), ν(z1) are all > 0. We have

f(x, y, z) = xMgp+Nfp1 y

Mgp−1+Nfp−11 f1(x1, y1, z1)

where

f1 = (z1 + c)γ1(d∑i=1

ai(z1 + c)γi−γ1) + x1y1H

is the strict transform of f in K[x1, y1, z1]. We thus have constructed a birationalextension T → T1 = K[x1, y1, z1]/(f1) such that T1 ⊂ V and mV ∩ T1 = (x1, y1, z1).


We have f1(0, 0, 0) = 0, so u = c is a root of

φ(u) =d∑i=1

aiuγi−γ1 = 0.

Let r1 = ord f1(0, 0, z1). Then r1 ≤ γd − γ1 ≤ r. If r1 < r we have obtained areduction, so we may assume that r1 = r. Thus γ1 = 0, γd = r, αd = βd = 0 (sinceγi ≤ r for 1 ≤ i ≤ d) and

φ(u) = ad(u− c)r.Hence γd−1 = r − 1 and ad−1 6= 0, by the binomial theorem. We have ν(z) =αd−1 + βd−1τ . Our conclusion is that there exist non-negative integers m and n suchthat ν(z) = m+ nτ . There exists c ∈ K such that

ν(z − cxmyn) > ν(z).

We make a change of variables in K[x, y, z], replacing z with z′ = z− cxmyn, and letf ′(x, y, z′) = f(x, y, z). We have ν(z′) > ν(z) and ord f ′(0, 0, z′) = r. Since radzr−1

is a minimal value term of ∂f∂z , we have ν(z) ≤ ν(∂f∂z ). We further have ∂f

∂z = ∂f ′

∂z′ .We now apply a transformation of the kind (105) with respect to the new variablesx, y, z′. If we do not achieve a reduction r1 < r, we have a relation ν(z′) = m1 + n1τwith m1, n1 non-negative integers, and we have

ν(z) < ν(z′) ≤ ν(∂f

∂z).

We have that there exists c2 ∈ K such that

ν(z′ − c2xm1yn1) > ν(z′).

Set z′′ = z′−c2xm1yn1 . We now iterate the procedure starting with the transformation(105). If we do not achieve a reduction r1 < r after a finite number of iterations, weconstruct an infinite bounded sequence (since ∂f

∂z 6= 0 in T )

m+ nτ < m1 + n1τ < · · · < mi + niτ < · · ·where mi, ni are non-negative integers, which is impossible. We thus must achievea reduction r1 < r after a finite number of interations. By induction on r, we canconstruct a birational extension T → T1 such that T1 ⊂ V and (T1)mV ∩T1 is a regularlocal ring.

valuations of rank 2 and dimension 0

Let 0 ⊂ p ⊂ mV be the distinct prime ideals of V . Let x, y ∈ L be such that x, yare a transcendence basis of L over K, ν(x) and ν(y) are positive and the residue ofx in K(Vp) is a transcendence basis of K(Vp) over K. Let w be a primitive elementof L over K(x, y) such that ν(w) > 0. Let T = K[x, y, w] ⊂ V . Let f(x, y, z) be anirreducible element in the polynomial ring K[x, y, z] such that

K[x, y, z]/(f) ∼= K[x, y, w]. (106)

By our construction, the quotient field of T is L and trdegK(T/p ∩ T ) = 1. Thusp ∩ T is a prime ideal of a curve on the surface spec(T ).

Set R0 = RmV ∩T . Let C0 be the curve in spec(T ) with ideal p∩R0 in R0. Corollary4.4 implies there exists a proper birational morphism π : X → spec(T ) which is a


sequence of blow ups of points such that the strict transform C0 of C0 on X is non-singular. Let X1 → X be the blow up of C0. Since X1 → spec(T ) is proper, thereexists a unique point a1 ∈ X1 such that V dominates R1 = OX1,a1 . R1 is a quotientof a 3 dimensional regular local ring (as explained in Lemma 5.4). Let C1 be thecurve in X1 with ideal p ∩ R1 in R1. (R1)p∩R1 dominates Rp∩R and (R1)p∩R1 is alocal ring of a point on the blow up of the maximal ideal of Rp∩R. We can iteratethis procedure to construct a sequence of local rings (with quotient field L)

R0 → R1 → · · · → Ri → · · ·

such that V dominates Ri for all i, Ri is quotient of a regular local ring of dimension3 and (Ri+1)p∩Ri+1 is a local ring of the blow up of the maximal ideal of (Ri)p∩Ri forall i. Furthermore, each Ri is a localization of a quotient of a polynomial ring in 3variables.

Since (R0)p∩R0 can be considered as a local ring of a point on a plane curve overthe field K(x) (as is explained in the proof of Lemma 5.10), by Theorem 3.12 (orCorollary 4.4) there exists an i such that (Ri)p∩Ri is a regular local ring.

After replacing the T in (106) with a suitable affine ring, we can now assume thatTp∩T is a regular local ring. If TmV ∩T is a regular local ring we are done, so supposethat TmV ∩T is not a regular local ring. Let p = π−1(p ∩ T ) where π : K[x, y, z]→ Tis the natural surjection. If g ∈ K[x, y, z] is a polynomial, we will denote ν(π(g)) byν(g). Let I ⊂ K[x, y, z] be the ideal I = (f, ∂f∂x ,

∂f∂y ,

∂f∂z ) defining the singular locus of

T . Since I 6⊂ p (as Tp∩T is a regular local ring) one of π(∂f∂x ), π(∂f∂y ), π(∂f∂z ) 6∈ p ∩ T .Thus

minν(∂f

∂x), ν(

∂f

∂y), ν(

∂f

∂z) = (0, n)

for some n ∈ N. Writef = Fr + Fr+1 + · · ·+ Fm

where Fi is a form of degree i in x, y, z of degree i and r = ord (f). After possiblyreindexing the x, y, z we may assume that 0 < ν(x) ≤ ν(y) ≤ ν(z).

Since K(V ) = K, there exist c, d ∈ k (which could be 0) such that ν(y − cx) >ν(x) and ν(z − dx) > ν(x). Thus we have a birational transformation K[x, y, z] →K[x1, y1, z1] (a quadratic transformation) where

x1 = x, y1 =y − cxx

, z1 =z − dxx

,

with inversex = x1, y = x1(y1 + c), z = x1(z1 + d).

In K[x1, y1, z1] we have f(x, y, z) = xr1f1(x1, y1, z1) where

f1(x1, y1, z1) = Fr(1, y1+c, z1+d)+x1Fr+1(1, y1+c, z1+d)+· · ·+xm−r1 Fm(1, y1+c, z1+d)

is the strict transform of f . Let T1 = K[x1, y1, z1]/(f1). T → T1 is birational, T1 ⊂ Vand m ∩ T1 = (x1, y1, z1). We calculate

∂f∂x = xr−1

1 (x1∂f1∂x1− (y1 + c)∂f1

∂y1− (z1 + d) ∂f∂z1 + rf1)

∂f∂y = xr−1

1∂f1∂y1

∂f∂z = xr−1

1∂f1∂z1


Hence

min ν(∂f

∂x), ν(

∂f

∂y), ν(

∂f

∂z) ≥ (r − 1)ν(x1) + min ν(

∂f1

∂x1), ν(

∂f1

∂y1), ν(

∂f1

∂z1).

Since we have assumed that TmV ∩T is not regular, we have r > 1. We further havethat ν(x1) > 0, so we conclude that

min ν(∂f1

∂x1), ν(

∂f1

∂y1), ν(

∂f1

∂z1) = (0, n1) < (0, n).

Let r1 = ord (f1) ≤ r. (T1)mV ∩T1 is a regular local ring if and only if r1 = 1. Thuswe may assume that r1 > 1.

We now apply to T1 a new quadratic transform, to get

min ν(∂f2

∂x2), ν(

∂f2

∂y2), ν(

∂f2

∂z2) = (0, n2) < (0, n1).

