Resolution of Singularities (Graduate Studies in Mathematics, Vol. 63)

Resolution of Singularities

Steven Dale Cutkosky

Graduate Studies in Mathematics Volume 63

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Walter Craig Nikolai Ivanov Steven G. Krantz

David Saltman (Chair) 2000 Mathematics Subject Classification. Primary 13Hxx, 14B05, 14317, 14E15, 32Sxx.

For additional information and updates on this book, visit

www.ams.org/bookpages/gsm-63

Library of Congress Cataloging-in-Publication Data Cutkosky, Steven Dale. Resolution of singularities / Steven Dale Cutkosky. p. cm.- (Graduate studies in mathematics, ISSN 1065-7339; v. 63) Includes bibliographical references and index. ISBN 0-8218-3555-6 (acid-free paper) 1. Singularities (Mathematics). I. Title. II. Series. QA614.58.C87 2004 516.3'5 dc22

2004046123

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To Hema, Ashok and Maya

Contents

Preface

vii

Chapter 1. Introduction §1.1. Notation

Chapter 2. Non-singularity and Resolution of Singularities §2.1. Newton's method for determining the branches of a plane curve §2.2. §2.3. §2.4. §2.5.

Smoothness and non-singularity Resolution of singularities Normalization Local uniformization and generalized resolution problems

Chapter 3. Curve Singularities §3.1. Blowing up a point on A2 §3.2. Completion §3.3. Blowing up a point on a non-singular surface §3.4. Resolution of curves embedded in a non-singular surface I §3.5. Resolution of curves embedded in a non-singular surface H

1

2

3 3 7

9 10 11

17

17 22 25 26 29

Chapter 4. Resolution Type Theorems §4.1. Blow-ups of ideals §4.2. Resolution type theorems and corollaries

37

Chapter 5. Surface Singularities §5.1. Resolution of surface singularities

45

37 40

45

v

vi

Contents

§5.2.

Embedded resolution of singularities

56

Chapter 6. Resolution of Singularities in Characteristic Zero 61 §6.1. The operator A and other preliminaries 62 §6.2. Hypersurfaces of maximal contact and induction in resolution 66 §6.3. Pairs and basic objects 70 §6.4. Basic objects and hypersurfaces of maximal contact 75 §6.5. General basic objects 81 §6.6. Functions on a general basic object 83 §6.7. Resolution theorems for a general basic object 89 §6.8. Resolution of singularities in characteristic zero 99 Chapter 7. Resolution of Surfaces in Positive Characteristic §7.1. Resolution and some invariants §7.2. r(q) = 2 §7.3. T(q) = 1 §7.4. Remarks and further discussion

105

Chapter 8. Local Uniformization and Resolution of Surfaces §8.1. Classification of valuations in function fields of dimension 2 §8.2. Local uniformization of algebraic function fields of surfaces §8.3. Resolving systems and the Zariski-Riemann manifold

133

148

Chapter 9. Ramification of Valuations and Simultaneous Resolution

155

Appendix. Smoothness and Non-singularity II §A.1. Proofs of the basic theorems §A.2. Non-singularity and uniformizing parameters §A.3. Higher derivations §A.4. Upper semi-continuity of v4 (2)

163

Bibliography

179

Index

185

105 109 113

130

133

137

163 169 171

174

Preface

The notion of singularity is basic to mathematics. In elementary algebra singularity appears as a multiple root of a polynomial. In geometry a point in a space is non-singular if it has a tangent space whose dimension is the same as that of the space. Both notions of singularity can be detected through the vanishing of derivitives. Over an algebraically closed field, a variety is non-singular at a point

if there exists a tangent space at the point which has the same dimension as the variety. More generally, a variety is non-singular at a point if its local ring is a regular local ring. A fundamental problem is to remove a singularity by simple algebraic mappings. That is, can a given variety be desingularized by a proper, birational morphisui from a non-singular variety? This is always possible in all dimensions, over fields of characteristic zero. We give a complete proof of this in Chapter 6. We also treat positive characteristic, developing the basic tools needed

for this study, and giving a proof of resolution of surface singularities in positive characteristic in Chapter 7. In Section 2.5 we discuss important open problems, such as resolution of singularities 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 ZarLski's original proof of local uniformization for surfaces in characteristic zero. This book has evolved out of lectures given at the University of Missouri and at the Chennai Mathematics Institute, in Chennai, (also known as Madras), India. It can be used as part of a one year introductory sequence

vii

Preface

viii

in algebraic geometry, and would provide an exciting direction after the basic notions of schemes and sheaves have been covered. A core course on resolution is covered in Chapters 2 through 6. The major ideas of resolution have been introduced by the end of Section 6.2, and after reading this far, a student will find the resolution theorems of Section 6.8 quite believable, and have a good feel for what goes into their proofs. Chapters 7 and 8 cover additional topics. These two chapters are independent, and can be chosen as possible followups to the basic material in

the first 5 chapters. Chapter 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 singularities for algebraic surfaces. This chapter provides an introduction to valuation theory in algebraic geometry, and to the problem of local uniformization. The appendix proves foundational results on the singular locus that we

need. On a first reading, I recommend that the reader simply look up the statements as needed in reading the main body of the book. Versions of almost all of these statements are much easier over algebraically closed fields of characteristic zero, and most of the results can be found in this case in standard textbooks in algebraic geometry. I assume that the reader has some familiarity with algebraic geometry

and commutative algebra, such as can be obtained from an introductory course on these subjects. This material is covered in books such as Atiyah and Mac Donald [13] or the basic sections of Eisenbud's book [37], and the first two chapters of Hartshorne's book on algebraic geometry [47], or Eisenbud and Harris's book on schemes [38]. I thank Professors Seshadri and Ed Dunne for their encouragement to

write this book, and Laura Ghezzi, Tai Ha, Krishna Hanamanthu, Olga Kashcheyeva and Emanoil Theodorescu for their helpful comments on preliminary versions of the manuscript.

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

Steven Dale Cutkosky

Chapter 1

Introduction

An algebraic variety X is defined locally by the vanishing of a system of polynomial equations fz E K[xi , ... , x.n], { = ... = f", = 0. J1

If K is algebraically closed, points of X in this chart are cr = (al, ... , o") C AK which satisfy this system. The tangent space Ta(X) at a point cr E X is the linear subspace of AK defined by the system of linear equations

Li=...=L.=U, where Li is defined by n

Li=Etgxt(a)(xj-aj) j We have that dim TQ(X) > dim X, and X is non-singular at the point. a if dim Ta(X) = dim X. The locus of points in X which are singular is a proper closed subset of X. The fundamental problem of resolution of singularities is to perform simple algebraic transformations of X so that the transform Y of X is nonsingular everywhere. To be precise, we seek a resolution of singularities

of X; that is, a proper birational morphism 4 : Y - X such that Y is non-singular. The problem of resolution when K has characteristic zero has been studied for some time. In fact we will see (Chapters 2 and 3) that the method of Newton for determining the analytical branches of a plane curve singularity extends to give a proof of resolution for algebraic curves. The first algebraic

proof of resolution of surface singularities is due to Zariski [86]. We give 1

1. Introduction

2

a modern treatment of this proof in Chapter 8. A study of this proof is surely the best introduction to the methods and ideas underlying the recent emphasis on valuation-theoretic methods in resolution problems. The existence of a resolution of singularities has been completely solved by Hironaka [52], in all dimensions, when K has characteristic zero. We give a simplified proof of this theorem, based on the proof of canonical resolution by Encinas and Villamayor ([40], [411), in Chapter 6. When K has positive characteristic, resolution is known for curves, surfaces and 3-folds (with char(K) > 5). The first proof in positive characteristic of resolution of surfaces and of resolution for 3-folds is due to Abhyankar [1], [4].

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

1.1. Notation The notation of Hartshorne [47] will be followed, with the following differences and additions. By a variety over a field K (or a K-variety), we will mean an open subset of an equidimensional reduced subscheme of the projective space lP'K. Thus an integral variety is a "quasi-projective variety" in the classical sense. A curve is a one-dimensional variety. A surface is a two-dimensional variety, and a 3-fold is a three-dimensional variety. A suhvariety 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(.1) _ spec(OX/I) C X. If Y is a subscheme of a variety X, we denote the ideal of Y in X by ly.. If W1, W2 are subschernes of a variety of X, we will denote the scheme-theoretic intersection of W1 and W2 by W1 W2. This is the subscheme

=V(Iw1

+Iw2)CW.

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

Chapter 2

Non-singularity and Resolution of Singularities

2.1. Newton's method for determining the branches of a plane curve Newton's algorithm for solving f (x, y) = 0 by a fractional power series can be thought of as a generalization of the implicit function y= theorem to general analytic functions. We begin with this algorithm because of its simplicity and elegance, and because this method contains some of

the most important ideas in resolution. We will see (in Section 2.5) that the algorithm immediately gives a local solution to resolution of analytic plane curve singularities, and that it can be interpreted to give a global solution to resolution of plane curve singularities (in Section 3.5). All of the proofs of resolution in this book can be viewed as generalizations of Newton's algorithm, with the exception of the proof that curve singularities can be resolved by normalization (Theorems 2.14 and 4.3). Suppose K is an algebraically closed field of characteristic 0, K[[x, y]] is a ring of power series in two variables and f E K[[x, y]] is a non-unit, such

that x { f . Write f = >_,j aj xay? with ate E K. Let

inult(f) = mini + j I aid # 0}, and

mult(f (0, y)) = min{ j I aoi # 01. 3

2. Non-singularity and Resolution of Singularities

4

Set ro = mult(f (0, y)) > mult(f ). Set i

So=min{ TO

-j

.

j
So = oo if and only if f = uy1'°, where u is a unit in K[[x, y]]. Suppose that I So < oo. Then we can write

f=

aijxiy' i+bo j>boru

with aoro # 0, and the weighted leading form

ojxiy2 = aoro yr0 + terms of lower degree in y

L60 (x, y) = i+doj=boro

has at least two non-zero terms. We can thus choose 0 36 cl E K so that

aijc=0.

L6o(1,cl)_ i+ doj=dnro

where qo, Po are relatively prime positive integers. We make a transformation Write do

x=xio, y=xi°(yl+ci). Then f = x1 TO.fi(xi, YO,

where (2.1)

aij(ci +yi)? +xiH(xi,yi)

fi(xi,yi) _ i+aoj=born

By our choice of cl, fi (0, 0) = 0. Set ri = mult(fl(0, yi)). We see that ri < ro. We have an expansion

fi = Eaij(1)xlyl Set

Sl=min{ rl - j j
and write Si = P with Pl, qi relatively prime. We can then choose c2 E K for fl, in the same way that we chose Cl for f, and iterate this process, obtaining a sequence of transformations

x=xi°, (2.2)

y=xi°(y1+ci),

xl =x21, yi = 4'(Y2 + c2),

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

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

5

As our first approximation, we can use our first transformation to solve for y in terms of x and y1: V = C1x6o + y1x60.

Now the second transformation gives us Y = Cix6o

6+s 6o+ + C2Xao +y2xvo

We can iterate this procedure to get the formal fractional series (2.3)

y0 + C3xao+ 11 C2X6°+i qo+qo4j + .. .

y=

Theorem 2.1. There exists an io such that 6i E N for i > io. Proof. ri = mult(fi(0, yi)) are monotonically decreasing, and positive for all i, so it suffices to show that ri = ri+1 implies ai E N. Without loss of generality, we may assume that i = 0 and re = rl. f1 (x1, y1) is given by the expression (2.1). Set g(t) = fi (0, t) _

aij (c1 + t)7. i+6uj=6oro

g(t) has degree ro. Since ri = ro, we also have mult(g(t)) = ro. Thus g(t) = aorotro, and

E aijt.7 = aoro(t - ,)r0. i+do j=6Oro

In particular, since K has characteristic 0, the binomial theorem shows that ai,r0 -10 0,

(2.4)

where i is a natural number with i + 6()(ro - 1) = born. Thus So E N.

0

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

p(t) = E bit' such that (2.3) becomes (2.5)

y =P(xm").

For n E N, set

n

bit'.

pn(t) _ i=1

Using induction, we can show that mult(f (tm, pm(t)) -' oo

2. Non-singularity and Resolution of Singularities

6

as n -+ oo, and thus f (t'n, p(t)) = 0. Thus

y=

(2.6)

bix-

is a branch of the curve f = 0. This expansion is called a Puiseux series (when ro = 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, since we cannot conclude (2.4). Theorem 2.1 is in fact false over fields of positive characteristic. See Exercise 2.4 at the end of this section.

Suppose that f E K[[x, y]] is irreducible, and that we have found a solution y = p(xm) 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 divides f in RI = K[[xm, y]]. Let w be a primitive era-th root of unity in K. Since f is invariant under the K-algebra automorphism 0 of Rl determined

I f in Rl for all j, by xm -* wxns and y - y, it follows that y and thus y = p(wixm) is a solution to f (x, y) = 0 for all j. These solutions are distinct for 0 < j < rn - 1, by our choice of m. The series rn-i (Y

g

- P(WjX

j=U

is invariant under 0, so g E K[[x, y]] and g I f in K[[x, y]], the ring of invariants of Rl under the action of the group Zm generated by 0. Since f is irreducible, M-I

f =u rij=o (y-p(Wjxm)) where u is a unit in K[[x, y]].

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

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

Exercise 2.4. 1. Construct a Puiseux series solution to r

.f (x, y) = y4 - 2x3y2 - 4x5y + over the complex numbers.

X6

- x7 = 0

2.2. Smoothness and non-singularity

7

2. Apply the algorithm of this section to the equation

f(x,y) = y" + yp+1 + x = 0 over an algebraically closed field k of characteristic p. What is the resulting fractional series?

2.2. Smoothness and non-singularity Definition 2.5. Suppose that X is a scheme. X is non-singular at P E X if OX,j, is a regular local ring.

Recall that a local ring R, with maximal ideal m, is regular if the dimension of m/in2 as an 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[xi, ... , x,,] is a polynomial ring over a field K,

fl,...,

E K[x1,...,xn]

We define the Jacobian matrix al xt

J(f; X) = J(fl, ... , fm.; x1, ... , xn.) _ m

m

Let I = (fl, ... , f,,.,) be the ideal generated by ft, . , f,,,., and define R = K[xi, ... , /1. Suppose that P E spec(R) has ideal 7n in K [xt , ... , and ideal n in R. Let K(P) = Ra/ i . We will say that J(f ; x) has rank l at P if the image of the l-th Fitting ideal 11(J(f ; x)) of l x l minors of J(f, x) in K(P) is K (P) and the image of the s-th Fitting ideal 1,(.l (f ; x)) in K(P) is (0) for s > 1. . .

Definition 2.6. Suppose that X is a variety of dimension s over a field K, and P E X. Suppose that U = spec(R) is an affine neighborhood of P such with I = (ft, ... , f,,,). Then X is smooth over K that R_' K[xl,... , if I (f ; x) has rank n - s at P. This definition depends only on P and X and not on any of the choices of U, x or f. We verify this is the appendix.

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

Theorem 2.8. Suppose that X is a variety over a field K and P E X. 1. Suppose that X is smooth over K at P. Then P is a non-singular point of X.

8

2. Non-singularity and Resolution of Singularities

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

In the case when K is algebraically closed, Theorems 2.7 and 2.8 are proven in Theorems 1.5.3 and 1.5.1 of [47]. Recall that an algebraically closed field is perfect. We will give the proofs of Theorems 2.7 and 2.8 for general fields in the appendix.

Corollary 2.9. Suppose that X is a variety over a perfect field K and P E X. Then X 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 is always separably generated over K (Theorem 13, Section 13, Chapter II, [92]).

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 (f 1, ... , f m) C AK is an s-dimensional affine variety, where I = (ft, ... , fm) is a reduced and equidimensional ideal. We interpret the closed points of X as the set of solutions to f1 = - = fm = 0 in Kn. We identify a closed point p = (a1, ... , a,a) E V(I) c AK with the maximal of K[xl,... , x,,]. For 1 < i < rn, ideal m = (xi - a,, ... , xn -

fi = fi (p) + Li mod m2, where

n

Li

= j=1

(p) (xa - aj)

But fi(p) = 0 for all i since p is a point. of X. The tangent space to X in A'K at the point p is Tp(X) = V(T1,...,Lm) C AK.

We see that dim T(X) = n - rank(J(f;x)(p)). Thus

s (by Remark A.9), and X is non-singular at p if and only if dimTp(X) = s.

Theorem 2.10. Suppose that X is a variety over a field K. Then the 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 T.5.3 [47]. We will prove Theorem 2.10 when K is perfect in the appendix. The general case is proven in the corollary to Theorem 11 [85].

Exercise 2.11. Consider the curve y2 - x3 = 0 in AK, over an algebraically closed field K of characteristic 0 or p > 3.

2.3. Resolution of singularities

J

1. Show that at the point p1 = (1, 1), the tangent space T1(X) is the line

-3(x

1) + 2(y - 1) = 0. 2. Show that at the point p2 = (0, 0), the tangent space Tp2 (X) is the entire plane A 2K'

3. Show that the curve is singular only at the origin P2-

2.3. Resolution of singularities Suppose that X is a K-variety, where K is a field.

Definition 2.12. A resolution of singularities of X is a proper birational morphism (/) : Y -> X such that Y is a non-singular variety.

A birational morphism is a morphism 0 : Y - X of varieties such that. there is a dense open subset u of x such that 0-1(U) - U is an isomorphism. if X and Y are integral and K(X) and K(Y) are the respective function fields of X and Y, / is birational if and only if 0* : K(X) --+ K(Y) is an isomorphism. X is proper if for every valuation A morphism of varieties cp : Y

ring V with morphism a : spec(V) -> X, there is a unique morphism 3 : Y such that 0 0,3 = a. If X and Y are integral, and K(X) is the function field of x, then we only need consider valuation rings V such that K C V C K(X) in the definition of properness. The geometric idea of properness is that every mapping of a "formal" curve germ into X lifts uniquely to a morphism to Y. One consequence of properness is that every proper map is surjective. spec(V)

The properness assumption rules out the possibility of "resolving' by taking

the birational resolution map to be the inclusion of the open set. of nonsingi.ilar points into the given variety, or the mapping of the non-singular points of a partial resolution to the variety. Birational proper morphisrns of non-singular varieties (over a field of characteristic zero) can be factored by alternating sequences of blow-ups and blow-downs of non-singular subvarieties, as shown by Abrainovich, Karu, Matsuki and Wlodarczyk [11]. We can extend our definition of a resolution of singularities to arbitrary schemes. A reasonable category to consider is excellent (or quasi-excel lent) schemes (defined in IV.7.8 [45] and on page 260 of [66]). The definition of excellence is extremely technical, but the idea is to give minimal conditions ensuring that the the singular locus is preserved by natural base extensions

such as completion. There are examples of non-excellent schemes which admit a resolution of singularities. Rotthaus [74] gives an example of a

2. Non-singularity and Resolution of Singularities

10

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

2.4. Normalization Suppose that R is an affine domain with quotient, field L. Let S he the integral closure of R in L. Since R is affine, S is finite over R, so that S is alline (ef. Theorem 9, Chapter V [92]). We say that spec(S) is the normalization of spec(R). Suppose that V is a valuation ring of L such that R C V. V is integrally closed in L (although V need not. he Noetherian). Thus S C V. The morphism spec(V) -+ spec(R) lifts to a morphisni spec(V) -i spec(S), so that spec(S) > spcc(R) is proper. If X is an integral variety, we can cover X by open affines spec(k) with normalization spec(S,). The spec(S2) 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 normalization of X is the disjoint union of the normalizations of the irreducible components of X. A ring R is normal if Rp is an integrally closed domain for all p E spec(R). If R. is an affine ring and spec(S) is the normalization of spec(R) we constructed above, then S is a normal ring.

Theorem 2.13 (Serre). A Noetherian ring A is normal if and only if A is Ri (Ai, is regular if ht(p) < 1) and S2 (depth A,, > min(ht(p), 2) for all p E spec(A)).

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

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

Proof. Let X be the normalization of X. All local rings ox, of points p E X are local rings of dimension 1 which are integrally closed, so Theorem 2.13 implies they are regular.

-

As an example, consider the curve singularity y2 :i:3 = 0 in A2, with affine ring R = K[x, y]/(y2 - x3). In the quotient field L of R we have the relation (N)2 - :r: - 0. Thus x is integral over H. Since R[z] = K[Y] is a regular ring, it must be the integral closure of R in L. Thus spec(R[k]) -, spec(R) is a resolution of singularities. Normalization is in general not enough to resolve singularities in dimension larger than 1. As an example, consider the surface singularity X defined

2.5. Local uniforinization and generalized resolution problems

11

by z2 - xy = 0 in AK, where K is a field. The singular locus of X is defined = (x, y, z), so the singular locus of by the ideal J = (z2 - xy, y, X. 2z).

X is the origin in A2. Thus R = K[x, y, z]/(z2 - xy) is Ri. Since z2 - xy is a regular element in K[x, y, z], R is Cohen-Macaulay, and hence R is S2 (Theorem 17.3 [66]). We conclude that X is normal, but singular.

Kawasaki [58] has proven that under extremely mild assumptions a Noetherian scheme admits a Macaulayfication (all local rings have maxiinal depth). In general, Cohere-Macaulay schemes are far from being nonsingular, but they do share many good homological properties with nonsingular schemes.