Thus after a finite number of quadratic transforms T → Tl, we construct a Tl suchthat (Tl)mV ∩Tl is a regular local ring which is dominated by V .Remark 8.4. Zariski proves Local Uniformization for arbitrary characteristic zeroalgebraic function fields in [80].

8.3. Resolving systems and the Zariski-Riemann manifold. Let L/K be analgebraic function field. A projective model of L is a projective K-variety whosefunction field is L.

Let Σ(L) be the set of valuation rings of L. If X is a projective model of L thenthere is a natural morphism πX : Σ(L)→ X defined by πX(V ) = p if p is the (unique)point of X whose local ring is dominated by V . We will say that p is the center of Von X. There is a topology on Σ(L) where a basis for the topology are the open setsπ−1X (U) for an open set U of a projective model X of L. Zariski shows that Σ(L) is

quasi-compact (every open cover has a finite subcover) in [82] and Section 17, ChapterVI [85]. Σ(L) is called the Zariski-Riemann manifold of L.

Let N be a set of zero-dimensional valution rings of L. A resolving system of N isa finite collection S1, . . . , Sr of projective models of L such that for each V ∈ N ,the center of V on at least one of the Si is a non-singular point.Theorem 8.5. Suppose that L is a two-dimensional algebraic function field over analgebraically closed field K of characteristic zero. Then there exists a resolving systemfor the set of all zero-dimensional valuation rings of L.

Proof. By local uniformization (Theorem 8.1), for each valuation W of L there existsa projective model XW of L such that the center of W on XW is non-singular. LetUW be an open neighborhood of the center of W in XW which is non-singular, andlet AW = π−1

W (UW ), an open neighborhood of W in Σ(L). The set AW |W ∈ Σ(L)is an open cover of Σ(L). Since Σ(L) is quasi-compact, there is a finite subcoverAW1 , . . . , AWr of Σ(L). XW1 , . . . , XWr is then a resolving system for the set of allzero-dimensional valuation rings of L.

Lemma 8.6. Suppose that L is a two-dimensional algebraic function field over analgebraically closed field K of characteristic zero, and

R0 ⊂ R1 ⊂ · · · ⊂ Ri ⊂ · · ·


is a sequence of distinct normal two dimensional algebraic local rings of L. Let Ω =∪Ri. If every one dimensional valuation ring V of L which contains Ω intersects Riin a height one prime for some i, then Ω is the valuation ring of a zero-dimensionalvaluation of L.

Proof. Ω is by construction integrally closed in L. Thus Ω is the intersection of thevaluation rings of L containing Ω (c.f. Corollary to Theorem 8, Section 5, ChapterVI [85]). Observe that Ω 6= L. We see this as follows. If mi is the maximal ideal ofRi, then m = ∪mi is a non-zero ideal of Ω. Suppose that 1 ∈ m. Then 1 ∈ mi forsome i, which is impossible.

We will show that Ω is an intersection of zero-dimensional valuation rings. Toprove this, it suffices to show that if V is a one-dimensional valuation ring containingΩ, then there exists a zero-dimensional valuation ring W such that V is a localizationof W and Ω ⊂W . Since V is one-dimensional, V is a local Dedekind domain and is analgebraic local ring of L. Let K(V ) be the residue field of V . Let mV be the maximalideal of V , and let pi = Ri ∩mV . By assumption, there exists an index j such thati ≥ j implies that pi is a height one prime in Ri. (Ri)pi is a normal local ring whichis dominated by V . Since both (Ri)pi and V are local Dedekind domains, and theyhave the common quotient field L, we have that (Ri)pi = V for i ≥ j (c.f. Theorem9, Section 5, Chapter VI [85]). Thus the residue field Ki of (Ri)pi is K(V ) for i ≥ j.The one dimensional algebraic local rings Si = Ri/pi for i ≥ j have quotient fieldK(V ) and Si+1 dominates Si. ∪Si is not K(V ) by the same argument we used toshow that Ω is not L. Thus there exists a valuation ring V ′ of K(V ) which dominates∪Si. Let π : V → K(V ) be the residue map. W = π−1(V ′) is a zero dimensionalvaluation ring of V and W contains Ω.

Since we have established that Ω is the intersection of zero-dimensional valuationrings, we need only show that Ω is not contained in two distinct zero-dimensional val-uation rings. Suppose that Ω is contained in two distinct zero-dimensional valuations,V1 and V2.

We will first construct a projective surface Y with function field L such that V1

and V2 dominate (unique) distinct points q1 and q2 of Y . Let x, y be a transcendencebasis of L over K. Let ν1, ν2 be valuations of L whose respective valuation rings areV1 and V2. If ν1(x) and ν2(x) are both ≤ 0, replace x with 1

x . Suppose that ν1(x) ≥ 0and ν2(x) < 0. Let c ∈ K be such that the residue of x in the residue field K(V1) ∼= Kof V1 is not c. After replacing x with x − c, we have that ν1(x) = 0 and ν2(x) < 0.We can now replace x with 1

x to get that ν1(x) ≥ 0 and ν2(x) ≥ 0. In this way,after possibly changing x and y, we can assume that x and y are contained in thevaluation rings V1 and V2. Since V1 and V2 are distinct zero-dimensional valuationrings, there exists f ∈ V1 such that f 6∈ V2 and g ∈ V2 such that g 6∈ V1 (c.f. Theorem3, Section 3, Chapter VI [85]). As above, we can assume that ν1(f) = 0, ν2(f) < 0,ν1(g) < 0 and ν2(g) = 0. Let a = 1

f , b = 1g , A be the integral closure of K[x, y, a, b]

in L. By construction, A ⊂ V1 ∩ V2. Let m1 = A ∩mV1 , m2 = A ∩mV2 . We havethat a ∈ m2 but a 6∈ m1 and b ∈ m1 but b 6∈ m2. Thus m1 6= m2. We now let Y bethe normalization of a projective closure of spec(A). Y has the desired properties.

Since Ri is essentially of finite type over K, it is a localization of a finitely generatedK-algebra Bi which has quotient field L. Let Xi be the normalization of a projectiveclosure of spec(Bi). Since Ri is integrally closed, the projective surface Xi has a pointai with associated local ring OXi,ai = Ri.


There is a natural birational map Ti : Xi → Y . We will now establish that Ticannot be a morphism at ai. If Ti where a morphism at ai, there would be a uniquepoint bi on Y such that Ri = OXi,ai dominates OYi,bi . Since V1 and V2 both dominateOXi,ai , V1 and V2 both dominate OYi,bi , a contradiction.

By Zariski’s main theorem (c.f. Theorem V.5.2 [45]) there exists a one-dimensionalvaluation ring W of L such that W dominates OXi,ai and W dominates the local ringof the generic point of a curve on Y . Let Γi be the set of one-dimensional valuationrings W of K such that W dominates OXi,ai and W dominates the local ring of thegeneric point of a curve on Y . Γi is a finite set since the valuation rings W of Γiare in 1-1 correspondence with the curves contained in π−1

1 (ai), where π1 : ΓTi → Xi

is the projection of the graph of Ti onto Xi. Since Ri+1 = OXi+1,ai+1 dominatesRi = OXi,ai , Γi+1 ⊂ Γi for all i. Since each Γi is a non-empty finite set, thereexists a one-diimensional valuation ring W of L such that W dominates Ri for all i,a contradiction.

Theorem 8.7. Suppose that L is a two-dimensional algebraic function field over analgebraically closed field K of characteristic zero, R is a two dimensional normalalgebraic local ring with qoutient field L and that

R→ R1 → R2 → · · · → Ri → · · ·is a sequence of normal local rings such that Ri+1 is obtained from Ri by blowingup the maximal ideal, normalizing, and localizing at a maximal ideal so that Ri+1

dominates Ri. Then Ω = ∪Ri is a zero-dimensional valuation ring of L.

Proof. By Lemma 8.6, we are reduced to showing that there does not exist a one-dimensional valuation ring V of L such that Ω is contained in V and mV ∩Ri is themaximal ideal for all i.

Suppose that V is a one-dimensional valuation ring of L which contains Ω. Let νbe a valuation of L whose associated valuation ring is V . Since V is one-dimensional,there exists α ∈ V such that the residue of α in K(V ) is transcendental over K.

Let mi be the maximal ideal of Ri. Assume that mV ∩R is the maximal ideal m0 ofR. Write α = ξ

η with ξ, η ∈ R. α 6∈ R sinceR/m0∼= K. Hence η ∈ m0 and also ξ ∈ m0

since ν(α) ≥ 0. Since R1 is a local ring of the normalization of the blow up of m0,m0R1 is a principal ideal. In fact if ζ ∈ m0 is such that ν(ζ) = min ν(f) | f ∈ m0,we have that ζR1 = m0R1. Thus ξ1 = ξ

ζ ∈ R1 and η1 = ηζ ∈ R1.

If we assume that mV ∩R1 = m1, we have that ξ1, η1 ∈ m1, α = ξ2η2

with ξ2, η2 ∈ R2

andν(ξ) > ν(ξ1) > 0, ν(ξ1) > ν(ξ2)ν(η) > ν(η1) > 0, ν(η1) > ν(η2).

More generally, if we assume that mi = mV ∩V for i = 1, . . . , n, we can find elementsξi, ηi ∈ Ri for 1 ≤ i ≤ n such that α = ξi

ηiand

ν(ξ) > ν(ξ1) > · · · > ν(ξn) > 0

ν(η) > ν(η1) > · · · > ν(ηn) > 0.Since the value group of V is Z, it follows that n ≤ min ν(ξ), ν(η). Hence for all isufficiently large mV ∩ Ri is not the maximal ideal. mV ∩ Ri must then be a heightone prime in Ri, for otherwise we would have mV ∩Ri = (0), which would imply thatV = L, against our assumption.


Remark 8.8. The conclusions of this theorem are false in dimension 3 (Shannon[70]).Theorem 8.9. Suppose that S, S′ are normal projective surfaces over an algebraicallyclosed field K of characteristic zero and T is a birational map from S to S′. LetS1 → S be the projective morphism defined by taking the normalization of the blow upof the finitely many points of T where T is not a morphism. If the induced birationalmap T1 from S1 to S′ is not a morphism, let S2 → S1 be the normalization of theblow up of the finitely many fundamental points of T1. We then iterate to produce asequence of birational morphisms of surfaces

S ← S1 ← S2 ← · · · ← Si ← · · ·

This sequence must be of finite length, so that the induced rational map Si → S′ is amorphism for all i sufficiently large.

Proof. Since Si is a normal surface, the set of points where Ti is not a morphism isfinite (c.f. Lemma 5.1 [45]). Suppose that no Ti : Si → S′ is a morphism. Thenthere exists a sequence of points pi ∈ Si for i ∈ N such that pi+1 maps to pi andTi is not a morphism at pi for all i. Let Ri = OSi,pi . The birational maps Ti givean identification of the function fields of the Si and S′ with a common field L. ByTheorem 8.7 Ω = ∪Ri is a zero-dimensional valuation ring. Hence there exists apoint q ∈ S′ such that Ω dominates OS′,q. OS′,q is a localization of a K-algebraB = K[f1, . . . , fr] for some f1, . . . , fr ∈ L. Since B ⊂ Ω there exists some i such thatB ⊂ Ri. Since Ω dominates Ri we necessarilly have that Ri dominates OS′,q. ThusTi is a morphism at pi, a contradiction to our assumption. Thus our sequence mustbe finite.

Theorem 8.10. Suppose that S and S′ are projective surfaces over an algebraicallyclosed field K of characteristic zero which form a resolving system for a set of zero-dimensional valuations N . Then there exists a projective surface S∗ which is a re-solving system for N .

Proof. By Theorem 8.9, there exists a birational morphism S → S obtained by asequence of normalizations of the blow ups of all points where the birational map toS′ is not defined. Now we construct a birational morphism S

′ → S′, applying thealgorithm of Theorem 8.9 by only blowing up the points where the birational map to Sis not defined and which are nonsingular points. The algorithm produces a birationalmap S

′ → S which is a morphism at all non-singular points of S′.

Let S∗ be the graph of the birational map from S to S′, with natural projections

π1 : S∗ → S and π2 : S∗ → S′. We will show that S∗ is a resolving system for

N . Suppose that V ∈ N . Let p, p′, p′ and p∗ be the centers of V on the respectiveprojective surfaces S, S′, S

′and S

∗.

First suppose that p′ is a singular point. Then p must be nonsingular, as S, S′ isa resolving system for N . The birational map from S to S

′is a morphism at p, and

π1 is an isomorphism at p. Thus the center of V on S∗ is a nonsingular point.Now suppose that p′ is a nonsingular point. Then p′ is a nonsingular point of S

′

and the birational map from S′

to S is a morphism at p′. Thus π2 is an isomorphismat p′ and the center of V on S∗ is a nonsingular point.


Theorem 8.11. Suppose that L is a two-dimensional algebraic function field overan algebraically closed field of characteristic zero. Then there exists a nonsingularprojective surface S with function field L.

Proof. The Theorem follows from Theorem 8.5 and Theorem 8.9, by induction on rapplied to a resolving system S1, . . . , Sr for the set of zero-dimensional valuationrings of L.

Remark 8.12. Zariski proves the generalization of Theorem 8.10 to dimension 3in [83], and deduces resolution of singularities for characteristic zero 3-folds fromhis proof of local uniformization for characteristic zero algebraic function fields [80].Abhyankar proves local uniformization in dimension three and characteristic p > 5,from which he deduces resolution of singularites for 3-folds of characteristic p > 5 [4].

It is a very interesting and evidently extremely difficult problem to generalize The-orem 8.10 to dimension ≥ 4.

Exercises1. Prove that all the birational extensions constructed in Section 8.2 are products

of blow ups of points and nonsingular curves.2. Identify where characteristic zero is used in the proofs of this chapter. All but

one of the cases of Section 8.2 extend without great difficulty to characteristicp > 0. Some care is required to ensure that K(x, y)→ L is separable. In [1],Abhyankar accomplishes this and gives a very different proof in the remainingcase to prove local uniformization in characteristic p > 0 for algebraic surfaces.

9. Ramification of valuations and simultaneous resolution of

singularities

Suppose that L is an algebraic function field over a field K, and trdegKL < ∞ isarbitrary. We will use the notation on valuation rings of Section 8.1.

Suppose that R is a local ring contained in L. We will say that R is algebraic (oran algebraic local ring of L) if L is the quotient field of R and if R is essentially offinite type over K. Let K(S) denotes the residue field of a ring S containing K witha unique maximal ideal. We will also denote the maximal ideal of a ring S containinga unique maximal ideal by mS .

Suppose that L → L∗ is a finite separable extension of algebraic function fieldsover K, and V ∗ is a valuation ring of L∗/K associated to a valuation ν∗ with valuegroup Γ∗. Then the restriction ν = ν∗ | L of ν∗ to L is a valuation of L/K withvaluation ring V = L ∩ V ∗. Let Γ be the value group of ν.

There is a commutative diagram

L → L∗

↑ ↑V = L ∩ V ∗ → V ∗

.

There are associated invariants (c.f. page 53, Section 11, Chapter VI [85]) namely,the reduced ramification index of ν∗ relative to ν,

e = [Γ∗ : Γ] <∞


and the relative degree of ν∗ with respect to ν,

f = [K(V ∗) : K(V )].

The classical case is when V ∗ is an algebraic local Dedekind domain. In this caseV is also an algebraic local Dedekind domain. If mV ∗ = (y) and mV = (x), thenthere is a unit γ ∈ V ∗ such that

x = γye.