A scheme which does not admit a resolution of singularities is given by the example of Nagata (Example 3 in the appendix to [68]) of a onedimensional Noetherian domain R. whose integral closure R is not a finite R.module. We will show that such a ring cannot have a resolution of singulari-

ties. Suppose that there exists a resolution of singularities 0 : Y - spec(R). That is, Y is a regular scheme and 4) is proper and birational. Let L be the quotient field of R. Y is a normal scheme by Theorem 2.13. Thus Y has an open cover by affine open sets U1, . , U, such that Uz spec(7z), with Tz = R[yli, ... , yzt] where gz; E L, and each T., is integrally closed. The integrally closed subring . .

T=r(Y,OY)=nTi i=1

of L is a finite R-module since 0 is proper (Theorem III, 3.2.1 [44] or Corollary 11.5.20 [47] if 0 is assumed to be projective).

Exercise 2.15. 1. Let. K = Z ,,(I), where p is a prime and I is an indeterminate. Let

R = K[x, y]/(xp + yP - t), X = spec(R). Prove that X is nonsingular, but there arc no points of X which are smooth over K. 2. Let K = Zp(t), where p > 2 is a prime and t is an indeterminate.

Let R = K[x, y]/(x2 + yt' -- t), X = spcc(R). Prove that X is non-singular, and X is smooth over K at every point except at the prime (yp - t, x).

2.5. Local uniformization and generalized resolution problems Suppose that f is irreducible in K[[x, y]]. Then the Puiseux series (2.5) of Section 2.1 determines an inclusion (2.7)

R = K[[x, y]]l (f (x, y)) - K[[t]]

2. Non-singularity and Resolution of Singularities

12

with x = t', y = p(t) the series of (2.5). Since the rn. chosen in (2.5) of Section 2.1 is the smallest possible, we can verify that t is in the quotient field L of R. Since the quotient field of K[[t]] is generated by 1, t, ... , t' 1 over h, we conclude that R and K[[t]] have the same quotient field. (2.7) is an explicit realization of the normalization of the curve singularity germ f (x, y) = 0, and thus spec(K[[t]]) , spec(R) is a resolution of singularities. The quotient field L of K[[t]] has a, valuation v defined for non-zero h E L by

v(h(t)) ='n E Z if h(t) = t"u, where u is a unit in K[[t]]. This valuation can be understood in R by the Puiseux series (2.6). More generally, we can consider an algebraic function field L over a field

K. Suppose that V is a valuation ring of L containing K. The problem of local uniformization is to find a regular local ring R, essentially of finite type over K and with quotient field L such that the valuation ring V dominates R (R C V and the intersection of the maximal ideal of V with R is the maximal ideal of R). The exact relationship of local uniformization to resolution of singularities is explained in Section 8.3.

If L has dimension 1 over K, then the valuation rings V of L which contain K are precisely the local rings of points on the unique non-singular projective curve C with function field L (cf. Section 1.6 [47]). The Newton method of Section 2.1 can be viewed 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 are non-Noetherian (see the exercise of Section 8.1). Zariski proved local uniformization for two-dimensional function fields over an algebraically closed field of characteristic zero in [86]. He was able to patch together local solutions to prove the existence of a resolution of singularities for algebraic surfaces over an algebraically closed field of characteristic zero. He later was able to prove local uniformization for algebraic function fields of characteristic zero in [87]. This method leads to an extremely difficult patching problem in higher dimensions, which Zariski was able to solve in dimension 3 in [90]. Zariski's proof of local uniformization can be considered as an extension of the Newton method to general valuations. In Chapter 8, we present. Zariski's proof of resolution of surface singularities through local uniformization. Local uniformization has been proven for two-dimensional function fields in positive characteristic by Abhyankar. Abhyankar has proven resolution of singularities in positive characteristic for surfaces and for three-dimensional varieties, [1],[3],[4].

2.5. Local imiformization and generalized resolution problems

13

There has recently been a resurgence of interest in local uniformization in positive characteristic, and significant progress is being tnar_ie. Some of the recent papers on this area are: Hauser [50], Heinzer, Rotthaus and Wiegand [51], Kuhhnann [59], [60], Piltant [71], Spivakovsky [77], Teissicr [78].

Some related resolution type problems are resolution of vector fields, resolution of differential forms and monomialization of morphisms. Suppose that X is a variety which is smooth over a peeled. field K, and D E Hom(1 (/k., OX) is a vector field. If p E X is a closed point, and x1i ... ; xn are regular parameters at p, then there is a local expression

D=a1(x)F +...+an(x)a, xn where ai(x) E Ox,p. We can associate an ideal sheaf TD to D on X by

ZL,p = (a,,.. -, a,,)

forpEX. There are an effective divisor FD and an ideal sheaf Jj such that V (Jv ) has codimension > 2 in X and ID = Ox(-Fn)Jn The goal of resolution of vector fields is to find a proper morphism 7r : Y -+ X so that if D' = 7r-1(D), then the ideal 1D' c Cry is as simple as possible. This was accomplished by Seidenberg for vector fields on non-singular surfaces over an algebraically closed field of characteristic zero in [75]. The best statement that can be attained is that the order of JD, at p (Definition A.17) is vp(JD,) < 1 I'or all 1) E Y. The basic invariant considered hi Ilie

proof is the order v,(Jv). While this is an upper semi-continuous function on Y, it can go up under a monoidal transform. However, we have that vy(JD,) < vp(JD)+l if 7r : Y ---> X is the blow-up of p. and q E 7r-'(p). This should be compared with the classical resolution problem, where a basic result is that order cannot go up under a permissible monoidal transform (Lemma 6.4). Resolution of vector fields liar surootli sur1'a4:es over a. perf'ecl,

field has been proven by Cano [19]. Cano has also proven a local theorem for resolution for vector fields over fields of characteristic zero [20], which implies local resolution (along a valuation) of a vector held. The statement is that we can achieve v4 (JD,) < 1 (at least locally along a valuation) after a morphism 7r : Y -' X. There is an analogous problem for differential forms. A recent paper on this is [21]. We can also consider resolution problems for morphisms f : Y --p X of varieties. The natural question to ask is if it is possible to perlorm monoidal transforms (blow-ups of non-singular subvarieties) over X and Y to produce a morphism which is a monomial mapping.

14

2. Non-singularity and Resolution of Singularities

Definition 2.16. Suppose that 4D : X -> Y is a dominant morphism of nonsingular irreducible K-varieties (where K is a field of characteristic zero). is monomial if for all p E X there exists an etale neighborhood U of p, iiniformizing parameters (xl,... , on U, regular parameters (yl,.... in 0Y,,h(p), and a matrix (a1j) of non-negative integers (which necessarily has rank m) such that yl = xall ...:l: a,n, n

1

ym

_=

,xaml ... xan+n

Definition 2.17. Suppose that c : X -' Y is a dominant morphism of integral K-varieties. A morphism 41 X1 -, Yi is a monomialization :

of 1D [27] if Lhere are sequences of blow-ups of non-singular subvarieties a : XI --+ X and Q : Y1 -+ Y, and a morphism ' : X1 - Y1 such that the diagram

XI y Yl 1

1

X-Y commutes, and IP is a monomial morphism.

If fi : 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 monomialization of ob [27]. A generalized multiplicity is defined in this paper, and it can go up, causing a very high complexity in the proof. An extension of this result to strongly prepared morphisms from n-folds to surfaces is proven in [33]. It is not known if monomialization is true even for birational morphisms of varieties of dimension > 3, although it is true locally along a valuation, from the following Theorem 2.18. Theorem 2.18 is proven when the quotient field of S is finite over the quotient field of R in [25]. The proof for general field extensions is in [28]. Theorem 2.18 (Theorem 1.1 [26], Theorem 1.1 [28]). Suppose that R C S are regular local 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 sequences of rrionoidal transforms R --+ if and S S' such that V dominates S', S' dominates R' and there are regular parameters (xl. ...., x,,,) in R', (yt, ..., y,-) in S', units 81i ... , 8,n E S' and an m x n matrix (ate) of non-negative integenss such that rank(aij) = m is maximal and x1

=

yill..... yaln8l,

xm = y1 m l ..... yimn bm,

2.5. Local Ilniformization and generalized resolution problems

15

Thus (since char(K) = 0) there exists an etale extension S' -* S", where S" has regular parameters such that x1, ... , are pure monomials in yl......V. The standard theorems on resolution of singularities allow one to easily find R' and S' such that (2.8) holds, but, in general, we will not have the essential condition rank(aij) = m. The difficulty of the problem is to achieve this condition. This result gives very simple structure theorems for the ramification of valuations in characteristic zero function fields [35]. We discuss some of these results in Chapter 9. A generalization of monomialization in characteristic p function fields of algebraic surfaces is obtained in [34] and especially in [35]. We point out that while it seems possible that Theorem 2.18 does hold in

positive characteristic, there are simple examples in positive characteristic where a monolnializat.kni does not exist. The simplest example is the map of curves

y -:I:p+J:p+1 in characteristic p.

A quasi-complete variety over a field K is an integral finite type Kscheme which satisfies the existence part of the valuative criterion for properness (Hironaka, Chapter 0, Section 6 of [52] and Chapter 8 of [261). The construction of a monomialization by quasi-complete varieties follows from Theorem 2.18. Theorem 2.19 is proven for generically finite morphisms in [26] and for arbitrary morphisms in Theorem 1.2 [28].

Theorem 2.19 (Theorem 1.2 [26],Theorem 1.2 [28]). Let K be a field of characteristic zero,

: X -> Y a dominant rrr.orphism, of proper K-varieties.

Then there exist birational morphisms of non-singular quasi-complete Kvarieties a : X1 -> X and f3 : Y1 - Y. and a monomial morphsmm %P X1 -+ Yl such that the diagram X1 1

X

Y1

fi

I Y

commutes and a and ,3 are locally products of blow-ups of non-singular subvarieties. That is, for every z (z X1, there exist affine neighborhoods Vi of z and V of x = cr(z) such that cr : V1 - V is a finite product of monoidal transforms, and there exist affine neighborhoods W1 of W(z), W of y = t3('Y(z)) such that f3 : W1 -> W is a finite product of monoidal transforms.

16

2. Non-singularity and Resolution of Singularities

A monoidal transform of a non-singular K-scheme S is the map T -> S induced by an open subset T of proj(®Z"), where I is the ideal sheaf of a non-singular subvariety of S. The proof of Theorem 2.19 follows from Theorem 2.18, by patching a finite number of local solutions. The resulting schemes inay not be separated. It is an extremely interesting question to determine if a monomialization exists for all morphisms of varieties (over a, field of characteristic zero). That

is, the conclusions of Theorem 2.19 hold, but with the stronger conditions that, cr. and it are products of monoidal transforms on proper varieties X1 and Yl.

Chapter 3

Curve Singularities

3.1. Blowing up a point on A2 Suppose that K is an algebraically closed field. Let

Ul = spec(K[, , tl) = AA,

U2 = spec(K[u, v]) = 41

.

Define a. K -algebra isomorphism

A: K[s,t]R = K[s,t. 1] - K[u,v]L, = K[u,v, -] by A(s) - 1, A(/) -- nv. We define a K-variety Be by patching U, to U2 on the open sets U2 - V(v) = spec(K[u,

and Ul - V (s) = spec(K[s. t]3)

by the isomorphism A. The K-algebra homomorphisms K[x, y] --> K[s, t]

defined by x --> st, y H t, and K[x, y] -> K[u, v]

defined by x u, y -> uv, are compatible with the isomorphism A, so we get a morphisin

7r:B(p)=BO -+U0=spec(K[x,y])-42. where p denotes the origin of UO. Upon localization of the above maps, we get isomorphisms

K[x, y],, - K[s, t]t and K[x, y],,

K[u, v]u. 17

3. Curve Singularities

18

Thus ir:U1-V(t)=Uo-V(y),7r:U2-V(u)=Uo-V(x),andirisan isomorphism over Uo - V(x, y) = Up - {p}. Moreover, {V (t.) c spec(K[s, t])} = spec(K[s]),

7r-1(p) fl U1 = V (st, t.)

it-1(p) rl U2 = V(u,uv)

{V(u) C spec(K[u,vj)} = spec(K[S]),

by the identification s - v. Thus 7r-1(p) = 1P1. Set E _ it-1(p). We have "blown up p" into a codimension 1 subvariety of B(p), isomorphic to ]PI. Set R = K[x, y], rri. _ (x, y). Then

U1 = spec(K[s, t]) = spec(K[y, y]) = spec(R[y]), U2 = spec(K[x, Y]) = spec(R[!]). We see that

B(p) = spec(R[x]) U spec(R[y]) - proj((D7n'). ,,.>o

Suppose that q C 7r-1(p) is a closed point. If q E 111, then its associated ideal mq is a maximal ideal of Rl = K[y, y] which contains (x, y)Ri = yR1. Thus mq = (y, - a) for some ay E K. If we set yl = ,y, xl = ? - cx, we see that there are regular parameters (x 1, yl) in OB(l),q such that x =3/1(x1 + a),

y = y1.

By a similar calculation, if q E 7r 1(p) and q E U2, there are regular parameters (x I, yl) in OB(r) q such that

x- x1,

V = xl (y1 + 03)

for some fl E K. If the constant a or 0 is non-zero, then q is in both U1 and U2. Thus the points in 7r-1(p) can be expressed (uniquely) in one of the forms

x=xi,y-x1(yi+a)with aEK, or x=xlyl,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 Y or Say Y E V. Then R[x] C V, and we have a morphism

spec(V) - spec(R[2]) C 8(p) X

which lifts the morphism spec(V)

spec(I?).

E V.

3.1. Blowing up a point on A2

19

More generally, suppose that S = spec(R) is an affine surface over a field L, and p E S is a non-singular closed point. After possibly replacing S with an open subset spec(R f), we may assume that the maximal ideal of p in R is rrtp = (:c, y). We can then define the blow up of p in S by (3.1)

rr : R(p) =

S. n>O

We can write B(p) as the union of two affine open subsets: 11(p) = spec(R[y]) U spcc(R[y]).

rr is an isomorphism over S - p, and 7r-1(p) ^-' P1 Suppose that S is a surface, and p E S is a non-singular point, with ideal sheaf mp c Os. The blow-up of p E S is 7r : B (p) = Proj ((Dmp) n>O

it is an isomorphism away from p, and if' U = spec(R) C S is an affine open neighborhood of p in S such that (x, ?1) = r(U, mp) C R

is the maximal ideal of p in R, then the map it : it 1(U) -i U is defined by the construction (3.1). Suppose that C C AK is a curve. Since K[x, y] is a unique factorization domain, there exists f E K [x, y] such that. V (f) = C. If q E C is a closed point, we will denote the corresponding maximal ideal of K[x,y] by mq. We set vq(C) = max{r I f E rn'} (more generally, see Definition A.17). vq(C) is both the multiplicity and the order of C at q.

Lemma 3.2. q c A2 is a non-Singular point of C if and only if vq(C) = 1.

Proof. Let R = K[x, y],n4, and suppose that mg = (x, y). Let T = R/ f R. If vq(C) > 2, then f E ' m2, and rngT/rn T mq/m2 has dimension 2 > 1, so that T is not regular and q is singular on C. However, if vq (C) = 1, then we have f =_ a3; + /

mod ?rt.,

where a,,Q E K and at least one of a and /3 is non-zero. Without loss of generality, we may suppose that 13 # 0. Thus y--a

modm4+(f),

3. Curve Singularities

20

(f). We see that

and x is a K generator of iagT/m,2T

dimly rq7'/ni T = 1 and q is non-singular on C.

11

Remark 3.3. This lemma is also true in the situation where q is a nonsingular point 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 rn,Q/mg, and since R is a unique factorization domain, there

is f E R such that f = 0 is a local equation or(" at q.

Let p be the origin in AK, and suppose that vp(C) = r > 0. Let 7r: B(p) -> A2 be the blow-up of p. The strict transform C of C in B(p) is the Zariski closure of it-1(C - p) ^_' C - p in B(p). Let E _ 7r-1(p) be the exceptional divisor . Set-theoretically, 7r-'(C) = C U E. For some aij E K, we have a finite sum

aijx'i/

f= i+j>r

with aij # 0 for some i, j with i + j -- r. In the open subset U2 = spcc(K[xi, yi)) of B(p), where

x = xi,

Y=1:017

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

f = xrft, where

fi =

ai7'C1

iIj>r xl 1' fi since vp(C) = r, so that fi = 0 is a local equation of the strict transform C of C in U2. In the open subset Ui = spcc(K[xi, y1]) of B(p), where x=T1391,

Y=?li,

yi = 0 is a local equation for E and

f = yifl, where

i -i+j-r f 1 - L aijx1y1 i+j>r

is a local equation of e in U1.

A2

3.1. Blowing rip a point on

21

As a scheme, we see that 7r-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

aijxV.

L= i-I-j=r

Suppose that q E ir-1(p). There are regular parameters (x1, yi) at q of one of the forms

3; = :t:1,'y = x1('Ji + (z) or x . - xlyl, y - yi

In the first case f, =

t

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

aij (yl + a) + xi52

.fl = i+j=r

for some polynomial Q. Thug vq(C,) <,r, and vq(C) = r implies

F, aij (yl

t

} = aoryl

i+j=r

We then see that

E aijyi = aor(yl i+j=r ,,

a)r

and

In the second case fl =

L = aor(y - ax)''. = 0 is a local equation of C at q, where

fi _

aijxj + 02

for some series Q. Thus vq(C) < r, and vq(C) = r implies

aijx] = aroxi i

l j=r

and

L = aroxr.

We then see that there exists a point q E it-1(p) such that vq(C) = r only when L = (ax t by)' for some constants a, b c K. Since r > 0, there is at Most one point q E 7r1(p) where the multiplicity does not drop.

Exercise 3.4. Suppose that K[xl,... , xn] is a polynomial ring over a perfect field K and f E K[x1i... , xn] is such that 1(0,... 0) = 0. Let ,

rn = (x1, ... ) xn), a maximal ideal of K[xl,... , xn]. Define ord f (0, xn

= r if f(0,...,0,xn) and xrn 1 ]'f(0,...,0,x,,,).

. . .

, 0, x,)

3. Curve Singularities

22

Show that R = (K[xi, ... , xn]/(f )) is a regular local ring

if

rod f(0,...,O,x,) - 1.

3.2. Completion Suppose that A is a local ring with maximal ideal m. A coefficient field of A is a subfield L of A which is mapped onto A/m by the quotient mapping A A/m. A basic theorem of Cohen is that an equicharacteristic complete local ring contains a coefficient field (Theorem 27 Section 12, Chapter VIII [92]). This leads to Cohen's structure theorem (Corollary, loc. cit.), which shows that an equicharacteristic complete regular local ring A is isomorphic to a formal power series ring over a field. In fact, if L is a coefficient field of A, and if (xl,... , x,) is a regular system of parameters of A, then A is the power series ring

A-L[[xi,...,xn]]. We further remark that. the completion of a local ring R is a regular local ring if and only if R is regular (cf. Section 11, Chapter VIII [92]). Lemma 3.5. Suppose that S is a non-singular algebraic surface defined over a field K, p E S is a closed point, it : B = B(p) - S is the blow-up of p, and

suppose that q E r 1(p) is a closed point such that K(q) is separable over K(p). Let Ri = Oq,p and R2 = OB,q, and suppose that Kl is a coefficient field of R1, (x, y) are regular parameters in R1. Then there exist a coefficient field K2 = Kl (a) of R2 and regular parameters (xl, yl) of R2 such that. fr"

.

Ri

R2

is the map given by

aijx'y2 -y i,j>o

aijx+3*(yl+a)', i,j>o

where aij E K1, or K2 = Ki and is the map given by nijxiN

E aijx 1 y1 i,j>0

i,j>O

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

°Sp-+R1

23

3.2. Completion

which induces

q E B(p) x,,7 spec(Ri) = spec(Rj [y])

U

spec(R.1[y]).

Let mq be the ideal of q in B(p) x5 spec(Ri). If q E spec(Rj[y]), then K(q)

Rj[x]/mq

K1[x]/(f(y))

for some irreducible polynomial f(Y) in the polynomial ring K.[X]. Since K(q) is separable over K1 ^_' K(p), f is separable. We have mq = (x, f (2)) C R1[y].

Let a E Rl [x]/mq be the class of We have the residue map o : R2 - L where I, = Ri[k]/rrnq. There is a natural embedding of K1 in R.2. We have a factoriratiou of

f (t) = (t - ?)y(t) in L[t], where t - a and y(t) are relatively prime. By Hensel's Lemma (Theorem 17, Section 7, Chapter VIII [92]) there is a c R2

such that vi(a) = a and f (t) = (t - a)-y(t) in R2[t], where 45(y(t)) = y(t). The subfield K2 - K 1 [or] of R2 is thus a coefficient field of R2, and mgR2 = (x, - a). Thus R2 = K2[[x, Y - a]]. Set xl = x, yl = x a. The inclusion

Rl = Ki[[x,y]] _, R2 = K2[[xu, y,]]

is natural. A series

1: aijxiyj with coefficients aij E K1 maps to the series i+j E aijxl (yl + a)i.

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](p_t). Suppose that R has a coefficient field L containing K. Let 0 : R R/m be the residue map. Since ¢ L is an isomorphism, there exists A E L such that AP = t. Thus (xP - t) _ (x - A)p in R. But this is impossible, since (;i'P - 1) is a generator of mR, and is thus irreducible. Theorem 3.6. Suppose that R, is a reduced affine ring over a field K, and A = Rp, where p is a prime ideal of R. Then the completion A = Rp of A with respect to its maximal ideal is reduced.

3. Curve Singularities

24

When K is a perfect, field, this is a theorem of Chevalley (Theorem 31, Section 13, Chapter VIII [92]). The general case follows from Scholie IV 7.8.3 (vii) [45]. However, the property of being a domain is not preserved render coin-

pletion. A simple example is f = y2 - x2 + x3. f is irreducible in C[x, y], but is reducible in the completion C[[x, y]]:

f = y2 - x2 - x3 = (y - x l + :r) (y + x I + x). The first parts of the expansions of the two factors are 1

y - x - 2x2+... and

y+x+ IX2+

.

Lemma 3.7 (Weierstrass Preparation Theorem). Let K be a field, and suppose that f E K[[xl,... , xn, y]] i. such that 0 < r = v(f (0, ... , 0, y)) = max{n I y" divides f (0, ... , 0. y)) < oc.

Then there exist a unit series u in K[[xl, ... , x,,., y]] and non-unit series such that ai E K[[xl,... ,

f =u(y"+aly"-1 +...+a"). A proof is given in Theorem 5, Section 1, Chapter VII [92]. A concept which will be important in this hook is the Tschirnhausen transformation, which generalizes the ancient notion of "completion of the square" in the solution of quadratic equations.

Definition 3.8. Suppose that K is a field of characteristic 1, > 0 and f E k [[:r. r , .... x.n, y]] has an expression

f

-yr+alyr-1+...+ar

with az E K[[:c: i , ... , x,x]] and p - 0 or p { r. The Tschirnhausen transfor-

mation of f is the change of variables replacing y with a1

Yl=y+ ry,. f then has an expression

f - (y)' +

b2(y+)r-2 +... + br

with bz E K[[xl,... , x,,,]] for all i.

Exercise 3.9. Suppose that (R,

is a. complete local ring and L1 C R is a field such that R./m, is finite and separable over L1. Use Hensel's Lemma

(Theorem 17, Section 7, Chapter VIII [92]) to prove that there exists a coefficient field L2 of R containing L1.

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

25

3.3. Blowing up a point on a non-singular surface We define the strict transform, the total transform, and the multiplicity (or order) vp(C) analogously to the definition of Section 3.1 (More generally, sec Section 4.1 and Definition A.17).

Lemma 3.10. Suppose that X is a non-singular surface over an algebiuic:ally closed field K, and C is a curve on X. Suppose that p e X and vp(C) = r. Let 7r : B(p) -r X be the blow-up of p, C the strict transform of C, and suppose that q E 7r-1(p). Then vq(C) < r, and if r > 0, there is at most one point q E it-1(p) such that vq(C) = r.

Proof. Let f = 0 be a local equation of C at

p.

If (.T, y) are regular

parameters in OX,p, we can write

f = E o-ijxibj as a series of order r in Ox,p = K[[x,,yJ]. Let

L = > aiix'yj i+i=r

be the leading form of f. If q E 7r (p), then OB(p),y has regular parameters (xi, yl) such that

x=xi,y=xi(yi+a), orx=xlyi,y=?!iThis substitution induces the inclusion K[[x, y]] = Ox,p - OR(p), = K[[xt, y1]].

First suppose that (x1, yi) are defined by x = x1, y - 2*1(y1 + (k). Then a local equation for C at q is fl = -4, which has the expansion

fl =

aii (1/1 + a)j + x152

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

0

Suppose that C is the curve y2 - x3 = 0 in AK. The only singular point of C is the origin p. Let 7r : B(p) - AK be the blow-up of p, C the strict transform of C, and E the exceptional divisor K = it-1(p). On U1 = spec(K[y, y]) C B(p) we have coordinates x1, yt with x = .r. i yl, y = yl. A local equation for E on Ut is yt = 0. Also,

y2 - xs = yi (1 - xiyl ). A local equation for C on U1 is 1 - x3yj

CnEnU1=0.

0, which is a unit on E n U1, so

26

3. Curve Singularities

On U2 = spec(K[x, c B(p) we have coordinates xl, yl with x = ;ci, x])for F, on 112 is Yr = 0. Also, y = xlyl. A local equation y2 -

X3=T2 (y1

- x1)

A local equation for C on U2 is yi -:r:r = 0, which has order < 1 everywhere. Thus C is non-singular, and

0'C

is a resolution of singularities. Finally,

r(C, O ) = K[x1, V11/(-y2, -1;1) 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 - xs in A 2. The only singular point of C is the origin p. Let 7r : B(p) -r A2 he the blow-up of p, E be the exceptional divisor, C be the strict transform of C. There is only one singular point q on C. Regular parameters at q are (xl, yl ), where x = .r,1, y = xl V1. C has the local equation yi - :c: i = 0.

In this example v?(C) = vr(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 singular points, 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-singular surface, first over an algebraically closed field K of cllaracteristic zero, and then over an arbitrary field.

3.4. Resolution of curves embedded in a non-singular surface I In this section we consider curves embedded in a non-singular surface over an algebraically closed field K of characteristic zero, and prove that their singularities can be resolved. The theorem is proved by passing to the completion of the local ring of the surface at a singular point of the curve. This allows us to view a local equation of the curve as a power series in two variables. Our main invariant is the multiplicity of the curve at a point. We know that this multiplicity

3.4. Resolution of curves embedded in a non-singular surface 1

27

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

Theorem 3.11. Suppose that C is a curve on a non-singular surface X over an algebraically closed field K of characteristic zero. Then there exists a sequence of blow ups of points A : Y -' X such that the strict transform C of C on Y is non-singular.

Proof. Let r = max(vp(C) I p E C}. If r = 1, C is non-singular, so we may assume that r > 1. The set {p E C I v(p) = r} is a subset of the singular locus of C, which is a proper closed subset of the 1-dimensional variety C, so it is a finite set. The proof is by induction on r. We can construct a sequence of projective morphisms

....-, X9l,-f... 54 XO - X,

(3.2)

where each 7r,,.+l : Xn 11 - Xn is the blow-up of all points on the strict

transform Cn of C which have multiplicity r on C,,. If this sequence is finite, then there is an integer n such that all points on the strict transform C'n of

C have multiplicity < r - 1. By induction on r, we can repeat this process to construct the desired morphism Y X which induces a resolution of C. We will assume that the sequence (3.2) is infinite, and derive a contra, diction. If it is infinite, then for all n E N there are closed points pn E C which have multiplicity r on Cn and such that 7rn+l(pn+1) = pn for all n. for all n. We then have an infinite sequence of completions of quadratic transforms of local rings Rj--+ Let R11 =

OXTM p,,.

Suppose that (x, y) are regular parameters in Ro = ox,p, and f = 0 is a local equation of C in Re = K[[x, y]]. f is reduced by Theorem 3.6. After a linear change of variables in (x, y), we may assume that v(f (0, y)) = r. By the Weierstrass preparation theorem, + ... + a, (x)), f = u(yr + aj(x)yr-r

where u is a unit series. We now "complete the r-th power of f ". Set al(x) (3.3) y+ r

Then yr+a1(x)y'-l+...+ar(x) (3.4)

_

+b2(x)J -2+...+b,(x)

for some series bi (x). We may thus assume that (3.5)

f = yr +

a2(x)yr-2

+ ... + ar(x).

3. Curve Singularities

28

Since f is reduced, we must have some ai(x) # 0. Set mult(ai(x)) l n = min r (3.6) J l i We have n > 1, since v(f) = r. Ox1,pi has regular parameters xl, yl such that

1. x=xi,y=xi(yi+(1)with aeK,or 2. x=xiyi,Y=y1 In case 2,

f =YU1, where

h=1+ a2(xlyl) Y

ar(xlyl)

yi is a local equation of the strict transform Cl of C at pl. fj is a unit, so that vp1(C1) = 0. Thus case 2 cannot occur. In case 1,

f=x'ifi where

a''(xj) xr1 f j = 0 is a local equation at pi for C1. If a # 0, then vpl (C1) < r - 1, so this case cannot occur. If a = 0, and vp, (Cl) = r, then f j has the form of (3.5), with n decreased

fl = (vi +(r)r+

+(t)T-2+...+

x1

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 (3.2) has infinite length.

The transformation of (3.3) is called a Tschirnhausen transformation (Definition 3.8). The Tschirnhausen transformation finds a (formal) curve of maximal contact H = V(11) for C at p. The Tschirnhausen transformation

was introduced as a fundamental method in resolution of singularities by Abhyankar.

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

Sn - Sn-1 -> ... - Si - spec(Osp) is a sequence of blow-ups of points pi E Si, such that the strict transform Ci of C in Si contains pi with vp= (Ci) = vp(C) for i < n, then the strict transform Hi. of H in Si contains pi.

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

.

29

Exercise 3.13. 1. Resolve the singularities by blowing up: a. x2 = x4 + y4, b. xy = x6 + y6, c. x3 = y2 + x4 + y4, d. i2y + 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 CC on S and numbers r2 E N such that

ID = r i .1O D has simple normal crossings (SNCs) on S if for every p E S there exist regular parameters (x, y) in 0s,p such that if f = 0 is a local equation for D at p, then f - unit xy' in C)s,r Find a sequence of blow-ups of points making the total transform of y2 - x3 = 0 a SNCs divisor. 3. Suppose that D is an effective divisor on a non-singular surface S. Show that there exists a sequence of blow-ups of points

such that the total transform 7r*(D) is a SNCs divisor.

3.5. Resolution of curves embedded in a non-singular surface II In this section we consider curves embedded in a non-singular surface, defined over an arbitrary field K.

Lemma 3.14. Suppose that. K is a field, S is a non-singular surface over K, C is a curve on S and p E C is a closed point. Suppose that. 7r : B =

B(p) - S is the blow-up of p, C is the strict transform of C on B and q E it-1(p) f1 C. Then L'9 (C) 5 VP (C),

and if vq(C) = vplC),

then K(p) = K(q).

Proof. Let R = ds,p, (x, y) be regular parameters in R. Since there is a natural embedding of 7r-1(p) in B x S spec(R), it follows that

q E B xs spec(R) = spec(R[y]) U spec(R[f]).

3. Curve Singularities

30

Without loss of generality, we may assume that q E spec(R[y]). Let Ki K(p) be a coefficient field of R, m c R[i] the ideal of q. Since R[F]/(x, y) K1 there exists an irreducible monic polynomial h(t) E K1 [t] such that mR[y] = (x,h(y)).

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

f=E

aijx2yj

i+j>r

with aij E K1. Thus f = xr fl, where fl = 0 is a local equation of the strict transform C of C in spec(R[x]) and

fi =

aij(y)j +x11 i+j=r

in R[y]. Let n = vq(C). Then h(y )'° divides

aij (y )j

i+j-r in KI [x]. Also,

aij(y)j) < r

deg(

i+j=r

x

in K, [x] implies n _< r and vq(C) = r implies deg(h(x)) = 1, so that h = - a with a E Ki and (by Lemma 3.5) there exist regular parameters (xi, yi) in mq c 0B(p),q such that 6s,p = Ki [[x, y]] -+ Ki [[xi, yil] = OB(P),q

is the natural Ki

K(p) algebra homomorphism x = x1,y = xi(yl + a).

Theorem 3.15. Suppose that C is a curve which is a subvariety of a nonsingular surface X over an infinite field K. Then there exists a sequence of blow-ups of A : Y --+ X such that the strict transform C of C on Y is non-singular.

Proof. Lot r = max{vv(C) I p E C}. If r = 1, C is non-singular, so we may assume that r > 1. The set {p E C I vv(C) = r} is a subset of the singular locus of C, which is a proper closed subset of the 1-dimensional variety C, so it is a finite set. The proof is by induction on r.

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

31

We can construct a sequence of projective morphisuls

X, t

(3.7)

X1-4X0

.X,

where each 7r,,,+1 : Xn,+l --> Xn, is the blow-up of all points on the strict transform Cn, of C which have multiplicity r on G. If this sequence is

finite, then there is an integer n such that all points on the strict. transform C of C have multiplicity < r 1. By induction on r, we can then repeat this process to construct the desired Y -+ X which induces a resolution of C. We will assume that the sequence (3.7) is infinite, and derive a contradiction. If it is infinite, then for all n E N there are closed points p,,, E C,,, which have multiplicity r on Cn and are such that 7rn.+1(pn+1) = pn. Let Rn = 0x,,,p for all n. We then have an infinite sequence of completions of quadratic transforms (blow-ups of maximal ideals) of local rings

R=Re->R1 -,.---,R,,, --,... We will deline (dpi E

1l I

such that (3.8)

apf. - 6pr.-1 - 1

for all i > 1. We can thus conclude that. pi has multiplicity < r on the strict transform Ci of C for some natural number i < by + 1. From this contradiction it will follow that (3.7) is a sequence of finite length.

Suppose that f e Ro = Ox,p is such that f = 0 is a local equation of C and (a:, y) are regular parameters in R = Ox,p such that r = mult(f (0, y)). We will call such (x, y) good parameters for f. Let K' be a coefficient field of R. There is an expansion

f=

i+j>r

with aij E K' for all i, J and aor 0 0. Define (3.9)

5(f;x,y)=111111

i

. Ii
r-3

and

o}.

We have S(f ; x, y) > 1 since (x, y) are good parameters We thus have an expression (with S = b(f; x, y))

J, _ E aijxly3 = Ls + E aix`y i+j6>rb

i+ja>rS

,

3. Curve Singularities

32

where

Lb =

(3.10)

ai jxiy7 = aoryr + A i+jb=r6

is such that aor 0 and A is not zero. Suppose that (x, y) are fixed good parameters of f. Define (3.11)

n

6P

= sup{6(f; x, y1) ;y = y1+E bix' with n E N and bz E K'} E

N U {oo}.

i=1

We cannot have by = oo, since then there would exist a series 00

bix

y = yi + i=1

such that 6(f ; x, y1) = oc, and thus there would be a unit series-, -y in R such

that

f=yyT. But then r = 1 since f is reduced in R, a contradiction. We see then that 6p E 7 N.

After possibly making a substitution n

y=111+

1bixi i=1

with bi E K', we may assume t. hat 6p = 6(f ; x, y). Let 6 = bp. Since vp, (C1) = r, by Lemma 3.14, we have an expression R1 = K'[[x1, y1]1,

where R --r R1 is the natural K'-algebra homomorphism such that either

x=x1, y=xi(yi+a) for some a E K', or

x=xlyl, y=yi We first consider the csse where x = xlyl, y = y1. A local equation of C1 (in spec(Ri)) is f1 = 0, where

fi = f = E aijxi + yin = ao,. + xig + y1h yl

for some fl, g, h

i+j-r

R1. Thus f j is a unit in R1, a contradiction.

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

33

Now consider the case where x = x1, y = xl (yl + a) with 0 A local equation of C1 is

a E K'.

fl °

I = E aij (yl + a)' + x152

i+j=r for some Il E R1. Since vpl (C,) = r. 1

E

aij(yl + a)7 = ao41.

i+j-r

Substituting t = yi + a, we have

aijtj = aor(t - a)r. i+j=r

If we now substitute t = y and multiply the series by xr, we obtain the leading form L of f,

E aijxZy? = aor(y - ax)' = a0yr + ... + (-1)raorarxr.

i+j-r Comparing with (3.10), we see that r = r8 and 6 = 1, so that L6 = L. But we can replace y with y-ax to increase 5, a contradiction to the maximality of b. Thus vp, (C1) < r, a contradiction. Finally, consider the case x = x1 i y = x1 y1. Set, 6' = 6 - 1. A local equation of C1 is

fl

= E aiixli+j-r"1 Y1

i+jS>r,i

+

l

i+j6'_rb'

ai-j+r,j lyl, i+j6'>r6'

where i = i + j - r. Since L4 (x1,y1) _

1 xLb(xl,xivi)

has at least two non-zero terms, we see that 6(f1; x1, yi) = 6' = b - 1. We will now show that f11,, = 6(f1; x1, y1). Suppose not. Then we can make a substitution bixi = yi - bxi + higher order terms in x1 ?l1 = yi with 0,E b E K' such that

6(fi;xi,yi) > b(fi;xl,yl) Then we have an expression

ai-j+r,jxi (yi - bxi) = aor(yi)r + i+0'-r6'

bijxi(yi )i, i+j6'>r6'

3. Curve Singularities

34

so that 6'=dEN,and ;j+r,jxil(?/l

Q'r

- bx)j = uov.(J

i+j6'=r6'

Thus

E i+j6=r6

z+J '

a1 =

ai

+r,j "I '11

aQr (yl

)r_ + bxttij'r1

1

7+j6'=r6'

If we now multiply these series by xi, we obtain aijx'y? = aor(y + bx6)r.

L6 ti 1-j6=r6

But we can now make the substitution y' = y - bx6 and see that

6(f;x,J) > 6(f;x,y) = 6p, a contradiction, from which we conclude that 6p1 = 6' = 6p - 1. We can then inductively define 6p; for i > 0 by this procedure so that (3.8) holds. The conclusions of the theorem now follow.

Remark 3.16. 1. This proof is a generalization of the algorithm of Section 2.1. A more general version of this, valid in arbitrary two-dimensional regular local rings, can be found in [5] or [73]. 2. Good parameters (x, y) for f which achieve 6p = 6(f ; x, y) are such that y = 0 is a (formal) curve of maximal contact for C at p (Definition 3.12).

Exercise 3.17. 1. Prove that the 6p defined in formula (3.11) is equal to 6r, = sup{6(f ; x, y) I (x, y) are good parameters for f }.

Thus 5, does not depend on the initial choice of good parameters, and 5p is an invariant of p. 2. Suppose that vp(C) > 1. Let it : B(p) - X he the blow-up of p, C be the strict transform of C. Show that there is at most one point q E it-' (p) such that vq(C) = vp(C). 3. The Newton polygon N (f ; x, y) is defined as follows. Let I be the ideal in R = K[[x, y]] generated by the monomials xays such that the coefficient a,rp of x° yI' in f (x, y) = F, a,,px"y0 is not zero. Set P(f; x, y) = {(a, (1) E Z2lxayO E I}. Now define N(f ; x, y) to be the smallest convex subset of R2 such that N(f ; x, y) contains P(f ; x, y) and if (a, [3) E N (f ; x, y), (s, t) E

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

35

, then (a + s, fi + t) E N(f ; x, y). Now suppose that (x, y) are good parameters for f (so that v(f (0, y)) = v(f) = r). Then (0, r) E N(f ; x, y). Let the slope of the steepest segment of N(f; x, y) be s(f; x, y). We have 0>s(f;x,y)>-1,since a,rp=0 if a +,0 < r. Show that

R2

1

a(f; x, y) =

- s(f; x, y)

Chapter 4

Resolution Type Theorems

4.1. Blow-ups of ideals Suppose that X is a variety, and J C Ox is an ideal sheaf. The blow-up of

Jis

7r : B =

B is a variety and -7r is proper. If X is projective then B is projective. 7r is an isomorphism over X - V (J), and JOB is a locally principal ideal sheaf.

If U C X is an open affine subset, and R = r(U,Ox), I = (fi,..., fm) C R, then

7r-1(U) = B(I) = proj((D r). n>o

If X is an integral scheme, we have m

proj(®r) = U spec(R[ f: , ... , fit n>O

i=1

Suppose that W C X is a subscheme with ideal sheaf Tw on X. The total transform of W, 7r*(W), is the subscheme of B with the ideal sheaf Z,,-(W) = XWOB-

Set-theoretically, lr*(W) is the preimage of W, 7r-,(W). 37

4. Resolution Type Theorems

38

Let U = X - V (J). The strict transform W of W is the Zariski closure

of it-1(W fl u) in B(J). Lemma 4.1. Suppose that R is a Noetherian ring, I, J C R are ideals and I = q, fl ... fl q,n is a primary decomposition, with primes pi = qi. Then 00

U (I :

J'r') = n

n=o

qi.

{i1-70;}

The proof of Letnnia 4.1 follows easily from the definition of a primary

ideal. Geometrically, Lemma 4.1 says that U°_o(I : Jn) removes the primary components qj of I such that V(qi) C V(J). Thus we see that the strict transform W of W has the ideal sheaf 00

I,,i, = U (IwOn : J" OB)n-0

ForgEB, Zyy,q = If E OB,q I M q" C IWOB,q for some n > 0).

The strict transform has the property that

W = B(JOw) = prol(®(JOw)"') n>O

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

Theorem 4.2 (Universal Property of Blowing Up). Suppose that I is an ideal sheaf on a variety V and f : W -- V is a morphism of varieties such that TOw is locally principal. Then there is a unique morphism g : W -, B(I) such that f = it o g. This is proved in Proposition 11.7.14 [47].

Theorem 4.3. Suppose that C is an integral curve over a field K. Consider the sequence (4.1)

Cn is obtained by blowing up the (finitely many) singular points on Cn. 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 closure of R in the function field K(C) of C. R is a regular ring. Let C = spec(R). All ideals in R are locally principal (Proposition 9.2, page 94 [13]). By Theorem 4.2, we have a factorization C -, Cn - ... - C1 -+ C

4.1. Blow-ups of ideals

39

for all n. Since C - C is finite, Ci+1 -> C2 is finite for all i, and there exist affine rings Ri such that C1 = spec(Rj). For all n we have sequences of inclusions

R -Ri --+ . ..--+ Rn-+ R. Here Rj qb Ri 1 1 for all i, since a maximal ideal min Ri is locally principal if and only if (Ri)m is a regular local ring. Since R is finite over R, we have that (4.1) is finite.