Since we thus haveΓ∗/Γ ∼= Z/eZ,

we see that e is the reduced ramification index.Still assuming that V ∗ is an algebraic local Dedekind domain, if we further assume

that K = K(V ∗) is algebraically closed, and (e, char(K)) = 1, then we have that theinduced homomorphism on completions with respect to the respective maximal ideals

V → V ∗

is given by the natural inclusion of K algebras

K[[x]]→ K[[y]]

where y = e√γy and x = ye. We thus have a natural action of Γ∗/Γ on V ∗ (a generator

multiplies y by a primitive e-th root of unity) and the ring of invariants by this actionis

V ∼= (V ∗)Γ∗/Γ.

In this chapter we find analogs of these results for general valuation rings. Thebasic approach is to find algebraic local rings R of L and S of L∗ such that V ∗

dominates S (S is contained in V ∗ and mV ∗ ∩S = mS) and V dominates R such thatthe ramification theory of V → V ∗ is captured in

R→ S, (107)

and we have a theory for R→ S comparable to that of the above analysis of V → V ∗

in the special case when V ∗ is an algebraic local Dedekind domain.This should be compared with Zariski’s Local Uniformization Theorem [80] (also

Section 2.5 and Chapter 8 of this book). It is proven in [80] that if char(K) = 0 and Vis a valuation ring of an algebraic function field L/K, then V dominates an algebraicregular local ring of L. The case when trdegKL = 2 is proven in Section 8.2.

We first consider the following diagram:

L → L∗

↑S

(108)

where L∗/L is a finite separable extension of algebraic function fields over L, and S isa normal algebraic local ring of L∗. We say that S lies above an algebraic local ringR of L if S is a localization at a maximal ideal of the integral closure of R in L∗.

Abhyankar and Heinzer have given examples of diagrams (108) where there doesnot exist a local ring R of L such that S lies above R (c.f. [33]). To construct anextension of the kind (107), we must at least be able to find an S dominated by V ∗

such that S lies over an algebraic local ring R of L. To do this, we consider theconcept of a monoidal transform.


Suppose that V is a valuation ring of the algebraic function field L/K, and supposethat R is an algebraic local ring of L such that V dominates R. Suppose that p ⊂ Ris a regular prime; that is, R/p is a regular local ring. If f ∈ p is an element ofminimal value then R[ pf ] is contained in V , and if Q = mV ∩R[ pf ], then R1 = R[ pf ]Qis an algebraic local ring of L which is dominated by V . We say that R → R1 is amonoidal transform along V . R1 is the local ring of the blow up of spec(R) at thenonsingular subscheme V (p). If R is a regular local ring then R1 is a regular localring. In this case there exists a regular system of parameters (x1, . . . , xn) in R suchthat if height(p) = r, then R1 = R[x2

x1, · · · , xrx1

]Q.The main tool we use for our analysis is the Local Monomialization Theorem 2.15.With the notation of (107) and (108), consider a diagram

L → L∗

↑ ↑V = V ∗ ∩ L → V ∗

↑S∗

(109)

where S∗ is an algebraic local ring of L∗.The natural question to consider is Local Simultaneous Resolution; that is, does

there exist a diagram (constructed from (109))

V → V ∗

↑ ↑R → S

↑S∗

such that S∗ → S is a sequence of monoidal transforms, S is a regular local ring, andthere exists a regular local ring R such that S lies above R?

The answer to this question is no, even when trdegK(L∗) = 2, as was shown byAbhyankar in [2]. However, it follows from a refinement in Theorem 4.8 [33] (strongmonomialization) of Theorem 2.15 that if K has characteristic zero, trdegK(L∗) isarbitrary and V ∗ has rational rank 1, then Local Simultaneous Resolution is true,since in this case (8) becomes

x1 = δ1ya111 , x2 = y2, · · · , xn = yn,

and S lies over R0.The natural next question to consider is Weak Simultaneous Local Resolution; that

is, does there exist a diagram (constructed from (109))

V → V ∗

↑ ↑R → S

↑S∗

such that S∗ → S is a sequence of monoidal transforms, S is a regular local ring, andthere exists a normal local ring R such that S lies above R?


This was conjectured by Abhyankar on page 144 [8] (and is implicit in [1]). IftrdegK(L∗) = 2, the answer to this question is yes, as was shown by Abhyankar in[1], [3]. If trdegK(L∗) is arbitrary, and K has characteristic zero, then the answerto this question is yes, as we prove in [27] and [33]. This is a simple corollary ofLocal Monomialization. In fact, Weak Simultaneous Local Resolution follows fromthe following theorem, which will be of use in our analysis of ramification.

Theorem 9.1. (Theorem 4.2 [33]) Let K be a field of characteristic zero, L analgebraic function field over K, L∗ a finite algebraic extension of L, ν∗ a valuation ofL∗/K, with valuation ring V ∗. Suppose that S∗ is an algebraic local ring of L∗ whichis dominated by ν∗ and R∗ is an algebraic local ring of L which is dominated by S∗.Then there exists a commutative diagram

R0 → R → S ⊂ V ∗

↑ ↑R∗ → S∗

(110)

where S∗ → S and R∗ → R0 are sequences of monoidal transforms along ν∗ such thatR0 → S have regular parameters of the form of the conclusions of Theorem 2.15, Ris a normal algebraic local ring of L with toric singularities which is the localizationof the blowup of an ideal in R0, and the regular local ring S is the localization at amaximal ideal of the integral closure of R in L∗.

Proof. By resolution of singularities [48] (c.f. Theorem 2.6, Theorem 2.9 [24]), we firstreduce to the case where R∗ and S∗ are regular, and then construct, by the LocalMonomialization Theorem, Theorem 2.15, a sequence of monoidal transforms alongν∗

R0 → S ⊂ V ∗

↑ ↑R∗ → S∗

(111)

so that R0 is a regular local ring with regular parameters (x1, . . . , xn), S is a regularlocal ring with regular parameters (y1, . . . , yn), there are units δ1, . . . , δn in S, and amatrix of natural numbers A = (aij) with nonzero determinant d such that

xi = δiyai11 · · · yainn

for 1 ≤ i ≤ n. After possibly reindexing the yi we may assume that d > 0. Let (bij)be the adjoint matrix of A. Set

fi =n∏j=1

xbijj = (

n∏j=1

δbijj )ydi

for 1 ≤ i ≤ n. Let R be the integral closure of R0[f1, . . . , fn] in L, localized at thecenter of ν∗. Since

√mRS = mS , Zariski’s Main Theorem (10.9 [4]) shows that R is

a normal algebraic local ring of L such that S lies above R.We thus have a sequence of the form (110).

In the theory of resolution of singularities a modified version of the Weak Simulta-neous Resolution Conjecture is important. This is called Global Weak SimultaneousResolution.


Suppose that f : X → Y is a proper, generically finite morphism of k-varieties.Does there exist a commutative diagram

X1f1→ Y1

↓ ↓X

f→ Y

(112)

such that f1 is finite, X1 and Y1 are complete k-varieties such that X1 is nonsin-gular, Y1 is normal and the vertical arrows are birational?

This question has been posed by Abhyankar (with the further conditions thatY1 → Y is a sequence of blowups of nonsingular subvarieties and Y , X are projective)explicitely on page 144 of [8] and implicitly in the paper [2].

We answer this question in the negative, in an example constructed In Theorem3.1 of [28]. The example is of a generically finite morphism X → Y of nonsingular,projective surfaces, defined over an algebraically closed field K of characteristic notequal to 2. This example also gives a counterexample to the principle that a geometricstatement that is true locally along all valuations should be true globally.

We now state our main theorem on ramification of arbitrary valuations.

Theorem 9.2. (Theorem 6.1 [33]) Suppose that char(K) = 0 and we are given adiagram of the form of (109)

L → L∗

↑ ↑V = V ∗ ∩ L → V ∗

↑S∗.

Let K be an algebraic closure of K(V ∗). Then there exists a regular algebraic localring R1 of L such that if R0 is a regular algebraic local ring of L which contains R1

such thatR0 → R → S ⊂ V ∗

↑S∗

is a diagram satisfying the conclusions of Theorem 9.1, then

(1) There is an isomorphism Zn/AtZn ∼= Γ∗/Γ given by

(b1, . . . , bn) 7→ b1ν∗(y1) + · · ·+ bnν

∗(yn).

(2) Γ∗/Γ acts on S⊗K(S)K and the invariant ring is

R⊗K(R)K ∼= (S⊗K(S)K)Γ∗/Γ.

(3) The reduced ramification index is

e =| Γ∗/Γ |=| Det(A) | .

(4) The relative degree is

f = [K(V ∗) : K(V )] = [K(S) : K(R)].


It is shown in [33] and [30] that we can realize the valuation rings V and V ∗ asdirected unions of local rings of the form of R and S in the above Theorem, and theunions of the valuation rings is a Galois extension with Galois group Γ∗/Γ

We now give an overview of the proof of Theorem 9.2, referring to [33] for details.ν∗ is our valuation of L∗ with valuation ring V ∗, and ν = ν∗ | L is the restiction ofν∗ to K. In the statement of the Theorem Γ∗ is the value group of ν∗ and Γ is thevalue group of ν. By assumption, we have regular parameters (x1, . . . , xn) in R0 and(y1, . . . , yn) in S which satisfy the equations

xi = δi

n∏j=1

yaijj for 1 ≤ i ≤ n.

of (8). We have relations

ν(xi) =n∑j=1

aijν∗(yj) ∈ Γ

for 1 ≤ i ≤ n. Thus there is a group homomorphism

Zn/AtZn → Γ∗/Γ

defined by 1. To show that this homomorphism is an isomorphism we need an exten-sion of Local Monomialization (proven in Theorem 4.9 [33]). 3. is immediate from1.R0 → S is monomial, and R0 → R is a weighted blowup. Lemma 4.4 of [33] Shows

that if M∗ is the quotient field of S⊗K(S)K and M is the quotient field of R⊗K(R)K,then M∗/M is Galois with Galois group Gal(M∗/M) ∼= Zn/AZn ∼= Zn/AtZn. Fur-thermore, (S⊗K(S)K)Zn/AZn ∼= R⊗K(R)K.

We see that R⊗K(R)K is a quotient singularity, by a group whose invariant factorsare determined by Γ∗/Γ. This observation leads to the counterexample to global weaksimultaneous resolution stated earlier in this section. This example is written up indetail in [28]. We will give a very brief outline of the construction. With the notationof (112), if a valuation ν of K(Y ) has at least two distinct extensions ν∗1 and ν∗2to K(X) which have nonisomorphic quotients Γ∗1/Γ and Γ∗2/Γ, then this presents anobstruction to the existence of a map X1 → Y1 (with the notation of (112)) which isfinite, and X1 is nonsingular. If there were such a map, we would have points p1 andp2 on X1, whose local rings R1 and R2 are regular and are dominated by ν∗1 and ν∗2respectively, and a point p on Y1 whose local ring R is dominated by p. If we chooseour valuation ν carefully, we can ensure that R is a quotient of R1 by Γ∗1/Γ and R

is also a quotient of R2 by Γ∗2/Γ. We can in fact choose the extended valutions sothat we can conclude that not only are these quotient groups not isomorphic, but theinvariant rings are not isomorphic, which is a contradiction.

An interesting question is if the conclusions of Local Monomialization (Theorem2.15) hold if the ground field K has positive characteristic. This is certainly true iftrdegK(L∗) = 1.

Now suppose that trdegK(L∗) = 2 and K is algebraically closed of positive charac-teristic. Strong monomialization (a refinement in Theorem 4.8 [33] of Theorem 2.15)is true if the value group Γ∗ of V ∗ is a finitely generated group (Theorem 7.3 [33]) orif V ∗/V is defectless (Theorem 7.35 [33]). The defect is an invariant which is trivial


in characteristic zero, but is a power of p in an extension of characteristic p functionfields.

Theorem 7.38 [33] gives an example showing that Strong Monomialization fails iftrdegK(L∗) = 2 and char(K) > 0. Of course the value group is not finitely generatedand there is defect in this example. This example does satisfy the less restrictiveconclusions of Local Monomialization (Theorem 2.15).

It follows from Strong Monomialization in characteristic zero (a refinement of The-orem 2.15) that Local Simultaneous Resolution is true (for arbitrary trdegK(L∗)) ifK has characteristic zero and rational rank ν∗ = 1. We consider this condition whentrdegK(L∗) = 2 and K is algebraically closed of positive characteristic. In Theorem7.33 [33] it is shown that in many cases Local Simultaneous Resolution does holdif trdegK(L∗) = 2, char(K) > 0 and the value group is not finitely generated. Forinstance, Local Simultaneous Resolution holds if Γ is not p-divisible. The example ofTheorem 7.38 [33] discussed above does in fact satisfy Local Simultaneous Resolution,although it does not satisfy Strong Monomialization.

10. Smoothness and non-singularity II

10.1. 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 )

DefineD(P ) to be the image of dP . SinceOAn,P is a regular local ring, dimK(P )mP /m2P =

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

dP : mP /(mP )2 → D(P )

is a K(P )-linear isomorphism.Theorem 10.1. Suppose that P ∈ An

K = spec(K[x1, . . . , xn]) is a closed point. ThendimK(P )D(P ) = n if and only if K(P ) is separable over K.

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 then inductively definefi(z1, . . . , zi) in the polynomial ring K[z1, . . . , zi] so that fi(α1, . . . , αi−1, zi) is theminimal polynomial of αi over K(α1, . . . , αi−1) for 2 ≤ 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 . Thusf1(x1), f2(x1, x2), . . . , fn(x1, . . . , xn) is a regular system of parameters in OAn

K,P , and

the images of f1(x1), f2(x1, x2), . . . , fn(x1, . . . , xn) in mP /m2P 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 and only if J(f ;x) has rank n at P , which holds if and onlyif the determinant

| J(f ;x) |=n∏i=1

∂fi∂xi

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

Corollary 10.2. 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 of P , and suppose thatu1, . . . , un ∈ mP . Then u1, . . . , un is a regular system of 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 10.3. Suppose that A is a regular local ring of dimension b with maximalideal m, p ⊂ A is a prime 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 short exact sequenceof k-vector spaces

0→ p/p ∩m2 ∼= p+m2/m2 → m/m2 → m′/(m′)2 = m/(p+m2)→ 0.(113)

The Lemma now follows since

dimkm′/(m′)2 ≥ a

and A′ is regular if and only if dimkm′/(m′)2 = a (Corollary 11.5 [12]).

We now give two related results, Lemma 10.4 and Corollary 10.5.Lemma 10.4. Suppose that A is a regular local ring of dimension n with maximalideal m, p ⊂ A is a prime such that A/p is regular, dim (A/p) = n − r. Then thereexist regular parameters u1, . . . , un in A such that p = (u1, . . . , ur) and ur+1, . . . , unmap to regular parameters in A/p.

Proof. Consider the exact sequence (113). Since A and A′ are regular, there existsregular parameters u1, . . . , un in A such that ur+1, . . . , un map to regular parametersin A′, and u1, . . . , ur are in p. (u1, . . . , ur) is thus a prime ideal of height r containedin p. Thus p = (u1, . . . , ur).

Corollary 10.5. Suppose that Y is a subvariety of dimension t of a variety X ofdimension n. Suppose that q ∈ Y is a nonsingular closed point of both Y and X. Thenthere exist regular parameters u1, . . . , un in OX,q such that IY,q = (u1, . . . , un−t),and un−t+1, . . . , un map to regular parameters in OY,q. If vn−t+1, . . . , vn are regularparameters in OY,q, there exist regular parameters u1, . . . , un as above such that uimaps to vi for n− t+ 1 ≤ i ≤ n.