Corollary 4.4. Suppose that X is a variety over a field K and C is an integral curve on X. Consider the sequence XnI'Xn_1

X,

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

Proof. The induced sequence

... _, C, --'.... -,Cl -4C 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 projective birational morphism. Then V = B(T) for some ideal sheaf I c

dw This is proved in Proposition 11.7.17 [47]. Suppose that Y is a non-singular subvariety of a variety X. The rnonoidal

transform of X with center Y is it : B(Iy) - X. We define the exceptional divisor of the monoidal transform 7r (with center Y) to be the reduced divisor with the same support. as 7r*(Y).

In general, we will define the exceptional locus of a proper birational morpism 0 : V --> W to be the (reduced) closed subset of V on which 0 is not an isotnorphisrn.

Remark 4.6. If Y has codimension greater than or equal to 2 in X, then the exceptional divisor of the rnonoidal transform 7r : B(Ty) ; X with center Y, is the exceptional locus of ir. However, if X is singular and not locally factorial, it may be possible to blow up a non-singular codimension I subvariety Y of X to get a uionoidal transform 7r such that the exceptional locus of it is a proper closed subset of the exceptional divisor of ir. For an

example of this kind, see Exercise 4.7 at the end of this section. If X is non-singular and Y has codimension 1 in X, then it is an isomorphism and

4. Resolution Type Theorems

40

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

Exercise 4.7. 1. Let K be an algebraically closed field, and X he the affine surface

X = spec(K[x, y, z]/(xy - z2)).

Let Y = V(x, z) C X, and let 7r : B -> X be the monoidal transform centered at the non-singular curve Y. Show that it is a resolution of singularities. Compute the exceptional locus of 7r, and compute the exceptional divisor of the monoidal transform 7r 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 7r : B --r X with center Y is a resolution of singularities in each of the following cases, and describe the exceptional locus and exceptional 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 monoidal transform of case a.

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 birational morphism 4): W -+ V such that W is non-singular.

Principalization of ideals. Suppose that V is a non-singular variety, and

I C Ov is an ideal sheaf. A principaliration of I is a proper birational V such that W is non-singular and IOW is locally morphism : W principal.

Suppose that, X is a non-singular variety, and I C OX is a locally principal ideal. We say that I has simple normal crossings (SNCs) at p E X if there exist regular parameters (xl, ... , xr,) in Ox,p such that Zp = xil xn,OX,p for some natural numbers al, . . . , a,,.

4.2. Resolution type theorems and corollaries

41

Suppose that D is an effective divisor on X. That is, D = r j El + - - + r,,E,,, where Ei are irreducible codimension 1 subvarities of X, and ri are natural numbers. D has SNCs if -TD = I has SNCs. Suppose that W C X is a subscheme, and D is a divisor on X. We say that W has SNCs with D at p if

I

1. W is non-singular at p, 2. D is a SNC divisor at p, 3. there exist regular parameters (xl, ... ,;r,,,,) in Ox,p and r < n such that Zw,p = (xl,... , x,) (or 2W,p = Ox,p) and TD,p = :[:11 ... X' t'X,p

for some ai E N.

Embedded resolution. Suppose that W is a subvariety of a non-singular variety X. An embedded resolution of W is a proper morphism it : Y - X which is a product of monoidal transforms, such that 7r is an isomorphism on an open set which intersects every component of W properly, the exceptional divisor of 7r is a SNC divisor D, and the strict transform W of W has SNCs with D.

Resolution of indeterminacy. Suppose that 0 : W -+ V is a rational map of proper K-varieties such that W is non-singular. A resolution of indeterminacy of 0 is a proper non-singular K-variety X with a birational morphism 0: X-* W and a morphism A: X- V such that A= 0 o V,. Lemma 4.8. Suppose that resolution of singularities is true for K-varieties of dimension n. Then resolution of iruleterrninacy is true for rational maps from K-varieties of dimension n.

Proof. Let q5: W -+ V be a rational reap of proper K-varieties, where W is non-singular. Lot U be a dense open set of W on which 0 is a morphism. Let ro be the Zariski closure of the image in W x V of the map U -> W x V defined by p H (p, 4'(p)). By resolution of singularities, there is a proper hirational morphism X - r,h such that X is non-singular. Theorem 4.9. Suppose that K is a perfect field, resolution of singularities is true for projective hypersurfaces over K of dimension n, and principalization

of ideaLs is true for non-singular varieties of dimension n over K. Then resolution of singularities is true for projective K-varieties of dimension n.

4. Resolution Type Theorems

42

Proof. Suppose V is an n-dimensional projective K-variety. If V1,..., V. are the irreducible components of V, we have a projective birational morphism from the disjoint union of the V to V. 't'hus it suffices to assume that V is irreducible. The function field K(V) of V is a finite separable extension of a rational function field K(xl i ... , xn) (Chapter II, Theorem 30, page 104 [92]). By the theorem of the primitive element,

K(V) ' K(xr,...,xn)[xn+l]/(f) We can clear the denominator of f so that

f - ail...in+1x1

21 .

in+1 .I%,,,+i

is irreducible in K[x1i ... , .:.,,+1]. Let

d = max{ii + -

- in+1 I aj,... a+1

001.

Set

F-E

i ... X2 n+1

the homogenization of f. Let V be the variety defined by F = 0 in Then K(V) q' K(V) implies there is a birational rational reap from V to V. That is, there is a birational morphism (0 : V - V, where V is a dense open subset of V. Let F0 be the Zariski closure of {(a, 0(a)) a E V} in V x V. We have birational projection morphisms 7rl : I'p - V and 7r2 : T'y - V. I'4,, is the blow-up of an ideal sheaf 9 on V (Theorem 4.5). By resolution of singularities for n-dimensional hypersurfaces, we have a pn+l.

resolution of singularities f : W' --> V. By principalization of ideals in non-

singular varieties of dimension n, we have a principalization g : W - W' for JOw' By the universal property of blowing up (Theorem 4.2), we have a morphism h : W --> r&. Hence 7r1 o h : W ---, V is a resolution of singularities.

D

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

Proof. By Theorem 3.15, resolution of singularities is true for projective plane curves over K. All ideal sheaves on a non-singular curve are locally principal, since the local rings of points are Dedekind local rings. D Note that Theorem 4.3 and Corollary 4.4 are stronger results than Corollary 4.10. However, the ideas of the proofs of resolution of plane curves in Theorems 3.11 and 3.15 extend to higher dimensions, while Theorem 4.3 does not. As a consequence of embedded 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, [73]).

4.2. Resolution type theorems and corollaries

43

Theorem 4.11. Suppose that S is a non-singular surface over a field K, and J C Os is an ideal sheaf. Then there exists a finite sequence of blow-ups

of points T - S such that JOT is locally principal. Proof. Without loss of generality, S is affine. Let

J= r(J,ds) _ (fl,...,fm) By embedded resolution or curve singularities on a surface (Exercise 3.13 of 1 Section 3.4) applied to m E S, there exists a sequence of blow-ups of points 7rl : S1 -> S such that f, f,,,, = 0 is a SNC divisor everywhere on S1.

Let {pl, ... , p,} be the finitely many points on S1 where Jds, is not locally principal (they are contained in the singular points of V( 51 sl)). Suppose that p E {pj, ... , p,.}. By induction on the number of generators of .1, we may assume that .IOg p = (f, g). After possibly multiplying f and g by units in Os,,p, there are regular parameters (x, y) in Os,,p such that f = x"yb, 9 = xCyd

Set. lp = (a - c) (b - d). (f, g) is principal if and only if tp > 0. By our assumption, tp < 0. Let ir2.: S2 -> S1 be the blow-up of p, and suppose that q F 7_1 (p). We will show that tq > tp. The only two points q where (f, g) may not, he principal have regular parameters (xi,Yi) such that

x=x1,y=xjyl orx=xlyt,'y=yt If x = xl, y = xly1, then "+h G

f = 1 tq

yl,

c+d d 9 = xi yr,

(a+b - (c+d))(b-d) - (b-(/)2+(a-c)(b-d) > tp.

The case when x _ xlyl, y - yl is similar. By induction on min{tp I Jds,,p is not principal} we can principalize J after a finite number of blow-ups of points.

Chapter 5

Surface Singularities

5.1. Resolution of surface singularities In this section, we give a simple proof of resolution of surface singularities in characteristic zero. The proof is by the good point algorithm of Abhyankar ([71, [621, [731).

Theorem 5.1. Suppose that S is a projective surface over an algebraically closed field K of characteristic 0. Then there exists a resolution of singularities Theorem 5.1 is a consequence of Theorems 5.2, 4.9 and 4.11.

Theorem 5.2. Suppose that S is a hypersurface of dimension 2 in a nonsingular variety V of dimension .4, over an algebraically closed field K of characteristic 0. Then there exists a sequence of blow-ups of points and non-singular curves contained in the strict transform Si of S, such that the strict. tran.9form Sn of S on Vn is non-singular.

The remainder of this section will be devoted to the proof of Theorem Suppose that V is a non-singular three-dimensional variety over an algebraically closed field K of characteristic 0, and S C V is a surface. For a natural number t, define (Definitions A.17 and A.20) 5.2.

Singt(S) = {p E V I vp(S) > t}. By Theorem A.19, Singa(S) is Zariski closed in V. 45

5. Surface Singularities

46

Let

r - max{t 1 Singt(S) 01 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 7r: B(p) - V of a closed point p E V. Suppose that U = spec(R) c V is an affine open neighborhood of p, and p has ideal 7rtp- (a:, y, z) C R. Then

7r-1(U) =proj((D7np) =

Y. Z

]) U spec(R.[y, y]) l.1

v]).

The exceptional divisor is E _ 7r-'(p)) 1P2. At each closed point q E it-1(p), we have regular parameters (x1, Y1, zl) of the following forms:

x=x1i y=xl(y1+a),

z=x1(21 with a, J3 E K, xl = 0 a local equation of E, or x = xly1, y = yl) z = yl(z1 + a) with cr E K, yi = 0 a local equation of E, or J: - :r,1z1,

J - 'y1z1,

z - z1,

z1 = 0 a local equation of E. We will now consider the blow-up 7r : B(C) -- V of a non-singular curve C C V. If p E V and U= spec(R) C V is an open affine neighborhood of p in V such that mp = (x, y, z) and the ideal of (7 is I = (x, y) in R, then

7r-1(U) = proj((DIn) = spec(R[y]) U spec(R[y]).

Also, it-1(p) = P1 and 7r-1(C fl U) ^_' (C fl U) x P1. Let E = 7r-1(C) be the exceptional divisor. E is a projective bundle over C. At each point q E 7r-1(p), we have regular parameters (x1, yi, zl) such that

x=x1, y=x1(y1+a), z=z1, where cr E K, x1 -- 0 is a local equation of E, or

x=xiYi,

Y =Y11

'Z =Z11

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

ir:B(W)=B(Tw)->V of a non-singular subvariety W of V above a closed point p E V, by passing to a formal neighborhood spec(Ov,p) of p and analyzing the map T : B(:t p) - spec(Ov,p). We have a natural isomorphism B(2w,p)

47

5.1. Resolution of surface singularities

B(Iw) xv spec(Ov,p). Observe that we have a natural identification of it-1(p) with '(p) We say that an ideal 1 C Ovp is algebraic if there exists an ideal J C Ov,p such that I = J. This is equivalent to the statement that there exists an ideal sheaf I C Ov such that Ip = I. If I C Ov,p is algebraic, so that there exists an ideal sheaf I C Ov such spec(Ovp) to a that ±p = I, then we can extend the blow-up 7 : B(I) blow-up 7r : B(I) - V. The maximal ideal mpOv is always algebraic. However, the ideal sheaf I of a non-singular (formal) curve in spec(Ov ;p) may not be algebraic. One example of a formal, non-algebraic curve is

I = (y - ex) C C[[x, y]] = OA,o. A more subtle example, which could occur in the course of this section, is the ideal sheaf of the irreducible curve

J= (y2 -xl -x3,z) C R=K[x,y,z], which we. studied alter Theorem 3.6. In R = K[[x, y, z]J we have regular parameters

x=y-x1+x, y=y+x 1+x, z=z. Thus

JR=(xy-N)=(x,z")n(?!,z) CR=K[[x,7/,z]]. In this example, we may be tempted to blow-up one of the two formal branches r = 0, z = 0 or y = 0, z = 0, but the resulting blown-up scheme will not extend to a blow-up of an ideal sheaf in A. A situation which will arise in this section when we will blow up a formal curve which will actually be algebraic is given in the following lemma.

Lemma 5.3. Suppose that V is a non-singular three-dimensional variety, p E V is a closed point and 7r : V7 = B(p) -- V i9 the blow-up of p with exceptional divisor F = 7r r (p) ?` p2. Let R = Ov,p. We have a commutative diagram of morphisms of schemes

B=B(mpf)

--+

W1 spec(R)

-

V1 =B(p)

17 V

such that U '(mp) - 7r-1(p) = E is an isomorphism of schemes. Suppose that I C R is any ideal, and f C On is the strict transform of 1. Then there exists an ideal sheaf J on V1 such that JOB = IEOB + I.

5. Surface Singularities

48

Proof. IOE is an ideal sheaf on E, so there exists an ideal sheaf 3 C Ov1 such that ZE C J and J/ZE ^_' WE. Thus 3 has. the desired property. Lemma 5.4. Suppose that V is a non-singular three-dimensional variety, S c V is a surface, C C Sing,.(S) is a non-singular curve, 7r : B(C) -+ V is the blow-up of C, and S is the strict transform of S in B(C). Suppose that p E C. Then vq(S) < r for all q E rr-1(p), and there exists at most one point q E 7r -'(p) such that vq(S) = r. In particular, if E = 7r-1(C), then either Sing, (9) fl E is a non-singular curve which maps isomorphically onto C, or Sing.,.(S) n E is a finite union of points.

Proof. By the Weierstrass preparation theorem and after a Tschirnhauscn

transformation (Definition 3.8), a local equation f = 0 of S in Os,p = Ii [[x, y, z]] has the form (5.1)

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

aZr1 E Zcr,. Thus ',p , and r!z = z E ZC,p. After a change of variables in x and y, we may assume that

But f e (I

TC,n implies

E

TC,p = (x, z). Then f E ZCp implies x1' az for all i. If q E 7r-1(p), then OB(C),q has regular parameters (x1, y, zi) such that, I

x=x1z1, z=z1 or

x=x1, z=x1(z1+a) for some a E 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+ay +

112

2(z1+a),.-2

x1

a* +...+ T x1

We have v(f1)
Proof. By the Weierstrass preparation theorem and after a Tschirnhausen transformation, a local equation f = 0 of S in OS,p = K[[x, y, z]] has the form (5.2)

f = z" + a2 (X, y)z'-2 + ... +

y).

5.1. Resolution of surface singularities

49

-'(p) and 1iq(S) > r, then OR((-has regular parameters (xl, y, zl) such that X - a:iyi, J = 'Ji, z=yrzi. If q E

or

y - x1(yr + a), z = X1 zl for some a E K, and v1(S) r. Thus Sing,.(S) fl E is contained in the X = s:1,

line which is the intersection of E with the strict transform of z = 0 in ti(Ci) xv spec((jvp). Definition 5.6. Sing,.(S) has simple normal crossings (SNCs) if 1. all irreducible components of Sing,.(S) (which could be points or curves) are non-singular, and 2. if p is a singular point of Sing,.(S), then there exist regular parameters (z:, y, z) in Ovp such that ZSing,.(S),p = (xy, z).

Lemma 5.7. Suppose that Sing,.(S) has simple normal crossings, W is a point or an. irreducible curve in Sing,.(S), : V' = B(W) -' V is the blow-

up of W and S' is the strict transform of S on V. Then Sing,.(S') has simple normal crossings.

Proof. This follows from Lemmas 5.4 and 5.5 and a simple local calculation F1 on W. Definition 5.8. A closed point p E S is a pregood point if, in a neighborhood of p, Sing,.(S) is either empty, a non-singular curve through p, or a a union

of two non-singular curves intersecting transversally at p (satisfying 2 of Definition 5.6).

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

X,, -> X,,.-r -> ... -' X1

spec(Ovp)

of blow-ups of non-singular curves in Sing,.(Se), where S, is the strict, trans-

form of S fl spec(Ovp) on X,, then q is pregood for all closed points q E Sing,.(S,,). In particular, Sing,.(S,,) contains no isolated points. A point which is not good is called bad.

Lemma 5.10. Suppose that all points of Sing,.(S) are good. Then there exists a sequence of blow ups V'=V"-,...-V1-'V

of non-singular curves contained in Sing,. (Se ), where Si is the strict t rans-

form of S in Vi, such that Sing,.(S') _ 0 if S' is the strict transform of S on V.

5. Surface Singularities

50

Proof. Suppose that C is a non-singular curve in Sing,.(S). Let 7r1 : V1 = B(C) -> V be the blow-up of C, and let Si be the strict transform of S. If Sing,.(S1) 0, we can choose another non-singular curve C1 in Sing,.(Si) and blow-up by 7r2 : V2 -- B(Ci) - V1. Let S2 be the strict transform of S1. We either reach a surface S,, such that Sing.,.(S,,,) = 0, or we obtain an infinite sequence of blow-ups

...-+V,->V1,-l ...-,V such that each V+1 - Vi is the blow-up of a curve Gi in Singr(SS), where Each curve Ci which is blown up must S; is the strict transform of S on map onto a curve in S by Lemma 5.5. Thus there exists a curve y C S such that there are infinitely many blow-ups of curves mapping onto y in the above sequence. Let R = OV,,y, a two-dimensional regular local ring. 1s,y is a height 1 prime ideal in this ring, and P = I.y,.y is the maximal ideal of R. We have dim R + trdegK R/P = 3 by the dimension formula. (Theorem 15.6 [661). Thus RIP has transcendence

degree 1 over K. Let t E R be the lift of a transcendence basis of RIP over

K. Then K [t] n P = (0), so the field K(t) C R. We can write R = AQt 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 ring of a non-singular point q on the K (t) surface spec(A). q is a point of multiplicity r on the curve in spec(A) with ideal sheaf Is,.y in R. The sequence -V,, X V spec(R)

V,x_1 x V spcc(R) ---

-

-

- - spec(R)

consists of infinitely many blow-tips of points on a K(t)-surface, which are of multiplicity r on the strict transform of the curve spec(R/Is,.y). But this is impossible by Theorem 3.15.

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

Proof. Let Bo = {isolated points of Sing,.(S)} U (singular points of Sing,.(S)}. Sing,.(S) - Bo is a non-singular 1-dimensional subscheme of V. Let, ir1

: V1 = B(Sing,(S) - BO) ._.t V - BO

be its blow-up. Let S1 be the strict transform of S, and let B1 = {isolated points of Sing,.(S1)} U {singular points of Sing,.(Si)}.

We can iterate to construct a sequence

...

(V,.t

- B,,z)

... , V,

5.1. Resolution of surface singularities

51

where ir : V,, - B,,, - V - T are the induced surjective maps, with

Uir1(Bi)U...Uirn(Bn) Let S be the strict transform of S on V. let C C Sing,.(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 (as in the proof of Lemma 5.10). irn induces by base change

S xs spee(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, S xs spec(Os,c) is non-singular. r uis there are no curves in which dominate C. For n. >> 0 there are thus no curves in Sing,.(S) so that n (V - Bn) is empty for large n, and all bad points of 9 are contained in a finite set T.

Theorem 5.12. Let

... -V,a-+V».-1 --,..._,VI -V

(5.3)

be the sequence where 7rn : V -, is the blow-up of all bad points on the strict transform. 1 of S. Then this sequence is finite, so that it terminates

after a finite number of steps with a V such that all points of Singr(S,,,) are good.

We now give the proof of Theorem 5.12. Suppose there is an infinite sequence of the form of (5.3). Then there is an infinite sequence of points p E such that for all n. We then have an infinite sequence of homomorphisms of local rings

... Ro = Ov,p - R1 = Ov1 p, - ....+ R., = O4np++ By the Weierstrass preparation theorem and after a Tshirnhausen transformation (Definition 3.8) there exist regular parameters (x, y, z) in Rp such that there is a local equation f = 0 for S in Rp of the form f = Zr + a2 (X, y)zr 2 + ... + ar(x, y) .

with v(ai(x, y)) ? i for all i. Since vp,(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

I

ai),

zi-1 ='1'i-Izi-1

with ai E K, or xi-i = xiyi,

yi-1 = yi,

zi-1 = yi-lzi-1,

and T

T

A = z9. + a2,i (xi, yi)zi

+ ... ar,i (xi, yi),

5. Surface Singularities

52

where

dj,i-1(ai,xi (yi+CMj))

aji(xi, yi) =

X or

for all j. Thus the sequence

K[[x, y] --y K[[xi, yi]] - .. . is a sequence of blow-ups of a maximal ideal, followed by completion. In K[[xi, yi]] we have relations aj = 'Yj,ixic1,, yid1,i aj,i-1

for all i and 2 < j < r, where yji is a unit. By embedded resolution of curve singularities (Exercise 3.13), there exists mo such that fj'-2 a j (x, y) = 0 is a SNC divisor in Ri for all i > mo. Thus for i > mo we have an expression k1i zr_2 + ... + izri (xi, yi)xori yibr; , 5(.4) fi = z'' + a (X,, 2i .y,) i, , i )x°j'i yi 1

where the aji are units (or zero) in Ri and aji + bji > j if aji # 0. We observe that, by the proof of Theorem A.19 and Remark A.21, TSillg,.(S;).pi

where J is the ideal in Ri generated by 3j+k+tf {

8xjaykzi

10< j

k+ l< r}.