Lemma 10.6. Let I = (f1, . . . , fm) ⊂ K[x1, . . . , xn] be an ideal. Suppose thatP ∈ V (I) and J(f ;x) has rank n at P . Then K(P ) is separable algebraic over K andn of the polynomials f1, . . . , fm form a system of regular parameters in OAn

K,P .

Proof. Let m be the ideal of P in K[x1, . . . , xn]. After possibly reindexing the fi, wemay assume that | J(f1, . . . , fn;x) |6∈ m. By the nullstellensatz, m is the intersectionof all maximal ideals n of K[x1, . . . , xn] containing m. Thus there exists a closedpoint Q ∈ An

K with maximal ideal n containing m 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 thatK(Q) is separable over K and (f1, . . . , fn) is a regular system of parameters in OAn,Q

by Theorem 10.1 and Corollary 10.2. Since I ⊂ m ⊂ n, we have P = Q.

Lemma 10.7. 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 . Let αi be the imageof xi in K(P ) for 1 ≤ i ≤ r. Then the determinant | J(t1, . . . , tn−r;xr+1, . . . , xn) |is nonzero at P if and only if α1, . . . , αr is a separating transcendence basis of K(P )over K and t1, . . . , tn−r is a regular system of parameters in OAn,P .

Proof. Suppose that | J(t1, . . . , tn−r;xr+1, . . . , xn | is nonzero at P . There exists0, s1, . . . , sn−r ∈ K[x1, . . . , xn] such that si ∈ K[x1, . . . , xn] for all i, s0 6∈ m andsis0

= ti for 1 ≤ i ≤ n − r. We then have that | J(s1, . . . , sn−r;xr+1, . . . , xn) | isnonzero at P . Let K∗ = K(α1, . . . , αr), 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−r

K∗ . We have K(P ) = K∗(Q) and| J(s∗1, . . . , s

∗n−r;xr+1, . . . , xn) |6= 0 at Q. By Lemma 10.6 K∗(Q) is separable alge-

braic over K∗ and (s∗1, . . . , s∗n−r) is a regular system of parameters in OAn−r

K∗ ,Q=

K∗[xr+1, . . . , xn]n. Since trdegKK(P ) = r, α1, . . . , αr is a separating transcen-dence 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 OAn

K,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 OAnK,P = K[x1, . . . , xn]m. Let

K∗ = K(α1, . . . , αr), n be the kernel of K∗[xr+1, . . . , xn]→ K(P ). n is the ideal ofa point Q ∈ An−r

K∗ . 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. By Corollary10.2, | J(t1, . . . , tn−r;xr+1, . . . , xn) | is 6= 0 at Q, and thus is 6= 0 at P .

Theorem 10.8. 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 and only if K(P ) isseparably generated over K.

Proof. Let αi, 1 ≤ i ≤ n be the images of xi in K(P ), m be the ideal of P inK[x1, . . . , xn].


Suppose that rank(J(t;x)) = n− r at P . After possibly reindexing the xi and tj ,we have | J(t1, . . . , tn−r;xr+1, . . . , xn) |6= 0 at P . Now K(P ) is separably generatedover K by Lemma 10.7.

Suppose that K(P ) is separably generated over K. Let ζ1, . . . , ζr be a separatingtranscendence 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 n be the kernel of the K-algebrahomomorphism K[x1, . . . , xn+r]→ K(P ) defined by xi 7→ αi for 1 ≤ i ≤ n, xi 7→ ζi−nif n < i. n is the ideal of a point Q ∈ An+r

K . Let Φ0(x),Φ1(x), . . . ,Φn−r(x) ∈K[x1, . . . , xn] be such that 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 theideal generated by Φ1, . . . ,Φn. I ⊂ n. We will show that Φ1, . . . ,Φn are in fact aregular system of parameters in OAn+r

K,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. Givensuch 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 nonnegative integer. F ∈ n implies 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, Ci ∈ 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 inK[x1, . . . , xn+r]n and ζ1, . . . , ζr

is a separating transcendence basis of K(Q) over K, | J(Φ1, . . . ,Φn;x1, . . . , xn) | isnonzero at Q by Lemma 10.7. Thus J(Φ1, . . . ,Φn−r;x1, . . . , xn) is of rank n − r atP , and by Lemma 10.7, α1, . . . , αn must contain a separating transcendence basis ofK(P ) over K.

Proof of Theorem 2.7.

Let s = dim X. There exists an affine neighborhood U = spec(R) of P in X suchthat R = K[x1, . . . , xn]/I, I = (f1, . . . , fm). Let mP ∈ spec(K[x1, . . . , xn]) be theideal of P in An

K . Suppose that dim (K[x1, . . . , xn]/mP ) = t.Suppose that K(P ) is separably generated over K and P is a nonsingular point

of X. By Theorem 10.8, dP : mP /m2P → D(P ) is an isomorphism. By Lemma 10.3,

dimK(P )dP (I) = n− s (take A = OAn,P , p = ImP in the statement of the Lemma, sothat b = n − t, a = s − t) which is equivalent to the condition that J(f ;x) has rankn− s at P .


Suppose that X is smooth at P so that J(f ;x) has rank n − s at P . ThendimK(P )dP (I) = n− s so that dimK(P )I +mP /m

2P ≥ n− s. Thus P is a nonsingular

point of X by Lemma 10.3.Remark 10.9. With the notation of the proof of Theorem 2.7, for any point P ∈V (I), we have

dimK(P )I +mP /m2P ≤ n− s

by Lemma 10.3, so that J(f ;x) has rank ≤ n− s at P .

Proof of Theorem 2.6

It suffices to prove that the locus of smooth points of X lying on an open affinesubset U = spec(R) of X of the form of Definition 2.5 is open. Let A = In−2(J(f ;x)).p ∈ U −V (A) if and only if J(f ;x) has rank greater than or equal to n−s at p, whichholds if and only if J(f ;x) has rank n− s by Remark 10.9.

Proof of Theorem 2.9 when K is perfect

Suppose that K is perfect. The openness of the set of nonsingular points of Xfollows from Theorem 2.6 and Corollary 2.8.

Let η ∈ X be the generic point of an irreducible component of X. Then OX,η is afield which is a regular local ring. Thus η is a nonsingular point. We conclude thatthe nonsingular points of X are dense in X.

10.2. Non-singularity and uniformizing parameters. Consider the sheaf of dif-ferentials Ω1

X/K . With the notation of Definition 2.5, we have, by the “second funda-mental exact sequence” (Theorem 25.2 [61]), a presentation

(K[x1, . . . , xn])mJ(f ;x)→ Ω1

K[x1,... ,xn]/K ⊗R→ Ω1R/k → 0

Theorem 10.10. Suppose that X is a variety of dimension r over a field K, andP ∈ X is a closed point such that X is nonsingular at P and K(P ) is separable overK.

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 anddf1, . . . , dfr generate Ω1

X/K,P as an OX,P module. Then f1, . . . , fr are regularparameters 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 1. of the Theorem follow from Theorem 10.8 when X = AnK .

In the general case of 1., by Corollary 10.5, there exist regular parameters (y1, . . . , yn)in A = K[x1, . . . , xn]m (where m is the ideal of P in spec(K[x1, . . . , xn]) such thatIA = (yr+1, . . . , yn) and (y1, . . . , yr) map to the regular parameters (y1, . . . , yr) in


B = A/IA. By the second fundamental exact sequence (Theorem 25.2 [61]) we havean exact sequence of B modules

IA/I2A→ Ω1A/K ⊗A B → Ω1

B/K → 0

This sequence is thus a split short exact sequence of free B modules, with Ω1B/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 regularparameters in R. Then dy1, . . . , dyr is a basis of the R module Ω1

X/K,P by 1. of thistheorem. 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 =∑rj=1 aijdyj for 1 ≤ i ≤ r. By Nakayama’s Lemma we have that the wi

generate Ω1X/K,P as an R-module. Thus | (aij) |6∈ m and (aij) is invertible over R, so

that f1, . . . , fr generate m.

Lemma 10.11. Suppose that X is a variety of dimension r over a perfect field K.Suppose that P ∈ X is a closed point such that X is nonsingular at P , and y1, . . . , yrare regular parameters in OX,P . Then there exists an affine neighborhood U = spec(R)of P in X such that y1, . . . , yr ∈ R, the natural inclusion S = K[y1, . . . , yr] → Rinduces a morphism π : U → Ar

K such that

Ω1R/K = dy1R⊕ · · · ⊕ dyrR, (114)

DerK(R,R) = HomR(Ω1R/K , R) =

∂

∂y1R⊕ · · · ⊕ ∂

∂yrR,

(with ∂∂yi

(dyj) = δij) and the following condition holds. Suppose that a ∈ U is a closedpoint, 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 10.10, there exists an affine neighborhood U = spec(R) of P in Xsuch that y1, . . . , yr ∈ R and dy1, . . . , dyr is a free basis of Ω1

R/K . Consider the naturalinclusion S = K[y1, . . . , yr] → R, with induced morphism π : spec(R) → spec(S).Suppose that a ∈ U is a closed point with maximal ideal m in R and b = π(a),with maximal ideal n in S. Suppose that (x1, . . . , xr) are regular parameters inB = Sn. Let A = Rm. By construction, Ω1

S/K is generated by dy1, . . . , dyr as anS module. Thus Ω1

A/K is generated by Ω1B/K as an A module. By 1. of Theorem

10.10, dx1, . . . , dxr generate Ω1B/K as a B module, and hence they generate Ω1

A/K asan A module. By 2. of Theorem 10.10, x1, . . . , xr is a regular system of parametersin A.


10.3. Higher derivations. Suppose that K is a field and A is a K-algebra. A higherderivation of A over K of length m is a sequence

D = (D0, D1, . . . , Dm)

if m <∞, and a sequenceD = (D0, D1, . . . )

if m =∞ of K-linear maps Di : A→ A such that D0 = id and

Di(xy) =i∑

j=1

Dj(x)Dj−i(y) (115)

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 by Et(f) =∑∞i=0Di(f)ti

is a K-algebra homomorphism with D0(f) = f . D = (D0, D1, . . . , Dm) is a higherderivation of A over K of length m precisely 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 10.12. Suppose that A = K[y1, . . . , yr] is a polynomial ring over a fieldK. For 1 ≤ i ≤ r, let εj be the vector in Nr with a 1 in the jth coefficient and 0everywhere else. For f =

∑ai1,... ,iry

i11 · · · yirr ∈ A and m ∈ N define

D∗mεj (f) =∑

ai1,... ,ir

(ijm

)yi11 · · · y

ij−mj · · · yirr .

Then D∗j = (D∗0 , D∗εj , D

∗2εj , . . . ) is a higher derivation of A over K of length ∞ for

1 ≤ j ≤ r. DefineD∗i1,... ,ir = D∗irεr D

∗ir−1εr−1

· · · D∗i1ε1for (i1, . . . , ir) ∈ Nr.D∗i1,... ,ir are called Hasse-Schmidt derivations [52]. If K has characteristic zero,

D∗i1,... ,ir =1

i1!i2! · · · ir!∂i1+···+ir

∂yi11 · · · ∂yirr

.

Lemma 10.13. Suppose that X is a variety of dimension r over a perfect field K.Suppose that P ∈ X is a closed point such that X is nonsingular at P , y1, . . . , yrare regular parameters in OX,P , and U = spec(R) is an affine neighborhood of P inX such that the conclusions of Lemma 10.11 hold. Then there are differential opera-tors Di1,... ,ir on R which are uniquely determined on R by the differential operatorsD∗i1,... ,ir on S of Example 10.12, such that if f ∈ OX,P , and

f =∑

ai1,... ,iryi11 · · · yirr

in OX,P ∼= K(P )[[y1, . . . , yr]], where we identify K(P ) with the coefficient field ofOX,P containing K. Then ai1,... ,ir = Di1,... ,ir (f)(P ) is the residue of Di1,... ,ir (f) inK(P ) for all indices i1, . . . , ir.

Proof. If K has characteristic zero we can take

Di1,... ,ir =1

i1! · · · ir!∂i1+···+ir

∂yi11 · · · ∂yirr

. (116)


Suppose that K has characteristic p ≥ 0. Consider the inclusion of K-algebras

S = K[y1, . . . , yr]→ R

of Lemma 10.11, 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 Example10.12. Ω1

R/S = 0 by 1. of Theorem 10.10, (114) and the “first fundamental exactsequence” (Theorem 25.1 [61]). Thus π is non-ramified by Corollary IV.17.4.2 [43].By Theorem IV.18.10.1 [43] and Proposition IV.6.15.6 [43] π is etale.

We will now prove that D∗j extends uniquely to a higher derivation Dj of R overK 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 imme-diate. 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 ,

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 formally etale(Definition IV.17.3.1 [43] and Definition 17.1.1 [43]) there exists a unique K-algebrahomomorphism h making a commutative diagram

Rg→ R[t]/(tm+1)

↑h

↑S

f→ R[t]/(tm+2).

Thus there is a unique extension of (D∗0 , D∗εj , . . . , D

∗(m+1)εj

) to a higher derivation ofR.

Set Di1,... ,ir = Dirεr · · · Di1ε1 .Let A = OX,P . Since K(P ) is separable over K, and by the exercise of Section

3.2, we can identify the residue field K(P ) of A with the coefficient field of A whichcontains K. Thus we have natural inclusions

K[y1, . . . , yr] ⊂ A ⊂ A ∼= K(P )[[y1, . . . , yr]].

We will now show that Dεj extends uniquely to a higher derivation of A over K for1 ≤ j ≤ r. Let m be the maximal ideal of A. Let t be an indeterminate. A[[t]] is acomplete 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-algebra homomorphism since Dεj is a higher

derivation. Et is continuous in the m-adic topology, as Et(ma) ⊂ na for all a. Thus


Et extends uniquely to a K-algebra homomorphsim A → A[[t]]. Thus Dεj extendsuniquely to a higher derivation of A over K.

We can thus extend Di1,... ,ir uniquely to A. Since K(P ) is algebraic over K, andDi1,... ,ir extends D∗i1,... ,ir , Di1,... ,ir acts on A as stated in the Theorem.

Remark 10.14. Suppose that the assumptions of Lemma 10.13 hold, and that K isan algebraically closed field. Then the following properties hold.

1. Suppose that Q ∈ U is a closed point and a is the corresponding maximal idealin R. Then a = (y1, . . . , yr) where yi = yi − αi with αi = yi(Q) ∈ K for1 ≤ i ≤ r, and K[[y1, . . . , yr]] = Ra = OU,Q.

2. The differential operators Di1,... ,in on R are such that if f ∈ R, and a is amaximal ideal in R, then with the notation of 1.,

f =∑

ai1,... ,iryi11 · · · yirr

in K[[y1, . . . , yr]] = Ra, where ai1,... ,ir = Di1,... ,ir (f)(Q) is the residue ofDi1,... ,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. Letyi = yi−αi for 1 ≤ i ≤ r. a = (y1, . . . , yr) by Theorem 10.10 2., and 1. follows. Thederivations ∂

∂y1, . . . , ∂

∂yrare exactly the derivations ∂

∂y1, . . . , ∂

∂yron S of Example

10.12, so they extend to the differential operators D∗i1,... ,ir on S of Example 10.12. 2.now follows from Lemma 10.13 applied to Q and y1, . . . , yr.

Lemma 10.15. Let notation be as in Lemmas 10.11 and 10.13. In particular wehave a fixed closed point P ∈ U = spec(R) and differential operators Di1,... ,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 supposethat (z1, . . . , zr) is a regular system of parameters in mQ.

Let Di1,... ,ir be the differential operators constructed on OW,Q by the constructionof Lemmas 10.11 and 10.13. Then

(Jm,U )mQ = Di1,... ,ir (f) | f ∈ I and i1 + · · ·+ ir < m.