It be one of (xi, yi, zi), (xi, zi), (yi, zi) or (Xiyi, zi) Sing,.(Si) has SNCs at pi unless v"J- = (xjyj, zi) is the completion of an ideal of an irreducible (singular) curve on Vi. Let E = rr= 1(pi_ 1) ^' iiD2. By construction, we have that xi = 0 or yi = 0 is a local equation of E at pi-1. Without loss of generality, we may suppose that xi = 0 is a local equation of E. zi = 0 is a local equation of the strict transform of z = 0 in V x v spec(R), so that (xi, zi) is the completion at qj of the ideal sheaf of a subscheme of E. Thus there is a curve C in Vi such that 2c',p, = (xi, zi) by Lemma 5.3. The ideal sheaf ,7 = (25ingr(si) : Tc) in Ov; is such that Thus ±Singr(Si),pi

(,7 fl T(,r)p, '" 2Sing,.(si),p,, so that (xiyi, zi) is not the completion of an ideal of an irreducible curve on V, and Sing,.(Sj) has simple normal crossings at pi-

We may thus assume that rno is sufficently large that Sing,.(Si) has SNCs everywhere on Vi for i > mo.

We now make the observation that we must have bji # 0 and aki # 0 for some j and k if i > m0. If we did have bji = 0 for all j (with a similar analysis if aji. = 0 for all j) in (5.4), then, since TSingr(S;),p; _

'

5.I. Resolution of surface singularities

53

where J is the ideal in Ri generated by &i+k+tf

10 < j

{t3xJ r3yjk a zz

k l l < r},

we have (with the condition that bji = 0 for all j) that v1rJ = (xi, zi). Since Singr(Si) has SNCs in Vi, there exists a non-singular curve C C Singr(Si) such that xi = zi = 0 are local equations of C in R. Let A : W --> V be the blow-up of C. If q E A-'(pi), we have regular parameters (u, v, y) in duq such that xi = u,

zi = u(v +P)

or

xi=uV,

zi=u.

If f = 0 is a local equation of the strict transform S' of Si in W, we have that vy(f')
min{ ' 12 < j < r} has decreased by 1. We see, using Lemma 5.4, that, after a finite number of blow-ups of non-singular curves in Sing,. 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 b ji # 0 and aki 0 0 in (5.4) for some

j andkifi>mu.

If follows that, for i > mo, we must have xi = xi+1,

yi = xi+lyi+i,

zi = xi+lzi+1

or

xi = xi+1yi+1,

Yi = Yi+1,

Z i = Yi+lzi+1

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

aj,i+l =aji + bji - j, bj,i+l = bji

(5.5)

or

aj,i+.l = aji, h j,i+I = a j,i + bji respectively, for all i > ono. Suppose that C E R. Then the fractional part of ( is t,

ifc =a+t with a E Z and 0 < t < 1. Lemma 5.13. For i > mo, pi is a good point on V if there exists j such that aji

0 and

5. Surface Singularities

54

1.

aji

<

k'

j

bki

<

a k i

j

for all k with 2 < k < r and ski

k

0, while

2.

?`}+{bji}<1. Proof. Since vri(Si) = r, aji +bji > j. Condition 2 thus implies that either aji > j or bji > j. Without loss of generality, aji > j. Then condition 1 implies that aki >_ k for all k, and ZSing,,(si),Pi C (xi, zi)ni

Since Sing,.(Sj) has SNCs, there exists a non-singular curve C C Sing,.(Si) Vi be the blow-up of C, such that fr,pi = (xi, zi). Let ir : V' = B(C) and let S' be the strict transform of S on V. By Lemma 5.7, Sing,.(S') has SNCs. A direct local calculation shows that there is at most one point q E 7r '(pi) such that vq(S') = r, and if such a point, q exists, then Ov',q

has regular parameter 4 (a:', j , z') such that

xi=x', Vi=y', zi =x'z'A local equation f = 0 of S' at q is fI

= (z'), +

a2i(X')a2i-2(J

)6'r, (zi)r-2 +

Thus we have an expression of the form (5.4), and conditions 1 and 2 of the statement of this lemma hold for f'. After repeating this procedure a. finite number of times we will construct V -, Vi such that v,,,(S) < r for all points a on the strict transform S of S such that a maps to pi, since V + must O drop by 1 every time we blow up.

Lemma 5.14. Let

k'

bji J

bja

1

Then

aj,k,i+1 > bjki,

and if bjki < 0 then 1

'jk,i+1 - Sjki > T4

Proof. We may assume that the first case of (5.5) holds. Then bki)2 8j,k,i+1 = bj.k,i

+

Cbji

j

_

k

5.1. Resolution of surface singularities

0. Thus

If oj,k,i < 0 we must have`-

ji

j

55

Ski

1

2

1

k) -j2k2>r4

Corollary 5.15. There exists ml such that i > ml implies condition 1 of Lemma 5.13 holds.

Proof. There exists ml such that i > ml implies ajki ? 0 for all j, k. Then the pairs (ak , bk}) with 2 < k < r arc totally ordered by <, so there exists a minimal element (

, 3 ).

Lemma 5.16. There exists an i' > mi such that we have condition 2 of Lemma 5.13,

OP Proof. A calculation shows that

implies C

With the above i', pi, is a good point, since pig 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 rnultiplicity of points of S. By Theorem 5.12, there exists a finite sequence are good of blow-ups of points V,,, -y V such that all points of points, where S,,, is the strict transform of S on V,,,. By Lemma 5.10, there exists a sequence of blow-ups of non-singular curves V. - V,,,, such that 0, where S,, is the strict, tra,nslorin of s on V,,. By descending induction on r we can reach the case where Sing2(Sn) = 0, so that S,a is non-singular.

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

5. Surface Singularities

56

Exercise 5.18. 1. Suppose that S is a surface which is a subvaricty of a non-singular three-dimensional variety V over an algebraically closed field K of characteristic 0, and p E S is a closed point. a. Show that a Tschirnhausen transformation of a suitable local equation of 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 regular parameters at p such that v(f) = v(f (0, 0, z)), then b 1 = 0 is a hypersurface of maximal contact (A.20) for f = 0 in a neighborhood of the origin. 2. Follow the algorithm to reduce the multiplicity of f = z''+xayb = 0, where a + b > r.

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

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

Definition 5.20. A resolution datum R = (E0, El, S, V) is a 4-tuple where E0, El, S are reduced effective divisors on the non-singular three-dimensional

variety V such that E - E0 U El is a SNC divisor. R is resolved at p c S if S is non-singular at p and E U S has SNCs at p. For r > 0, let Sing,. (R) = {p E S I vp(S) > r and R is not resolved at p}. Observe that Sing,.(R) = Sing, (S) if r > 1, and Sing, (?Z) is a proper Zariski closed subset, of S for r > 1. For P E S, let

i7(p) = the number of components of Et containing p.

We. have 0 t}.

5.2. Embedded resolution of singularities

57

Sing,..r()Z) is a Zariski closed subset of Sing,.(R) (and of S). For the rest of this section we will fix

r = v(R) = v($, E) = max{vp(S) I p E S, R is not resolved at p},

t = y(R) - the maximum number of components of El containing a point of Sing,.(R).

Definition 5.21. A permissible transform of R is a monoidal transform 7r : V' -- V whose center is either a point of Sing,.,t (R) or a non-singular curve C C Sing,.,t(R) such that C makes SNCs with E. Let F be the exceptional divisor of 7r. Then ir* (E) U F is a SNC divisor. We define the strict transform R' of R to be R' = (Es, Ei, S'), where S' is the strict transform of S. and we also define

EO = *(EO)red + F, E' = strict transform of El if v(S', 7r" ((Eo + E'r )red + F) = v(R), EO' = 0, El' = 7r*(EO + El)red + F if v(S', 7r*((EO + Ei)red + F) < v(R).

Lemma 5.22. Witlr, the notation of the previous definition,

v(R') < v(R) and

v(R') = v(R) implies q(V) < ri(R). The proof of Lemma 5.22 is a generalization of Lemmas 5.3 and 5.4.

Definition 5.23. p E Sing,,,, (R) is a pregood point if the following two conditions hold:

1. In a neighborhood of p, Singr.t (R) is either a non-singular curve through p, or two non-singular curves through p intersecting transversally at p (satisfying 2 of Definition 5.6). 2. Sing,,,.()Z) makes SNCS with E at p. That is, there exist. regular parameters {x, y, z} in Ovp such that the ideal sheaf of each component, of E and each curve in Sing, t(R) containing p is generated by a subset of {x, y, z} at p. p c S is a good point if p is a pregood point and for any sequence it:Xn-+Xn_1-...-,Xr

-,spcc(Ov,p)

of permissible monodial transforms centered at curves in Sing,.,t (R4 ), where Rz is the transform of R on Xi, then q is a pregood point for all q E 7r-1(p).

A point p E Sing, t(R) is called bad if p is not good. Lemma 5.24. The number of bad points in Sing,.,t(7Z) is finite.

5. Surface Singularities

58

Lemma 5.25. Suppose that all points of Sing,.,t()Z) are good. Then there exists a sequence of permissible ionoidal transforms, rr : V' - V, centered at non-singular curves contained in Singr,t(Si), where Si is the strict transform of S on the i-th monoidal transform, such that if R' is the strict transform of 7Z on V', then either

v(R') < v(R) 07

v(R') = v(R) and 77(R') < 77(R).

The proofs of the above two lemmas are similar to the proofs of Lemmas

5.11 and 5.10 respectively, with the use of embedded resolution of plane curves.

Theorem 5.26. Suppose that. R is a resolution datum such that Eo = 1D. Let 7r

Irl

be the sequence where rrn is the monoidal transform centered at the union of bad points in Sing,,,t(Rj), where Ri is the transform of 7Z on Vi. Then this sequence is finite, so that it terminates in a V, such that all points of Sing,.,t(R,,,) are good.

Proof. Suppose that (5.6) is an infinite sequence. Then there exists an infinite sequence of points pn E Vn such that rrn+1(pn+i) = pn for all n and

p,,, is a bad point. Let R.,, = 0v,,,p for n > 0. Since the sequence (5.6) is infinite, its length is larger than 1, so we may assume that t < 2. Let f = 0 he a local equation of S at po, and let. gk, 0 < k < 1, be local equations of the components of El containing pp. In RO = °v,p there are regular parameters (x, y, z) such that v(f (U, 0, z)) = r and v(gk(0, U, z)) = 1 for all k. This follows since for each equation this is a non-trivial Zariski open condition on linear changes of variables in a set of regular parameters. By the Weierstrass preparation theorem, after multiplying f and the gk by units and performing a Tschirnhausen transformation on f, we have expressions (5.7)

+ E.i=2 aj(r, y)zr-i, fgk ==z --z' ¢k(x, y), (or gk is a unit)

in 1?. z = 0 is a local equation of a formal hypersurface of maximal contact

for f = 0. Thus we have regular parameters (x1i y1i z1) in RI = 6v,,pr defined by x = x1,

y = x1(y1 + a),

z = x1z1,

with c E K, or

x=xfy1, Y=Y1, z=y1zi.

5.2. Embedded resolution of singularities

59

The strict transforms f1 of f and gkl of 9k have the form of (5.7) with aj replaced by i (or 2), cbk with P- (or P-k). After a finite number of A

-51

monoidal transforms centered at points pi with v(ff) = r, we have forms in Ri = OV,,p, for the strict transforms of f and gk:

fi Ski

e

uji hji r-j Ej-2 ai (xi yi) xi yi zi Ckx yd,, / r zk + 1ki(xio yi)xi i I

Zri +

such that aj, ski are either units or zero (as in the proof of Theorem 5.12). We further have that local equations of the exceptional divisor EE of V --> V are one of xi = 0, yi = 0 or xiyi = 0. This last case can only occur if t < 2.

Note that our assumption 77(pi) = t implies ski is not a unit in Ri for

1
We can further assume, by an extension of the argument in the proof of curves in Singrt(Ri) are non-singular. Theorem 5.12, that all By Lemma 5.14, we can assume that the set

T=

\%

'

bla) ' I aj

01

f

U l (Cki, dki) I Oki -A 01

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

l

.70"

1+

1 bao" J

< 1'

where (a!, b) is the minimum of the7 (,'). Recall that, if t > 0, ? 10 Jo i=1 gi = 0 makes SNCs with the exceptional divisor Ep of V - V. These conditions imply that, after possibly interchanging the ski, interchanging xi, Vi and multiplying xi, yi by units in Ri, that the .qki can be described by one of the following cases at. pi: 1. t = 0.

2. t=1,

b. gli=zi+xzy'`,A> 1,/1> 1. C. g1i = zi 3. t = 2,

a. gig = zi, 92i = zi + xi b. 9i1 = zi + xi, 9i2 = zi + Exi, where a is a unit in Ri such that the residue of a in the residue field of Ri is not 1. C. 9i1 = zi + xi, gig = zi + ex%'y , where e. is a unit in Ri, A > 1

and A+µ>2. If p > 0 we can take E = 1. We can check explicitly by blowing up permissible curves that pi is a good point. In this calculation, if t = 0, this follows from the proof of

5. Surface Singularities

60

Theorem 5.12, recalling that xi = 0, yi = 0 or xiyi = 0 are local equations of Eo at pi. If t = 1, we must remember that we can only blow up a curve if it lies on g1 = 0, and if t = 2, we can only blow up the curve gl = g2 = 0. Although we are constructing a sequence of permissible monoidal transforms

over specA.,p=), this is equivalent to constructing such a sequence over spee(Ov1,p;), as all cu veg blown up in this calculation are actually algebraic, since all irreducible curves in Sing,.,t(R.i) are non-singular.

We have thus found a contradiction, by which we conclude that the sequence (5.6) has finite length. The proof of Theorem 5.19 now follows easily from Theorem 5.26, Lemma 5.25 and descending induction on (r, t) E N x N, with the lexicographic order.

Remark 5.27. This algorithm is outlined in [73] and [62]. Exercise 5.28. Resolve the following (or at least make (v(R),,q(R)) drop in the lexicographic order): 1. 1Z = (0, El, S), where S has local equation f = z = 0, and El is the union of two components with respective local equations gl =

z+xy, 92 = z+x; 2. R = (0. El, S), where S has local equation f = z3 + x7y7 = 0 and El is the union of two components with respective local equations

91 = z+xy, 92 _ z+y(:r2+y3).

Chapter 6

Resolution of Singularities in Characteristic Zero

The first proof of resolution in arbitrary dimension (over a field of characteristic zero) was by Hironaka [52]. The proof consists of a local proof, and a complex web of inductions to produce a global proof. The local proof uses sophisticated methods in commutative algebra to reduce the problem of reducing the multiplicity of the strict, transform of an ideal after a permissible sequence of monoidal transforms to the problem of reducing the multiplicity of the strict transform of a hypersurface. The use of the Tschfrnhausen transformation to Gnd a hypersurface of maximal contact lies at the heart of this proof. The order v and the invariant it (of Chapter 7) are the principal invariants considered.

De Jong's theorem [36], referred to at the end of Chapter 7, can be refined to give a new proof of resolution of singularities of varieties of characteristic zero ([9], [16], [70]). The resulting theorem is not as strong as the classical results on resolution of singularities in characteristic zero (Section

6.8), as the resulting resolution X' -+ X may not be an isomorphism over the non-singular locus of X. In recent years several different proofs of canonical resolution of singularities have been constructed (Bierstone and Milman [14], Bravo, Encinas and Villamayor [17], Enemas and Villamayor [40], [41], Enemas and Hauser [39], Moh [67], Villamayor [79], [80], and Wlodarczyk [84]). These papers

61

6. Resolution of Singularities in Characteristic Zero

62

have significantly simplified Hironaka's original proofs in [47] and [55], and represent a great advancement of the subject. In this chapter we prove the basic theorems of resolution of singularities hi characteristic 0. Our proof is based on the algorithms of [40] and [41]. The final statements of resolution are proved in Section 6.8, and in Exercise 6.42 at the end of that section. Throughout this chapter, unless explicitely stated otherwise, we will assume that all varieties are over a field K of characteristic zero. Suppose that X1 and X2 are subschemes of a variety W. We will denote the reduced set-theoretic intersection of Xi and X2 by Xl fl X2. However, we will denote the scheme-theoretic intersection of Xl and X2 by Xl X2. X1 - X2 is the subseheme of W with ideal sheaf Tx, +Tx2.

6.1. The operator L and other preliminaries Definition 6.1. Suppose that K is a field of characteristic zero, R = K[[xi, . , is a ring of formal power series over K, and J = (fl, ... , f,) C . .

R is an ideal. Define

M +C8 I1
IL(J)=OR(J)_(fi,

,L(J) is independent of the choice of generators f; of .1 and regular parameters ;rf in R.

Lemma 6.2. Suppose that W is a non-singular variety and J C Ow is an ideal sheaf. Then there exists an ideal A(J) = Aw(J) C Ow such that

L.(J)Ow,q = 0(J) for all closed points q E W.

Proof. We define L(J) as follows. We can cover W by affmc open subsets U = spec(R) which satisfy the conclusions of Leinnia A.11, so that illRIK we define

dy1R E}3 ... ED dy,,R. If r(U, J)

r(U, A

= (f4, aye

1 < i < r, I -< j < n).

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

eters in Ow,q. Let U = spec(R) be an affine neighborhood of q as above. Let A = Owq. Then {dyl, . , dy,,,} and {dx1, ... , dx,,} are A-bases of SlA/k by Lemma A.11 and Theorem A.10. Thus the determinant . .

Kftj yt) J (q)

0

6.1. The operator A and other preliminaries

63

from which the conclusions of the lemma follow.

Given J C Ow and q E W, we define v.j(q) = vq(J) (Definition A.17). If X is a subvariety of W, we denote vx - vex. We can define AT(J) for r E N

inductively, by the formula A (J) = A(A''-1(J)). We define. A°(J) - J.

Lemma 6.3. Suppose that W is a non-singular variety, (0) 0 J C Ow is an ideal sheaf, and q E W (which is not necessarily a closed point). Then:

1. vq(J) = b > 0 if and only if vq(A(J)) = b - 1. 2. vq(.l) = b > 0 if and only if vq(Ab-1(J)) = 1. 3. vq(J) > b > 0 if and only if q E V(Ab-1(J)). 4. vj : W - N is an upper semi-continuous function. Proof. The leninia follows from Theorem A.19.

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

Maxf ={gEXI f(q)-rnaxf}. We have Maxv,j = V(Ab-1(J)) if b - maxvj. Suppose that X is a closed subset of a n-dimensional variety W. We shall denote by R(1)(X) C X (6.1) the union of irreducible components of X which have dimension n - 1.

Lemma 6.4. Suppose that W is a non-singlar variety, J C Ow is an ideal sheaf, b = max v.7 and Y C Max vj is a non-singular subvariety. Let Let D be the 7r : W1 -, W be the rn.onoidal transform with center Y. exceptional divisor of 7r. Then:

1. There is an ideal sheaf J1 C Ow, such that J1 = TDbJ and TD { J1.

2. If q c W1 and p = 7r(q), then v-71(q) < vj(p) 3. max vj > max v7r1 .

Proof. Suppose that q E W1 and p = 7r(q). Let A be the Zariski closure of {q} in W1, and B the Zariski closure of {p) in W. Then A -, B is proper. By upper semi-continuity of vj, there exists a closed point x E B such that vi(x) = vj(p). Let y E 7r-1 (x) n A be a closed point. By semi-continuity

6. Resolution of Singularities in Characteristic Zero

64

of v f , we have L -j, (q) < vj-, (y), so it suffices to prove 2 when q and p are closed points. By Corollary A.5, there is a regular system of parameters 1 11 , . . in Ow,p such that yi = y2 = = yr = 0 are local equations of Y at p. By Remark A.18, we may assume that K = K(p) = K(q). Then we can make a linear change of variables in ,yl,... , y,, to assume that there are regular .

parameters -,.y 1, ... , y,,, ir1 OW,,q such that y1 = 711, Y2 = ?Jl?!2, ... , yr =Fill,, yr+1 = "1r+1,

yn = yn-

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

R - Ow,p = K[[yi,..... n] and

S = Owi,q = K[[j1...... yn]]. Y C Max vj implies .n(J) - h, where rt is the general point of the componen

of Y whose closure contains p. Suppose that f E JR. Let t = vR(f) > b. Then there is an expansion

f=

ail,...,inyl

. y1i

in R, with ai,,...,.,, c K and ai...... in = 0 if it + + i,. < t. Further, there exist i1,. .. , in with i1 + + i,, = t such that ai...... i # 0. In S, we have

t

f = V1 h! where

fl

i1+...+.,r-t

ai1,...,in311

...

yi»n

is such that y1 does not divide fl. Thus we have vg(fl) < vn(f). In JS; and since there exists f E R with VR(f) = b.

particular, we have yb 1 1` JS. Thus

y

J1 = ZDb.1

satisfies 1 and

u, (q)

v.7 (p) = b.