Proof. When K has characteristic zero, this follows directly from differentiation and(116).

Assume that K has characteristic zero. By Lemma 10.13 and Theorem IV.16.11.2[43], U is differentiably smooth over spec(K) and Di1,... ,ir | i1, . . . , ir < m is anR-basis of the differential operators of order less than m on R. Since dz1, . . . , dzris a basis of Ω1

RmQ/K(by Lemma 10.11), Di1,... ,ir | i1, . . . , ir < m is an RmQ -basis

of the differential operators of order less than m on RmQ by Theorem IV.16.11.2 [43].The Lemma follows.

10.4. Upper semi-continuity of νq(I).Lemma 10.16. Suppose that W is a nonsingular variety over a perfect field K and K ′

is an algebraic field extension of K. Let W ′ = W×KK ′, with projection π1 : W ′ →W .Then W ′ is a nonsingular variety over K ′. Suppose that q ∈ W is a closed point.Then there exists a closed point q′ ∈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′ .

(2) If I ⊂ OW,q is an ideal, then

(IOW ′,q′) ∩ OW,q = I.

Proof. The fact that W ′ is nonsingular follows from Theorem 2.7 and the definitionof 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 neighborhood of q inW . There is a presentation R = K[x1, . . . , xn]/I. We further have that π−1

1 (U) =spec(R′), with R′ = K ′[x1, . . . , xn]/I(K ′[x1, . . . , xn]). Let m be the ideal of q inK[x1, . . . , xn], n be the ideal of q′ in K ′[x1, . . . , xn]. By the exact sequence of Lemma10.3, there exists a regular system of parameters f1, . . . , fn in K[x1, . . . , xn]m, suchthat Im = (fr+1, . . . , fn)m and f i maps to fi for 1 ≤ i ≤ r. By Corollary 10.2, thedeterminant | J(f1, . . . , fn;x1, . . . , xn) | is nonzero at q, so it is nonzero at q′. ByCorollary 10.2, f1, . . . , fn is a regular system of parameters in K ′[x1, . . . , xn]n. Thusf1, . . . , 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 faithfully flatover OW,q, and (IOW ′,q′) ∩ OW,q = I (by Theorem 7.5 (ii) [61]).

Suppose that (R,m) is a regular local ring containing a field K of characteristiczero. Let K ′ be the residue field of R. Suppose that J ⊂ R is an ideal. We define theorder of J in R to be

νR(J) = max b | J ⊂ mb.If J is a locally principal ideal, then the order of J is its multiplicity. However, orderand multiplicity are different in general.Definition 10.17. Suppose that q is a point on a variety W and J ⊂ OW is an idealsheaf. We denote

νq(J) = νOW,q (JOW,q).If X ⊂W is a subvariety, we denote

νq(X) = νq(IX).

Remark 10.18. Let notations be as in Lemma 10.16 (except we allow K to be anarbitrary, not necessarily perfect field). Observe that if q ∈ W and q′ ∈ W ′ are suchthat 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 totally ordered set.f : X → I is said to be an upper semi-continuous function if for any α ∈ I,

q ∈ X | f(q) ≥ αis a closed subset of X.

Suppose that X is a subvariety of a nonsingular variety W . Define

νX : W → N

by νX(q) = νq(X) for q ∈W . The order νq(X) is defined in Definition 10.17.


Theorem 10.19. Suppose that K is a perfect field, W is a nonsingular variety overK 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 affine subset ofW such that the conclusions of Lemma 10.11 hold on W . Set I = Γ(U, I).

With the notations of Lemmas 10.11 and 10.13, define an ideal Jm,U ⊂ R by

Jm,U = Di1,... ,ir (g) | g ∈ I and i1 + · · · ir < m.By Lemma 10.15, this definition is independent of choice of y1, . . . , yr satisfying theconclusions of Lemma 10.11, so our definition of Jm,U determines a sheaf of ideals Jmon W .

We will show thatV (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 , and suppose

that P ∈ Y is a closed point such that Y is nonsingular at P .By Corollary 10.5, Lemma 10.11 and Lemma 10.15, there exists an affine neighbor-

hood U = spec(R) of P in W such that there are y1, . . . , yr ∈ R with the propertiesthat y1 = . . . = yt = 0 (with t ≤ r) are local equations of Y in U and y1, . . . , yrsatisfy the conclusions of Lemma 10.11.

For g ∈ IP , consider the expansion

g =∑

ai1,... ,iryi11 · · · yirr

in OW,P ∼= K(P )[[y1, . . . , yr]], where we have identified K(P ) with the coefficientfield of OW,P containing K and (with the notation of Lemma 10.13)

ai1,... ,ir = Di1,... ,ir (g)(P ).

η ∈ V (Jm) is equivalent to Di1,... ,ir (g) ∈ (y1, . . . , yt) whenever g ∈ IP and i1 +· · ·+ ir < m, which is equivalent to

ai1,... ,ir = Di1,... ,ir (g)(P ) = 0

if g ∈ IP and i1 + · · ·+ it < m, which holds if and only if

g ∈ ImY,P ∩ OW,P = ImY,Pfor all g ∈ IP , where the last equality is by Lemma 10.16. Localizing OW,P at η, wesee that νη(I) ≥ m if and only if η ∈ V (Jm).

Definition 10.20. Suppose that W is a nonsingular variety over a perfect field Kand X ⊂W is a subvariety.

For b ∈ N, defineSingb(X) = q ∈W | νq(W ) ≥ b.

Singb(X) is a closed subset of W by Theorem 10.19. Let

r = maxνX(q) | q ∈W.Suppose that q ∈ Singr(X). A subvariety H of an affine neighborhood U of q in W iscalled a hypersurface of maximal contact for X at q if


(1) Singr(X) ∩ U ⊂ H and(2) If

Wnπn→ · · · →W1

π1→W

is a sequence of monoidal transforms such that for all i, πi is centered at anonsingular subvariety Yi ⊂ Singr(Xi), where Xi is the strict transform of Xon Wi, then the strict transform Hn of H on Un = Wn ×W U is such thatSingr(Xn) ∩ Un ⊂ Hn.

With the above assumptions, a nonsingular codimension one subvariety H of U =spec(OW,q) is called a formal hypersurface of maximal contact for X at q if 1. and 2.above hold for U = spec(OW,q).Remark 10.21. Suppose that W is a nonsingular variety over a perfect field K andX ⊂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 sheaf whose supportis Singb(X), then JqT is a reduced ideal whose support is Singb(OX,q).

Proof. Let U = spec(R) be an affine neighborhood of q in W satisfying the conclusionsof Lemma 10.13 (with p = q), and Let notation be as in Lemma 19.13 (with p = q).With the further notation of Theorem 10.19 (and I = IX) we see that Γ(U, J) =√Jb,U . JqT is a reduced ideal by Theorem 3.5.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. Now induction in the formula(115) shows that Di1,... ,ir (gh) ∈ P b−i1,... ,ir and Di1,... ,ir (g) ∈ P (b−i1−···−ir) ⊂ P ifi1 + · · ·+ ir < b. Thus JqT ⊂ P .

Now suppose that P ∈ spec(T ) and JqT ⊂ P . Let S = OW,q. If Jq = P1 ∩ · · · ∩Pmis a primary decomposition, Then all primes Pi are minimal since Jq is reduced. ByTheorem 3.5 there are pimes Pij in T and a function σ(i) such that PiT = Pi1 ∩ · · · ∩Piσ(i) is a minimal primary decomposition. Thus JqTPij = PiTPij = PijTPij for all i, j.We have ∩ijPij ⊂ P so Pij ⊂ P for some i, j. IX,q ⊂ P bi SPi implies IX,q ⊂ P bijTPijimplies IX,q ⊂ P bTPij . Thus νTP (IX,qTP ) ≥ b and P ∈ Singb(OX,q).

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Steven Dale Cutkosky, Department of Mathematics, University of MissouriColumbia, MO 65211, [email protected]

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