Suppose that W is a non-singuular variety and J C Ow is an ideal sheaf.

Suppose that Y C W is a non-singular subvariety. Let Y1,. .. , Y be the distinct irreducible (connected) components of Y. Let 7r : W1 --> W be the monoidal transform with center Y and exceptional divisor D. Let D =

D1+

where Ti arc the distinct irreducible (connected) components

6.1. The operator A and other preliminaries

65

of D, indexed so that ir(Di) = Y. Then by an argument as in the proof of Lemma 6.4, there exists an ideal sheaf 71 C OW1 such that

Jowl

(6.2)

-I

i ....TSmJ1,

where J1 is such that for 1 < i < m we have 2D, f J1 and T_i = v,h (J), with 77i the generic point of Y. JI is called the weak transform of J.

There is thus a function c on W1, defined by c(q) = 0 if q c(q) = ci if q E Di, such that (6.3)

JOw1

=

U and

2 J1.

In the situation of Lemma 6.4, J1 has properties 1 and 2 of that lernlna.. 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 ideal sheaf J1. We see that the weak transform X of X is much easier to calculate than the strict transform x of x. We have inclusions X C X C 7r*(X).

Example 6.5. Suppose that X is a hypersurface on a variety W, r maxvx and Y C l'vIax'ix is a non-singular subvaricty. Lot it : W1 - W be the monoidal transform of W with center Y and exceptional divisor E. In this case the strict transform X and the weak transform X are the same scheme, and 7r* (X) = X + r F,.

Example 6.6. Let X C W = A3 be the nodal plane curve with ideal A3 IX = (z, y2 - a;3) c k[x, y, z]. Let m = (x, y, z), and let it : B = B(m) be the blow-up of the point m. The strict transform X of X is non-singular. The most interesting point in X is the point q E 7r-1(m) with regular parameters (xi i ?!1, r1) such that X

X1,

J = xi'Ji,

z =X121

Then

2

2X,4 = (zi, xiyt - xl ), 2X,4 =

(Z1, Y2

-xi).

In this example, X,X and 7r*(X) are all distinct.

6. Resolution of Singularities in Characteristic Zero

66

6.2. Hypersurfaces of maximal contact and induction in resolution Suppose dial. W is a non-singular variety, J C Ow is an ideal sheaf and b E N. Define

Sing(./, b) = {q E W vi(q) > b}. Sing(J, b) is Zariski closed in W by Lemma 6.3. Suppose that Y C Sing(J, b) is a non-singular (but not necessarily connected) subvariety of W. Let ir1

W1 -, W be the monodial transform with center Y, and let D1 - ,r-1(Y) be the exceptional divisor. Recall (6.3) that the weak transform .I1 of J is defined by JOW'. = ,7D1 Jl,

where c is locally constant on connected components of D1. We have c >_ b by upper semi-continuity of vj. We can now define Ji by JOw,

(6.4)

.Jn, Ji,

so that

Jl = 1n, b7l. Suppose that (6.5)

W W..-1

W

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

Ji -1Ow; = AjJi and Di is the exceptional divisor of 7ri. We will say that (6.5) is a resolution of (W, J, b) if Sing(J, b) = 0.

Definition 6.7. Suppose that r = max vj, q E Sing(J, r). A non-singular codimension 1 subvariety H of an affine neighborhood U of q in W is called a hypersurface of maximal contact for J at q if

1. Sing(J, r) fl U C H, and 2. if

Wn-4 ...-,W1-,W is a sequence of monoidal transforms of the form of (6.5) (with b = r), then the strict transform IIn of II on Un = Wn x W U is such that Sing(In, r) fl Un C Hn.

6.2. Hypersurfaces of maximal contact and induction in resolution

67

With the assumptions of Definition 6.7, a non-singular codimension I subvariety H of U = spec(OW,q) is called a formal hypersurface of maximal contact for J at q if 1 and 2 of Definition 6.7 hold, with U = spec(Owq). If X C W is a codimension 1 subvariety, with r = max vx, q E max vy, then our definition of hypersurface of maximal contact for J = Ix coincides

with the definition of a hypersurface of maximal contact for X given in Definition A.20. In this case, the ideal sheaf J,, in (6.5) is the ideal sheaf of the strict transform X,,, of X on W. Now suppose that X C W is a singular hypersurface, and K is algebraically closed. Let

r = max{vx} > 1 be the set of points of maximal multiplicity on X. Let q E Sing(Zx,r) be a closed point of maximal multiplicity r. The basic strategy of resolution is to construct a sequence Wn-*...-,Wi-,W

of monoidal transforms centered at non-singular subvarieties of the locus of

points of multiplicity r on the strict transform of X so that we eventually reach a situation where all points of the strict transform of X have multiplicity < r. We know that under 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 is much more difficult.

Notice that the desired sequence Wn -+ Wi will be a resolution of the form of (6.5), with (.Ii, b) = (Ixi, r), where Xi is the strict transform of X on Wi. Let q E Sing(ZX, r) be a closed point. After Weierstrass preparation and

performing a Tschirnhausen transformation, there exist regular parameters (xi, ... , x,s, y) in R= Owq such that there exists f E R such that f = 0 is a local equation of X at q, and (6.6)

ar(xi, ... ,

f = yr +

xn)yr-2

i=2

Set H = V (V) C spec(R). Then H = spec(S), where S = K[[xl,... 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 C Sing(Tx, r) is a subvariety containing q. Let J = TZ,q. Write J = Pi fl . fl P,., where the Pi are prime ideals of the same height

in R. Let Ri = RP;. By semi-continuity of vX, we have f E P''Rp, for all

6. Resolution of Singularities in Characteristic Zero

68

i. Since ey : X.p; --4 H.pt, we have (r - 1)!y =

E JRpp for all i. Thus

y E n JRp; = J and Z n spec(R) C if. Now we will verify that if Y C Sing(IX, r) is a non-singular subvariety,

: W1 -> W is the monoidal transform with center Y, X1 is the strict transform of X and ql is a closed point of X1 such that 7r(gl) = q which has maximal multiplicity r, then ql is on the strict transform H1 of H (over the formal neighborhood of q). Suppose that ql is such a point. That is, qi E 7r-'(q) n Sing(7x1, r). We have shown above that y E ly. Without 7r

changing the form of (6.6), we may then assume that there is s < n such that

xl = ... = xs =y = 0 are (formal) local equations of Y at q, and there is a formal regular system of parameters x1(1), ... , xn(1),y(1) in OWL,gl such

that -I

Zi (1),

xi y

x1(1)xi (1) for 2 < i xi(1)y(1),

a; i

< s,

.

a:i(1)forv
Since Y C Sing(IX, r), we have qj E (xi, ... , xs)i for all i. Then we have a (formal) local equation fi = 0 of X1 at ql, where

1: L' fl = X = y(1)r +

r

(6.8)

xy(1)r-i

z_2

y(1)r i-

ai(1)(xl(1),...,xn(1))?/(1)r-i

i-2

and yl(1) - 0 is a local equation of H1 at ql, while x1(1) = 0 is a local equation of the exceptional divisor of it. With our assumption that fl also has multiplicity r, fl has an expression of the same form as (6.6), so by induction, H is a (formal) hypersurface of maximal contact. We see that (at least over a formal neighborhood of q) we have reduced the problem of resolution of X to some sort of resolution problem on the (formal) hypersurface II. We will now describe this resolution problem. Set

I = (a2 2'...'a;) C S=Kl[xl,...,xn1j Observe that the scheme of points of multiplicity r in the formal neighborhood of q on X coincides with the scheme of points of multiplicity > r! of I,

Sing(f R, r) = Sing(I, r!). However, we may have vR(I) > r!.

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

6.2. Hypersurfaces of maximal contact and induction in resolution

69

a morphism H1 -+ H, where Hl is the strict transform of H. By direct verification, we see that the transform 11 of I (this transform is defined by (6.4)) satisfies

I,S1 =x1-1 rISI where b'1 - K[[rl(l.), ... , c,,(1)]] - 6x, q and this ideal is thus, by (6.8), the ideal I1S1 =

which is obtained from the coefficients of fi, our local equation of X1, in the

exact same way that we obtained the. ideal I from our local equation f of X. We further observe that vs, (I, Si) > r! if and only if vq, (Xi) ? r. We see then that to reduce the multiplicity of the strict transform of X over q, we are reduced to resolving a sequence of the form (6.5), but over a formal scheme.

The above is exactly the procedure which is followed in the proof of resolution for curves given in Section 3.4, and the proof of resolution for surfaces in Section 5.1. When X is a curve (dim 14' = 2) the induced resolution problem on the

formal curve 11 is essentially trivial, as the ideal I is in fact just the ideal generated by a monomial, and we only need blow up points to divide out powers of the terms in the monomial. 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 points of 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 finitely many points in the locus of points of maximal multiplicity on X. The potentially worrisome problem of extending a resolution of an object over a formal germ of a hypersurface to a global proper morphism over W was not a great difficulty, since the blowing up of a point on the formal surface H always extends trivially, and the only other kind of blow-ups we had to consider were the blow-ups of non-singular curves contained in the locus of maximal multiplicity. This required a little attention, but we were able to fairly easily reduce to a situation were these formal curves were always algebraic. In this case we first blew tip points to reduce to the situation where the ideal I was locally a monomial ideal. Then we blew up more points to make it a principal monomial ideal. At this point, we finished the resolution process by blowing up non-singular curves in the locus of maximal multiplicity on the strict transform of the surface X, which amounts to dividing out powers of the terms in the principal monomial ideal. In Section 5.2, we extended this algorithm to obtain embedded resolution

of the surface X. This required keeping track of the exceptional divisors

70

6. Resolution of Singularities in Characteristic Zero

which occur under the resolution process, and requiring that the centers of all monoidal transforms make SNCs with the previous exceptional divisors. We introduced the i invariant, which counts the number of components of the exceptional locus which have existed since the multiplicity last dropped. In this chapter we give a proof of resolution in arbitrary dimension (over fields of characteristic zero) which incorporates all of these ideas into a general induction statement. The fact that we must consider some kind of covering by local hypersurfaces which are not related in any obvious way is incorporated in the notion of General Basic Object. In the algorithm of this chapter Tschirnhausen is replaced by a method of Ciraud for finding algebraic hypersurfaces of maximal contact for an ideal J of order b, by finding a hypersurface in Lb-1(J).

6.3. Pairs and basic objects Definition 6.8. A pair is (Wo, Eo), where Wo is a non-singular variety over a field K of characteristic zero, and E0 = {D1i . , DT} is an ordered . .

collection of reduced non-singular divisors on Wo such that D1 + a reduced simple normal crossings divisor.

+ D,. is

Suppose that (WO, Eo) is a pair, and Yo is a non-singular closed subvariety of Wo. Yo is a permissible center for (Wo, Eo) if Yo has SNCs with E0. Let it : W1 -+ W be the monoidal transform of W with center Yo and excep-

tional divisor D. By abuse of notation we identify Di (1 < i < r) with its strict transform on W1 (which could become the empty set if Di is a union of components of Yo). By a local analysis, using the regular parameters of the roof of Lemma 6.4, we see that D1 + + D, -i- D is a SNC divisor on Wl .

Now set E1 = {D1,.. . , D,., D,.+1 = D}. We have thus defined a new pair (W1, El), called the transform of (Wo, Eo) by the map it. The diagram (W1, Ei)

(Wo, Eo)

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

Definition 6.9. Suppose that (Wo, Eo) is a pair with Eo = Dl,..., D,.}, and Wo is a non-singular subvariety of Wo. Define scheme-theoretic intersec-

tions Di = Di Wo for all i. Suppose that Di are non-singular for 1 < i < r and that D1 + + Dr is a SNC divisor. Set Eo = {D1, ... , D,.} Then (Wo, Eo) is a pair, and (Wo, Eo) -+ (Wo, Eo)

is called an immersion of pairs.

6.3. Pairs and basic objects

71

Definition 6.10. A basic object is (Wo, (Jo, h), Eo), where (Wo, Eo) is a pair, Jo is an ideal sheaf of Wo which is non-zero on all connected components of Wo, and b is a natural number. The singular locus of (Wo, (Jo, b), Eo) is

Sing(Jo,b) = {q E Wo I vq(Jo) 2: b) = V(Ob-1(Jo)) C Wo.

Definition 6.11. A basic object (Wo, (Jo, b), Eo) is a simple basic object if vq(./) - b if q E Sing(J, b). Suppose that (Wo, (Jo, b), Eo) is a basic object and Yo is a non-singular closed subvariety of Wo. Yo is a permissible center for (WO, (Jo, b), En) if Yo is permissible for (WO, Eo) and Yo C Sing(Jo, b). Suppose that Yo is a permissible center for (WO, (.In, b), Eo). Let 7r : Wi -+ Wo be the monoidal transform of Wo with center Yo and exceptional divisor D. Thus we have a transformation of pairs (W1, El) -p (Wo, En). Let ./r be the weak transform of Jo on W1, so that there is a locally constant function cl on D (constant on each connected component of D) such that the total transform of Jo is

JoOw, = 2j JI . We necessarily have cl > b, so we can define

Jl = Z nbJodW, =

ZD-b jl

We have thus defined a new basic object (W1, (J1, b), E1), called the transform of (WO, (Jo, b), Eo) by the map 7r. The d iagrarrr

(Wi,(J1,b),E1)

(Wo,(Jo, b),Eo)

will he 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 (6.9)

(Wk, (Jk, b), Ek) - ...

(Wo, (Jo, b), Et))

is a sequence of transforinat.ions.

Definition 6.12. (6.9) is a resolution of (Wo, (Jo, b), Eo) if Sing(Jk, b) = 0.

Definition 6.13. Suppose that (6.9) is a sequence of transformations. Set Jo. = Jo, and let Ji+1 be the weak transform of Ji for 0 < i < k - 1. For

0
w-ordi : Sing(Ji, b) -+ by

w-ordi(q) = vq(Jj) b

7L

6. H.esolntion of Singularities in Characteristic Zero

72

and define 1

ordi : Sing(Ji, b) -- b 7L by vy(Ja)

ordi(q) = b

If we assume that Yi C Max w-ordii C Sing(Ji, b) for 1 < i < kin (6.9), then max w-orde > > max w-ordk by Lemma 6.4.

Definition 6.14. Suppose that (6.9) is a sequence of transformations such that Y C Max w-ordi C Sing(J21 b) for 0 < i < k. Let ko be the smallest index such that maxw-ordk0_1 > maxw-ordk,, = = maxw-ordk. In particular, ko is defined to be 0 if rnax w-ordo max w-ordk. Set Ek = EA U Ek , where Ez is the ordered set of divisors D in Ek which are strict transforms of divisors of Eko and F.k - F.k - Ek . Define tk : Sing(Jk, b) -> Q x Z, where Q x Z has the lexicographic order by tk(q) = (w-ordk(q), ri(q)), Card{D C Ek I q E D} if w-ordk(q) < maxw-ordk,, (q) Card{D E E; (I E D} if w-ordk(q) = max w-ordk0 .

-

This definition allows us to inductively define upper semi-continuous functions to, t1.... , tk on a sequence of transformations (6.9) such that Y C Maxw-ordi C Sing(Ji, b) for I < i < k. Lemma 6.15. Suppose that (6.9) is a sequence of transformations such that Y C Max ti C Max w-ordi for 0 < i < k. Then

max tk < ... < max tk. Lemma 6.15 follows from Lemrna 6.4 and a local analysis, using the regular parameters of the proof of Lemma 6.4. Definition 6.16. Suppose that (Wo, (Jo, b), Eo) is a basic object, and (6.10)

(Wk, (Jk, b), Ek) --+ ... - (Wo, (Jo, b), Eo)

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 (6.10)). Suppose that (Wk, (Jk, b), EA..) is a monomial basic object (relative to a sequence (6.10)). Let E0 = {D1,..., D,.} and let Dz be the exceptional divisor of iri : Wi - Wi_1, where i = i + r. Then we have an expansion (in a neighborhood of Sing(Jk, h)) of the weak transform Jk of .1:

6.3. Pairs and basic. objects

73

where the di are functions on Wk which are zero on W1, -- Di and d1 is a locally constant function on D-. for 1 < i < k. Suppose that B = (Wk, (Jk, b), Ek) is a monomial basic object (relative to (6.10)). We define

r(B) : Sing(Jk, b) -IIRf =ZxQxZN by

r(B)(q) _ (-1P1(q), r2(q), I'3(q)), where

p such that there exist i 1 , . .. , ir

r1(q) = min

with r < i , ... , ip

r + k, such that. dil (q) + ... + clzT (q) > b for some q E Di, n ... n Dip

,l.l(q)+.dn(4) with r
r2(q) = max

di, (q) + ... + clip (q) > b, q E Di1 n ... n Din

r3(q) = Max

(i1i

... ,'ir 0, ...) such that r1(q) =

.(q)

d;l (g) t .--dQy b

with Observe that 1'(B) is upper semi-continuous, and MaxIF(B) is non-singular.

Lemma 6.17. Suppose that B = (Wk, (Jk, b), Fk) is a monomial basic object (relative to a sequence (6.10)). Then Y. = Max I'(B) is a permissible center for B, and if (Wk+1, (Jk+l, b), Ek+1) -> (Wk, (Jk, b), Ek) is the transformation induced by the monoidal transform Irk I 1 : Wk-+l -> Wk with center Y k.., then B1 = (Wk+1, (Jk+l, b), EA.+t) is a monomial basic object and

rnaxr(B) > maxr(B1) in the lexicographic order.

Proof. Max r(B) is certainly a permissible center. We will show that

maxr(B) > max1 (Bj).

We will first assume that max(-r1) < -1. Let. i = i + r. There exist locally constant functions di on W such that dz(q) = 0 if q 0 Di and d? is locally constant on D7, such that

Let Dz be the strict transform of D on Wk+1 and let Dk+1+l be the exceptional

divisor of the blow-up irk+t : Wk+1 -' Wk of Yk = Max r(B). Then Jk+1 =

1

Di

k+i ... Zd-Dk+1

,

6. Resolution of Singularities in Characteristic Zero

74

where for 1 < i < k we have dt(q) = di(irk+1(q)) if q E Dz and d7(q) = 0 if q ¢ D. We have dk+1 (q) = di1(7rk+1(q)) + ... + div (Irk+1(q))

-b

ifgEDk+1, anddk+l(q)=0if q Dki1' For q E Sing(Jk+l,b) we have r(Bl)(q) = r(B)(7rk+1(q)) if Irk+1(q) Maxr(B), so it suffices to prove that r(B1)(q) < r(B)(irk+1(q)) if q E Sing(Jk+l, b) and irk+1 (q) E Max r(B). Suppose that

r(B1)(q) _ (-p1, w1, (j1, ... , jp1, 0, ...)) and

r(B)(7rk+l(q)) = maxr(B) = (-p, w, (il, ... , ip, 0, ...)). We will first verify that (6.11)

If k + 1 V {j1,

p1 = r1 (q) ? r1(irk+1 (q)) = p jp1 }, then the inequality

b < dj1(q) + ... + d jp, (q) < d,1(7rk+1(q)) + ... + djr1(7rk+1(q))

implies pi > p.

Suppose that k + I E { j1, ... , jp1 }, so that j1 = k + I. If p1 < p, there exists ik E {i1, ... , ip} such that ik ¢ {j1, ... , jpl ). Let

d= ij1(q)+... +djnl(q) Then

rl = dj2(q)+...+(ljP1(q)+di1(7fk+1(q))+...+di,(7rk+1(q))-b (di1(irk+1(q)) + ... + dik-1 (7Ck+1(q)) + dik (irk+l (q)) + ... + (Iir, (irk 11(q)) - b) +dik(7rk+1(q)) + dj2(7rk+1(q)) + ... + d 1 (irk+1(q)) dik(7fk-hl(q)) + dj2(7rk+l(q)) + ... + dj,1(Irk+1(q)) i 1

< <

b,

a contradiction. Thus p1 > p.

Now assume that pl = p. We will verify that w1 < w. If k + 1 {ii,...,jp1}, we have dd1(q)+...+dd,1(q)

w1

<

di1(irk+1(q))+....+dia (1rk+1(q))

< w. Ifk:+1 E{jl,...,j,1}, then j1=k+1. We must have dj2(irk+l(q)) + ... + d,,, (Irk+1(q)) < b,

6.4. Basic objects and hypersurfaces of maximal contact

75

so that

w1b = (dil (Irk+1(q)) + ... + dig (ik+l (q))

-

b) + djs (q) + ... + djrl (q) < di1(7rk+1(q)) + ... + dip (1rk+1(q)) + (dj2(lrk+1(q)) + ... + djv, (lrk+I (q)) - b)

< wh. Thus w1 <w.

Assume that p = p1 and w = w1. We have shown that k + 1 ¢ {j,. .. , j,1 } in this case. Since Dil n ... n Dir = 01

we must have (j1i..., jp,0,...) < It remains to verify the lemma in the case when max(-r,(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 open cover of W. We then have associated basic objects BA

is a (finite)

= (Wa, (JA, b), EA)

for \ E A obtained by restriction to WA. Observe that Sing(J", b) = Sing(J, b) n WA. If 131 = (W1, (J1, b), E1) -+ (W, (.1, b), E) is a transformation induced by a monoidal transform it : W1 -+ W, then there are associated transformations

Bi = (Wi , (Ji , b), Ei)

(WA, (JA, b), E)), where Wj = 7r-1(Wa) and J1 = J1 I W11. In particular, W,\) is an open cover of W1.

Definition 6.18. Let W1 = W x A}. with projection it : W1 -+ W. Suppose that (W, (J, b), E) is a basic object. Then there is a basic object (W1, (J1, b), F1), where J1 = JOw1. If E = {D1,. .. , D,.} and Dg = 7r-r (Di) for 1 < i < r, then E1 = { D , ... , D'}. The diagram (W11(Ji, 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 C W is a non-singular hypersurface. Suppose that Zw,, C Ab-1(J). Then Sing(J,b) C V(Ob-1(J)) C Wh, and vq(AL-1(J) = 1 for any q E Sing(J,b). Conversely, if (W, (J, b), E) is a simple basic object and q E Sing(J, b), then vq(Ob-1(J)) = 1, so there exist an affinc neighborhood U of q in W and a non-singular hypersurface Uh C U such that U n Sing(J, b) C Uh.

6. Resolution of Singularities in Characteristic Zero

76

Lemma 6.20. Suppose that (Wi, (Ji, 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 .

WV)OW1 C TD and

LWi(J)OW1 C AW.i(.I1).

Proof. Let Y C Sing(J, b) be the center of 7r : Wr -' W. The first inclusion follows since vq(aW'(J)) > i if q E Y, so that TD divides AWx(J)OWI. Now we will prove the second inchision. The second inclusion is trivial if i = h, since JOW, = ZDJ1 by definition. We will prove the second inclusion

by descending induction on i. Suppose that the inclusion is true for some i > 1. Let q e D be a closed point,, p = 7r(q). Let K' = K(q). By Lemma A.16, it, suffices to prove the inclusion on W XK K', so we may assume that K(q) = K. Then there exist regular parameters.ro, xl,... , x, in R = OW,p such that I ,r, = (xo, x%1 , . , x,, ), with in C n; xo, p , .. , IM, xm-H1I ... , xra, is a regular system of parameters in R1 = Ow ,q, and TD,q = (xo). It suffices to show that for f E aW('-1)

f

(JI)q. W, i-1 E Lib-(i-1)

(6.12)

0

If f E aW `(J)p C Ab (i-1)(J)p, then the formula follows by induction, so we may assume that f = D(g) with g E tW7(J)p, D an R-derivation. Set D' = xoD. D' is an R1-derivation on RI = OWt,q since D'( 1i) E Ow1,q for 1 < i < in. By induction, aWl(i-1)(Jl)q

4 E ' OW(J)OWi,q C LW1 (J1)q C

x0

Tn

Thus

D'( q) E Z WWli-1)(Jl)q,

n

0

I)'(g)=D(1 -iD(xo)9, x0

0

0

which implies

xf 1 0

and (6.12) follows.

D(g) = D'(9) + iD(xo) 0

0

Z0

E

awii-1)(Ji)q,

6.4. Basic objects and hypersurfaces of maximal contact

77

Lemma 6.21. Suppose that (W, (J, b), E) is a simple basic object, and Wh C W is a non-singular hypersurface such that 1wh C Lb-1 (J). Suppose that

(W1, (Ji, b), El) - (W, (.I, b), E) is a transformation with center Y. Let (Wh)1 be the strict transform of Wh on W1. Then Y C Wh, (Wh)1 is a non-singular hypersur face in W1, and Z(Wh)l c Lb-1(J1).

Proof. Let D be the exceptional divisor of rr : W1 --* W. Suppose that q E D is a closed point, and

p = rr(q) E Y C Sing(J, b) = V(Ab-1(J)) C Wh.

Recall that Y C Sing(J, b) by the definition of transformation. Let f = 0 be a local equation for Wh at p, and g = 0 a local equation of D at q. Then 9 = 0 is a local equation of (Wh)1 at q, and 9 E Ab-1(Jl)q by Lemma 6.20. Thus

Sing(Ji, b) = V(Ab-1(J1)) C

V(f ). 0

Definition 6.22. Suppose that B = (W, (J, b), E) is a simple basic object and W- is a non-singular closed subvariety of W such that 7W C 2-' (J). Then the coefficient ideal of B on W is b-1

C. (J) _

Ai (.1) tn Oyl.. i=o

Lemma 6.23. With the notation of Lemma 6.21 and Definition 6.22, Sing(J, b) - Sing(C(J), b!) C Wh C W. Proof. It suffices to check this at closed points q E Wh. By Lemma A.16 we may assume that K(q) = K. Then there exist regular parameters x 1 ,.. . , xn at q in W such that xl = 0 is a local equation for Wh at q. Let 1? = 6Wq = K[[xl, ... , x,-,]], S = 6wh,q = K[[x2, ... , with natural projection 11-* S. It suffices to show that (6.13)

vR(J) > b if and only if vs(A (JS)) > h -

j

for j = 0,1, ... , b - 1. Suppose that f E J C R. Write

f=

aixi.,

i>O

with ai E S for all i. We have vR(f) > b if and only if vs(aj) > b - j for

j=0,1,...,b-1.

6. Resolution of Singularities in Characteristic Zero

78

We also have aj E Ai (J) S, since a1 is the projection in S of 37 we have the if direction of (6.13). The ideal t2 (J) is spanned by elements of the form ail

Thus C,

I .... +-i. f

axll ... axn

with f E J and it +

+ i,s < j, and the projection of ci...... in in S is ail+...+i"a al. aX2

it

... axn .

Thus we have the only if direction of (6.13).

O

Theorem 6.24. Suppose that (W, (J, b), E) is a simple basic object, and {W'"}AEA is an open cover of W such that for each A there is a non-singular

hypersurface Wh C WA such that Zwh C Aii-1(JA) and Wh has simple normal crossings with V. Let Eh be the set of non-singular reduced divisors on Wh obtained by intersection of E with W. Recall that R(1)(Z) denotes the codimension 1 part of a closed reduced subscheme Z, as defined in (6.1).

1. If R = R(1)(Sing(J, b)) # 0, then R is non-singular, open and closed in Sing(J, b), and has simple normal crossings with E. Furthermore, the monoidal transform of W with center R. induces a transformation

(Wi, (Ji, b), Ei) - (19, (J, b), E) such that R(1)(Sing(J1, b)) = 0. 2. If R(1)(Sing(J, b)) = 0, for each A E A, the basic object (Wh, (C(J'), b!), Eh) has the following properties: a. Sing(JA, b) = Sing(C(JA), b!). b. Suppose that (6.14)

(Wq, (.Is, b), EQ) -->

-, (W, (J, b), E)

is a sequence of transformations and restrictions, with induced sequences for each A (W,'. (.I,\, b), E.,) :... -, (W\, (J", b), F,A).

Then for each .A there is an induced sequence of transformations and restrictions ((Wh)s, (C(JA)., b!), (Eh)s) -' (Wh, (C(.IA), b!), Eh)

6.4. Basic objects and hypersurfaces of maximal contact

79

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

- (WA,E\)

(W.1, E.1)

(6.15)

T

((Wh)'\, (Eh)\)

-- ...

T

(W1',\1 Fh)

such that

Sing(C(J")i, b!) = Sing(JJ , b)

for all i.

Proof. We may assume that. W\ = W. First suppose that T = R(1) (Sing(J, b)) 0. Then T C Sing(J, b) C Wh implies T is a union of connected components of Wh, so T is non-singular, open and closed in Sing(J, b), and has SNCs with E. Suppose that q c T. Let f = 0 be a local equation of T at q in W. Since (W, (J, b), E) is a simple basic object, Jg = fbOWq. If 7r1 : W1 - W is the monoidal transform with center T, then (J1)Q, = OW,q, for all q1 E Wi such that ir1(gi) E T, and necessarily, R(1)(Sing(Ji,b)) = 0.

Now suppose that R(1)(Siug(J,b)) = 0, and (6.14) is a sequence of transformations and restrictions of simple basic objects. Suppose that (W1, (J1, b), E1) -, (W, (J, b), E) is a transformation with center Yo. By Lemma 6.21 we have Yo C Wh, the strict transform (Wh)1 of Wh is a non-singular hypersurface in W1, and Z(W,,), C

!-'-1(Jl).

Since Wh has SNCs with E, Y c Wh and Y has SNCs with E, it follows that. Y has SNCs with E U Wt,. Thus (W,)1 has SNCs with El, and if (Eh)1 = El (Wh)1 is the scheme-theoretic intersection, we have a closed immersion of pairs ((WI,) 1, (Eh)1) - (W1, E1). Since, the case of a restriction is trivial, and each (W1, (Ji, b), Ei) is a simple basic object, we thus can conclude by induction that we have a diagram of closed immersions of pairs (6.15) such that (6.16)

z(Wh)J c

Ob-1(Ji)

for all j. We may assume that the sequence (6.14) consists only of transformaWh,_1. It remains to show tions. Let Dk be the exceptional divisor of Wk

that (6.17)

Sing(C(J)j, M) = Sing(J7, b)

6. Resolution of Singularities in Characteristic Zero

80

for all j. By Leninia A.16 and Remark A.18 we may assume that K is algebraically closed. For k. > 0 and 0 < i < h - 1, set _ 1 [Ah_1(.J)] r

k-1

Dk

LL

OW,,

so that b

b,

ll

CV) k = 4.=1

0(W,,)k.

[nb-i(J)k

We will establish the following formulas (6.18) and (6.19) for 0 < k < s

and0
[Ab-i(J)Ik C Suppose that q E Sing(C(Jo)k, b!) is a closed point. Set Rk = dw,,,q, Sk = 6(w,,)t

that

Ab-2(Jk)

Then there are regular parameters (xk,l, ... , Xk,,,,, zk) in Rk such (zk), and there are generators { fk} of JkRk such that Z(W,,),.,q =

fk = E ak,azk

(6.19)

a

with tak,,, E Ii [[xkl , ... , ;t:k,,,]] = Sk such that for 0 < a < b - 1

(ak.a)V a E C(J)kSk Assume that the formulas (6.18) and (6.19) hold. (6.18) implies C(.I)k C C(Jk), so by Lemma 6.23, Sing(Jk, b) C Sing(C(J)k, b!). Now (6.19) shows that Siitg((.(.J)k, M) C Sing(Jk, b), so that Sing(C(J)k, M) = Sing(JJ, b), as desired.

We will now verify formulas (6.18) and (6.19). (6.18) is trivial if k = 0, and (6.19) with k = 0 follows from the proof of Lemma 6.23.

We now assume that (6.18) and (6.19) are true for k = t, and prove them for k = t + 1. Since [b_i(J)] c Ab-i(Jt), t

we have

[b_i(J)]

[Ab_i(J)J (+1

t

1

t

,

b-i(Jt)

(ZDe+t )i

where the last inclusion follows from the second formula of Lemma 6.20. We have thus verified formula (6.18) for k = t + 1. Let qt+i E Sing(C(Jo)t+i, b!) be a closed point, and qt the image of qt+1 in {Sing(C(Jo)t, b!).

By assumption, there exist regular parameters (xtl,... , xt,,, zt) in Rt which satisfy (6.19). Recall that we have reduced to the assumption that

6.5. General basic objects

81

K is algebraically closed. Let Y be the center of Wt-1I - Wt. By (6.16), xtn]] C Rt there exist regular parameters (x1, ... ,Win) in St = K[[xtl, such that (xl, ... , yn, Zt) are regular parameters in Rt, there exists r with r < n such that x1 = . = x,. - zt = 0 are local equations of Yt in °Wt,gt, and Rt+1 = dwt+i,qt+i has regular parameters (xt+1,1, .. , xt+l,n, zt+l) such ,

that. xt+1,1,

71 Zt

:rt+1,1xt+1,i for 2 < i < r, xt+l,lzt+1,

-Ti

xt+l,i for r < i < n.

:i'i

In particular, xt-hl,l - 0 is a local equation of the exceptional divisor D11 and zt+1 = 0 is a local equation of (Wh)t-i-1. We have generators .ft+1 =

ft b

=

t+11

at i

1

l,cxztt+l

a

of Jt+lRt+1 = -xt+i,i 1 JtRt+1 such that at+1,« _

at.,, b-cv

xt,1

if a < b, and (at+1,a) F-7 =

(at,.) ib" E (Zi I

(xt 1)L1

1

+t )b1

C(J)tst+1 = C(J)t+lst+i.

H

6.5. General basic objects Definition 6.25. Suppose that (Wo, Eo) is a pair. We define a general basic object (GBO) (.To, Wo, Eo) on (WO, Eo) to be a collection of pairs (Wi, Ei) with closed sets Fi C Wi which have been constructed inductively to satisfy the following three properties:

1. FocWo. 2. Suppose that Fi C Wt has been defined for the pair (Wi, Ei). If 7ri+1 : iri+1 (F1).

(Wi+1, Ei+1) -' (Wi, E1) is a restriction, then Fi+1 =

1

3. Suppose that Fi C Wi has been defined for the pair (W1, E1). If 7ri+1 : (Wi+1, Ei.11) -, (Wi, E1) is a transformation centered at Yi C Fi, then Fi+1 C Wi+1 is a closed set such that Fi - Yi Fi 11 - 7ri+1(Yi) by the map lri+1 The closed sets Fi will be called the closed sets associated to (FO, We, Eo).

6. Resolution of Singularities in Characteristic Zero

82

If

(WI,EI) --; (Wu,Eu) is a transformation with center Yo C FO, then we can use Lite pairs (Wi, Ei) and closed subsets FFz c Wi in Definition 6.25 which map to (W1iEI) by a sequence of restrictions and transformations satisfying 2 and 3 to define a general basic object (.Fl, WI, El) over (WI, E1 ). We will say that Yo is a permissible center for (.Fo, Wo, Eo) and call ()II, WI, E1) - (.Fo, Wo, Eo)

a transformation of general basic objects. If

(WI,E1)-'(Wo,Eo) is a restriction, then we can use the pairs (W1, E1) and closed subsets Fi defined in Definition 6.25 above which map to (WI, E1) by a sequence of restrictions and transformations satisfying (2) and (3) to define a general basic object (.F1, W1, El) over (WI, EI). We will call WI,Ei)-

(To,Wo,Eo)

a restriction of general basic objects. A composition of restrictions and transformations of general basic objects (6.20)

(.Fr, Wr, E,.) -> ... -r (.Fo, Wu, Eo)

will be called a permissible sequence for the GBO (To, Wo, Eo). (6.20) will be called a resolution if Fe, = 0.

Definition 6.26. Suppose that (Wo, Eo) is a pair, and (.Fo, WO, Eo) is a. GBO over (Wo, Eo), with associated closed sets {Fi}. Further, suppose that { N'}A(A is a finite open covering of WO and d E N. Suppose that we have a collection of basic objects {BB}ACA with Bo = (Wo , (ao, b".), k'\).

We say that {Bo }AFA is a d-dimensional structure on the GBO (.Fo, Wo, Eo) if for each A E A

1. we have an immersion of pairs

j0 : (Wo,Eo) -i

(Wo'plo),

where Wo has dimension d, and 2. if (Fr, Wr, Er) -3 ... -, (Yo, Wo, Eo)

6.6. Functions on a general basic object

83

is a permissible sequence for the GBO (Fo, We, Eo) (with associated

closed sets F C W for 0 < i < r), then the sequence of morphisms of pairs

(W,',Er) -+ ...

(Wo \Eo) induces diagrams of immersions of pairs (W,a,E,1,) T

(l ,ET)

(Wo,Eo)

- ...

-,

T

(Wo+Eo)

such that (W,A,

bA), E,a.) - ... -, (WoA, (ao, bA), Eo )

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

F'' =FfWA =Sing(a%,b")

for0
There is a function ordd : FO - Q such that for all A E A, the composition FF Fo 0 Q is the function ord\ : FF -+ Q defined by Definition 6.13 for

F' = Sing(aA,b\). Proof. Let {Bo }AEA, where BO\ = (W-o\, (a0-\, ba), .to \), be the given d-dimen-

sional structure on (.Fo, Wo, Eo). Let Eo = {D1, ... , D,.}. We have closed immersions jQ - Wo --r W01 for all A E A.

Suppose that qo E Fo is a closed point, and A, ,6 E A are such that there = j0/3(q0'7) = qo. We must show

are qo E 17VO and qo E W0'3 such that jo (qa)

that vy,\ (ao) bA

=vi0(a0 )

We will prove this by expressing these numbers intrinsically, in terms of the GBO (.Fo, Wo, Eo), so that they are independent of A. We will carry out the calculation for BO'_

Let (.F1, Wi, El) i (.Fo, WO, Eo) be the restriction where Wl = WO x Ah .

Let qi = (qo, 0) E W1. Let L1 = qo x Al; then Li c Fi = 7r- 1(Fo).

6. Resolution of Singularities in Characteristic Zero

84

For any integer N > 0 we can construct a permissible sequence of pairs 7IN ... 53 (WI' E1) (WO, (6.21) (WN, EN)

Eo)

where for i > 1, Wi+1 --> Wi is the blow-up of a point, qi. qi is defined as follows. Let Li be the strict transform of L1 on Wi, and let D1 be the exceptional divisor of 7ri, where i = i + r. Then qi = Li n Di. We have q1 E L1 C Fl. We must have L2 C F2 by 3 of Definition 6.25. Thus ire induces a transformation of GBOs (.F2, W2, E2) -' (.F1, W1, El), and q2 E F2. By induction, we have for any N > 0 a permissible sequence (6.22)

...

(.FN, WN, EN)

!

(F1, W1, El) 54 (Fo, Wo, Eo)

(6.22) induces a permissible sequence of basic objects with qi = (qo, 0) and centers qj for i > 1: (6.23)

(WN,(aA,bA),EN)

...

(Wo,(ao,b"),En)

Observe that Li C Wa for all i -> 1, so that qi E W? for all i > 1. Set b' = vg (ao).

Suppose that (x1, ... , xd) are regular parameters at qo in WO A. Then (:rrl,... , xd,1) are regular parameters at q1 in Wj (where Al = spec(K[t])), and thus there are regular parameters (x1(N), X2 (N), ... , xd(N), t) in Wn, at qN which are defined by xi = xi(N)tN-1 for 1 < i < d. t. = 0 is a local equation of the exceptional divisor DA, of WN

Suppose that f E Otyo expression

qo

-

WN 1.

and vqo (f) = r. Let K' = K (q(,)\). We have an

ail,...,idx1 ... x E K'[[x1,... , xdJJ = OWo ,gA,

f= i l +...id> r

where ail,---,id E K'. Thus in K'[[xi (N), ... , xd(N), t]] = OW,\ qN we have an expansion

f

E t(N-1 ),

at, ,...,idx1(N)u1 ...

ail,...,idx1(N)zl ... xd(N)zd + tN-1.lN i1.}....+ia=r

Thus we have an expression f = t(N-1)r f N in Oyl,N,gN, where t f IN.

Since vq (ao) = b', we have

aoO 1A =2nd 1lKN;

6.6. Functions on a general basic object

where I

N

85

f K. We thus have an expression aN = Zi5 i)(z' W)KN,

(6.24)

A

and vqN (aN) > br since qN E FN fl WN = Sing(aN, bA).

Under the closed immersion WN C WN we have (for N > 1) D DN. WN, where DN has dimension d and is irreducible. Since FN C WN, it follows that dim(FN - DN) = dim(FN - DN) < dim DN = d, and the following three conditions are equivalent:

1. dim(FN DN) = diui(FN - DN) - d. 2. DN C FN.

3. (N-1)(b'-bA)>bA. Thus = 1 holds if and only if b' - b" = 0, which holds if and only if, for any N, dim(FN - DN) < d. Thus the condition yqo (au)

=

ba

b' = 1 ba

is independent of A. Now assume that b' - bA > 0. Then, for N sufficently large,

(N - 1)(b - bA) > bA, and FN DN = DN is a permissible center of dimension d for (.FN, W,- , EN) . We now define an extension of (6.22) by a sequence of transformations (6.25) 1N+S

(.FN+S, WN+.S, EN+S) -,

... "N+1 -' (.FN, WN, EN)'1N_, ... - (.Fo, Wo, Eo),

where 1rN+i is the transformation with center YN+i = FN+i Dam, and we assume that cline YN+i - d for i = 0, 1, ... , S - 1. Observe that. -

(6.26)

dimYN+i = d if and only if DN++i - l

.

-

WN+i C FN+i.

We thus have YN+i = DN+t. (6.26) induces an extension of (6.23) by a sequence of transformations (627)

(WNA

(aN+S, YI ), EN+S) "- S ...

"!1 (WN, (aN, bA), EN)

...

(WW ,

(au, b), Eo).

The centers of 7ri in (6.26) are D for j = N,... , N+S-1. Thus WN+i+i WN+i is the identity for i = 0,... , S - 1.

6. Resolution of Singularities in Characteristic Zero

86

We see then that aN+f = 2(N-1)(b-bA)-ibaKM

for i = 1, ... , S. The statement that the extension (6.27) is permissible is precisely the statment that (N - 1)(b' - b'') > Sb", which is equivalent to the statement that dim FN+z DN+s = d for i = 0, 1, . . , S - 1. This last .

statement is equivalent to

L(N-1))(b'-ba)

.5'
where [xj denotes the greatest integer in x. Thus f.N is determined both by N and by the GBO (.Fa,Wo,Eo). The limit

PN _b bA1 is defined in terms of the GBO, hence is independent of A.

0

Theorem 6.28. Let (F0, WO, Eo) be a GBO which admits a d-dimensional structure, and fix notation as in Definition 6.26. Let F; be the closed sets associated to the GBO. Let FA = Sing(as ,&\), and let w ord , tL be the functions of Definition 6.13 defined for F?`. Consider any permissible sequence of transformations

(Fr,WT,Er) -p ... -' ()"o, Wo,Eo) 1. Then for 1 < i < r, there are functions w-ordd : Fz -i Q such that the composition

j, w-ordd FA->Fj is the function w-ordz : Fi' -, Q. 2. Suppose that the centers Y of 7r;+1 are such that YY C Max w-ordd. Then there are functions td : FF -, Q x Z such that the composition a

F''F `.QxZ to

is the function t1 : Fj' -* Q x Z. Proof. Let Eo = ID,,..., D,3}. The functions tj are completely determined by w-ord, with the constraint that Y C Max w-ordd for all i, so we need only show that w-ordd can be defined for GBOs with d-dimensional structure. The functions w-ordd are equal to ordd, which extend to ordd on FO by Proposition 6.27. We thus have defined w-ordd = ordd. Let D, be the exceptional divisor of irk, where i = i + s, and let Dz = Wa . Dt.

6.6. Functions on a general basic object

87

.'

For each A and i we have expressions

ast - DA

(6.28)

DA

where aL is the weak transform of ao on Wti and the e j are locally constant functions on b2 We further have that r

az

a

t

Da

Suppose that 77i-1 is a generic point of an irreducible component Y' of Y such that 77i-1 is contained in Wta (and thus necessarily in W''), and that qi E Di is a point which we do not require to be closed, but is contained in b4 = W,` Di, and lri(gi) is contained in the Zariski closure of rhi_ 1. From (6.28), we deduce that (6.29)

(aa)

t qi

(q=)

1.D;

... Zc=

ro-1

1

k

i (qf)Z

q2

Thus (6.30)

c (qi) _ (w-ordti(??il) + b-\

' j=1

b7

v+r, r

(ZDA)

-1.

We further have

ordi(gi) - w-ordi (qi) +

TAE 7(q;)

Suppose that qi' E Fi, and define qj' for 0 < j < i -- 1 by q.7 = 7rj+] (q?+ ) To simplify notation, we can assume that q'j E Yj for all j. Let Yl be the irreducible component of Yj which contains qj, and let 771

be the generic point of Yj for 0 < j < i - 1. Suppose that A E A is such that qi' E W. We have associated points qj

and 77, in W' for0<j
... , qi} and {rio, 771, ..., 71i-1 } with qj, rJj E W` for 0 < j < i

and

{q ,q ,...,qty} and {71( '),i ...... Jxl-,} with gq,rj' E W] for 0 < j < i.

6. Resolution of Singularities in Characteristic Zero

88

From equation (6.30) and induction we. see that the values cjWj)

(6.31)

ba

for j = 1,...,i

and w-ordi (gi) are determined by:

1. The rational numbers (6.32)

ordg (qo'), ordi (qi ), ... , ordz (q'),

(6.33)

ordu(7)u),ordi(rl1),...,ordz 1(77i-i)

2. The functions (6.34)

Y H(gg,j) _ {

1

if q's E YS and Ys C Dy locally at qy,

0

if q9

Ys or Y9

Dy locally at q.

Our theorem now follows from Proposition 6.27, since w-ordi (g,,) is com-

pletely determined by the data (6.32), (6.33) and (6.34).

Proposition 6.29. Let 130 = (.To, Wo, E0) be a GBO which admits a ddimensional stricture. Fix notation as in Definition 6.26. Let

Fja=Sing(ai,ba). Consider a permissible sequence of transformations

B,. _

W, E,,)

i ...

0, Wo, I,0)

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

rrr,.(B..): F;.-AIM=7GxQxZN such that the composition FT

Fr

-r-+' IM

is the function of Definition 6.16.

Proof. Suppose that, Eo = [DI,..., D,,}. The function IF on the basic objects ((V,a, (ai,, b>'), E;.) depends only on the values of

for 1 < i < r

(with the notation of the proof of Theorem 6.28). Thus the conclusions follow from (6.31) of the proof of Theorem 6.28.

6.7. Resolution theorems for a general basic object

89

6.7. Resolution theorems for a general basic object Theorem 6.30. Let (.Fo , W0, Eo) be a GBO with associated closed sets {Fd} which admits a d-dimensional structure. Fix notation as in Definition

6.26. Let Fl = Sing(ai , ba) = P,51 n Consider a permissible sequence of transformations

(yd, Wr, Ed) ' ; ...

(6.35)

(moo , Wo,

Ed)

such that the center Y of iri+l satisfies Y, C Max t.t. C Max w ordi C Fi

for i = 0.1... , r - 1 and inaxw-ordr > 0. This condition implies w-ordi(q) < w-ordi_1(7ri(q)), ti(q) < ti_i(iri(q))

forgEF1 andl max w-ordi > . . . > max w-ord,.,

maxtp>maxtl>...>maxtr. Let R(1)(Max t.,.) be the set of points where. Maxtr has dimension d - 1.

Then R(1) is open and closed in Maxtr and non-singular in. W. 1..

If H(1)(Maxtr) # 0, then R(1) is a permissible center for (Td, W',., Ed) with associated transformation

(rr+l W,I, d1

Irr+1

d

f

fir >W,, r)

and either R(1) (Max tr+i) _ 0 or max tr > max tr+1 2. If R(1)(Maxtr) = 0, there is a GBO (.Fd-1,Wr,Ed-1) with (d- 1)dimensional stricture, and associated closed sets {F,d-1} such that

if (6.36)

(FT I , WN5 EN 1) - ...

()-d-1 Wr,Ed-1) r r

is a resolution (FN 1 = 0), then (6.36) induces a permissible sequence of transformations

(-F

,

WN, EN)

,

W", Ed)

such that the center Y of it 1_1 satisfies

Y C Max ti C Max w-ordi C Fd

for r < i < N - 1, maxtr -

= maxtN_1, Maxti = Fa-i for i = r..... N - 1, and one of the following holds: a. FN = O.

b. P. 0 0 and max w-ordN = 0. c. FN 0 0, max w-ordN > 0 and max tN_i > max I.N.

6. Resolution of Singularities in Characteristic Zero

90

We will now prove Theorem 6.30. Let En = {Dl,... , D9}. Let Di be the exceptional divisor of rri, where i = s + i. Set bT - O(max w-ordr) > 0. Let ro be the smallest integer such that maxw-ordro = maxw-ord,. in (6.35). There exist an open cover (WQ) of WO and a d-dimensional structure {B' }AeA of (.r0, WO, En), where BLA) _ (Wi , (aA, b'`), EE) for A E A. Thus for A E A, we have sequences of transformations (ylrr', (ar, bA), Er) '

...

(WO A, (ao, bA), Eo ),

where the center Yip of W+1 -> W' is contained in Max ti. There is an expression

ara = D ...

D, ar r

We now define a basic object (B,')-' - (Wr , ((a.)-\, (b)-\), P,,\)

on TV* by

if b,' >0,

ar

(6.37)

(ar)e _

(11 )

(A)0 b,.

a=rl

(ZDi

if b, < b),

ifb'`>b',

br ba. (b-\

... 1

-

ba.)

if ba. < ba.

Lemma 6.31. 1. Sing((a;.)a, (b')') = (Maxw-ordr) fl W. is a simple basic object. 2. (W,\, ((ac)!, 3. A transformation

(l'+i

,

((ar+I

(b')''), k,\+ I )

(W, , ((a.)", (b')'), ET )

rrrith center Y,` induces a transforrrtation

(W +1, (aA 1, bA), EE+i) -' (Wj , (ar, bA), kr )

such that the center YA C Max w-ord,. fW, . 4. Sing((ar+l)'\, (b')a) = 0 if and only if one of the following holds: a. uiax w-ordr > max w-ordr+l, or b. max w-ordr = max w-ordr+1 and (Max w-ordr) fl Wa = 0.

5. If max word, = max w-ordr+i then Sing((ar+1)'\, (b')A) = Max w-ordr+l f1W +1,

and (ar+i)" is defined in terms of ar+1 by (6.37).

6.7. Resolution theorems for a general basic object

91

Proof. 1. For q E WA we have q E Max w-ord, if and only if vq(aA) > bT and vq(aT) > bA. These conditions hold if and only if vq((a')A) > (b)A. 2 follows since vq (aa.) < b,a. for q E W,?

.

3 is immediate from 1. If ba > bA, we have (6.38)

I.X

(')"

r+1

r+1

I 1 = 2Dr ar = Zv

a;.

(ar)t

Suppose that ba < bA. Then /

bA - bT

J1

lurI1)

A....L +1 A

_CA

+

1

(6 . 39)

)b"-ba.

(Z_bT a

=

D

Di

+

(c

r

l +1 na

D1"

r+1

bA

ba

1

;7+-r

where 7'

c9(77)vn(TDa) + b' ' - ba

r+1(q) = j=1

if 77 is the generic point of the component of Ya containing lrr+1(q)

If we set K =Tcni caI 1 (the ideal sheaf on l'', ), we see that (6.39) A

r+1

is equal to (4 + K)-T,1J'

(6.40)

b.)b.

_

Now 4 and 5 follow from 1 and equations (6.38), (6.39) and (6.40).

At q E Max tr n W,A there exist N = rl(q) hypersurfaces Dz , ... , D i\ (E,. )a such that

U E

gEDln...fl fl axw-ord,. 54 0. Thus there exists an affine neighborhood Ur 'o of q such that (6.41)

MaxtrnUT'O =D1'On.. nD

(The superscript A, 0 denotes restriction to UT 'O.) Now define a basic object.

(Br

(Up', ((ar) " , (b")" ),

by

(6.42)

(a',')'`_"

= (ar )'',m+

(b")",m

= (bl)A

(2DA,m)(V)'"16

+... + (7

k,)(b"),,mI

4N

6. Resolution of Singularities in Characteristic Zero

92

Lemma 6.32.

Max t, n 0'0

1.

2. (U,'", ((a;.') "9', (b")A. ), 3. A transformation

is a simple basic object. ((a")-',0,

{(ar+1)a,

(Er

induces a transformation , ba Ea,fi) (Ur,m (a" ba) FA,k -'r (Ua,4 ( r+1 1, r+1 r+1'(a-"-O

with center YrA

r+ r>

>

such that Y"'' C Max tr. 4. Sing((a°(b")A,O) = 0 if and only if one of the following conditions holds:

a. max tr > max tr+1, or b. max tr = max tr+1 and Max tr+1 n U, +1 = 0. 5. If max tr = max tr+1 and Max t,.+ i n W + 1 # 0, then

Sing((ar+1)a, (b")'

)

= Maxw-ordr+1 nU+O

and (a;'+1) A'O is defined in terms of aa.+1 and (a; +1)" by (6.85) and (6.37).

Proof. 1. For q E Ui 'O we have q E Max tr if and only if q E Max w-ordr and i(q) = N, which hold if and only if vq((a;.)'') > (b')A (by 1 of Lemma

6.31) and v,(-TAO) > I for 1 < j < N, which holds if and only if q E Dij Sing((ar),\,O, (b")-\,O).

2 follows since vq((a'')A?o) < (b")A'O for q E U"0.

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 c Max t,.+, n r , then q must be on the strict

transforms of the hypersurfaces Dz 'O, ... , D.

0

We now observe that the conclusions of Lemmas 6.31 and 6.32, which are formulated for transformations, can be naturally formulated for restrictions

also. Then it follows that the (B!)"0 define a CBO with d-dimensional structure on (Wr, (E' I)

W, ( )+), with associated closed sets F" = Max ti. If Y is a permissible center for (6.43), then Y makes simple normal (6.43)

crossings with ET E. The local description (6.41) and (6.42) then implies that

6.7. Resolution theorems for a general basic object

93

Y makes simple normal crossings with Er. Thus Y C Max tr is a permissible center for (.x'd, Wr, Ed). If the induced transformations are 1, Wr+1, (E

1) 1) _()r , Wr, (Ed) + )

and

(fir+Wr+l,Erd-1) -' (

,W,, %rd),

then either max tr > Max tr+1, in which case Fr'+1 = 0, or max tr = max tr+1 and Fr+1 = Max tr+1.

We will now verify that the assumptions of Theorem 6.24 hold for the simple basic object ((a")A'O, (b" )A,O), (E;. )A,*). Define (by (6.37)) (B'.(,)A in the same way that we defined (B'.)A. Now 5 of Lemma 6.31 implies (B'.)" is obtained by a sequence of transformations of the simple basic object (B'ro)A. (E; )+ is the exceptional divisor of the product of these transformations. Since (BT0)A is a simple basic object, we can find an open cover {V,' of W,' with non-singular hypersurfaces (Vh) u" in V,,'* such that

Let (Vh),.'* be the strict transform of (Vh),''o0 in W\10. By Lemma 6.21, (Vh)r'w is non-singular,

and we have that (Vh),a.'o makes SNCs with (E+),\,P. Let Ur' " = V''' n U. For fixed ). and {Ur'n'''} is an open cover of Ur ,0. For q E Ur'O''p (t(b/)A_1((a.)A.t)) C

((a)q

(1(uh)X'"')q C

by (6.42). Thus the assumptions of Theorem 6.24 hold for and Assertion 1 of Theorem 6.30 now follows from Theorem 6.24 applied to our GBO of (6.43), with d-dimensional structure (D,'.')a'O.

If R(])(Maxtr) is empty, then by Theorem 6.24, applied to the GBO of (6.43), and its d-dimensional structure, (6.43) has a (d - 1)-dimensional structure, and we set -1' Wr, Ed-1) o ( ( !, Wr, (Ed., )+) Given a resolution (6.36), by Lemma 6.32, we have an induced sequence (J

such that the conclusions of 2 of Theorem 6.30 hold.

6. Resolution of Singularities in Characteristic Zero

94

Lemma 6.33. Let 11 and 12 be totally ordered sets. Suppose that

(WN, EN) 4 ... '; (We, Eo)

(6.44)

is a sequence of transformations of pairs together with closed sets Fi. C Wi such that the center Y, of Sri+l is contained in Fi for all i, it+1(Fi+1) C F1

for all i, and FN = 0. Assume that for 0 < i < N the following conditions hold:

1. There is an upper semi-continuous function gi Fi -, I. 2. There is an upper semi-continuous function gi : Maxgi -' 3. Y = Max,q' in (6.44).

12-

4. For anygEl i, with 0
if 1ri+1(q) E Yi, if 7ri+1(q) V Yi

This condition implies that

maxgo > max91 > ... > maxgN-1, and if max gi = max gi+1, then iri+i (Maxgi+i) C Max gi.

5. If maxgi = maxgi+l and q E Maxgi+l, with 0 < i < N - 2, then g (1ri+1 (q)) >

(q)

g'(7ri+1(q)) _ gi+1(q)

if 7ri+1(q) E Y, if iri+1(q) V Yi

Then there are upper semi-continuous functions

fi:Fi-b1 X12 defined by

(6.45)

AM

() , [Ilax gz)

ifqEYi, if

f+ where in the second case we can view q as a point on Wi}1, as Wi+1 -' Wi is an isomorphism in a neighborhood of q. For q E Fi+1, (6.46)

fi(7ri+1(q)) > fi(q) fi(7ri+1(q)) = fi+1(q)

if 7ri+1(q) E Y1,

max fi = (maxgi,maxg')

and Max fi = MaxgjI.

if ni+1(q)

Y,

Proof. We first show, by induction on i, that the fi of (6.45) are well defined functions. Since FN = 0, YN_ 1 = FN_ 1. By assumption 3,

Maxg'N-1 = MaxgN-1 = FN 1, so we can define

fN-1(q) _ (gN-1(q),g,_1(q)) = (maxgN-1,maxg' -1)

forgEFN_1.

6.7. Resolution theorems for a general basic object

95

Suppose that q E Fr. If q E Yr, we define

fr(q) = (gr(q),9r(q)) = (maxgr,maxgr). Suppose that q ¢ Yr. As 7r;+1(Fi+1) c Fi for all i, we have an isomorphism Fr - Yr Fr+1 - D14 1, where Dr 1.1 is the reduced exceptional divisor of Wr+1 + W,.. Since FN = 0, we can define fr(q) = fr+1(q) for q E Fr - Yr. The properties (6.46) are immediate from the assumptions and (6.45). It remains to prove that fr is upper semi-continuous. We prove this by descending induction on r. Fix a = (al, a2) E Ii x 12. We must prove that

{q E F, f,-(q) > a} is closed. If wax fr < cr, then the set is empty. If max fr > a, then I

{q E Fr I fr(q)

a} = Maxg'r U7rr+1({q' E Fr+1 I .fr+1(q') > a}),

which is closed by upper semi-continuity of fr+1 and properness of lrr+l.

LI

Theorem 6.34. Fix an integer d > 0. There are a totally ordered set Id and functions f id with the following properties:

1. For each GBO (.Fg , Wo, Eo) with a d-dimensional structure and associated closed subsets Fd there is a function fo : Fo --> Id

with the property that Max fo is a permissible center for (Wo, Eo). 2. If a sequence of transformations with permissible centers Y (6.47)

(F,Wr,Ed) - ...

(.,Wo,Eo)

and functions fd : Fit -.> I,i, i = 0, ... , r - 1, have been defined with the property that Y = Max fd, then there is a function f d : Fd -- Id such that Max fd is permissible for (Wr, Ed). 3. For each GBO (Fg, Wo, Eo) with a d-dimensional structure, there. is an index N so that the sequence of transformations (6.48)

(,F'N,WN,En+)

... -> (0,Wo,Eo)

constructed by 1 and 2 is a resolution (FN = 0). 4. The functions fd in 2 have the following properties:

-

a. If q c Fi with U < i <- N - 1, and if q ¢ Yi, then fd(q) f+1(q)

b. Yi=Maxfd forO Max fi >

> Max fN-1

c. For 0 < i < N - 1 the closed set Max f id is smooth, equidimensional, and its dimension is determined by the value of maxfd.

6. Resolution of Singularities in Characteristic Zero

96

Proof. We prove Theorem 6.34 by induction on d. First assume that d = 1. Set 11 to be the disjoint union I1 -- QxZU{oo}, where Q x Z is ordered lexicographically, and oo is the maximum of I. Suppose that (Fo, Wo, Eo) is a GBO with one-dimensional structure, and with associated closed sets F' . Define fo = t1 (tl is defined by Theorem

6.28). The closed set F is (locally) a codimension > 1 subset of a onedinmensional subvariety of Wo, so dim FO = 0. Thus R(1.)(Maxt0') # 0, and we are in the situation of 1 of Theorem 6.30. If we perform the permissible transformation with center Max to, (.F11, W1: El) -' (Yn, Wo, F o ),

we have max to > max ti. We can thus define a sequence of transformations (,To,Wo,Eo

where each transformation has center Max t , f2' = tz

: Fi1

/ i and

maxtp> >maxtl. Since there is a natural number b such that max t! E 1 Z x 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 are a totally ordered set Id-1 and functions fid-1 satisfying the conclusions of the theorem. Let Id be the disjoint union

Id =QxZUIMU{oo}, where IM is the ordered set of Definition 6.16. We order Id as follows: Q x 7Z

has the lexicographic order. If a E Q x 7L and /3 E 1,y, then 0 < a, and oc is the maximal element of I. Define Id = Id x Id-I with the lexicographic ordering. Suppose that (YOd, , Wo, Ed) is a GBO of dimension d, with associated

closed sets F 1. Define go : FF

Id by go(q) = tg(q),

and go : Max go - Id-1 by oc

9o(q) = { f'-1(q)

if q E R(1)(Maxtd), if q g R(1)(Maxto),

where JO 'd-1 is the function defined by the GBO of dimension d - 1 (7-1 WO , EOd-1)

of 2 of Theorem 6.30.

6.7. Resolution theorems for a general basic object

97

Assume that we have now inductively defined a sequence of transformations (6.49)

(.F 'd, Wr, Ed)

...

":

(.F , Wo, Eo )

and we have defined functions

gi':Maxg:-)Id-1 satisfying the assumptions 1-5 of Lemma 6.33 (with I1 = Id and 12 = Id-1) for i = 0, ... , r - 1. In particular, the center of 1rq+1 is Yz = Max g2.

If F, #0l,defineg,.:Fr ->I' by gr(q)

J r(13r)(q) if w-ordd(q) = 0,

= 1 td (q) 71

if w-ordd(q) > 0, r

where F(Br) is the function defined in Proposition 6.29 for the GBO Br = (.Fd, Wr, Ed). Define g; : Max g,. -> Iii- i by g'r(q) =

if w-ordd(q) = 0, if q E R.(1)(Maxtd) and w-ordr(q) > 0, fd-1(q) if q 0 R(1)(Maxtd) and w-ord,.(q) > 0, oc 00

where fd-' is the function defined by the GBO of dimension d - 1

(rr -1, W,

Ed-1)

in 2 of Theorem 6.30. Observe that if maxw-ordr > 0, then the center Y.z of

7ri+I in (6.49) is defined by Y = Max gi c Max td for 0 < i < r - 1. Thus (if maxw-ordr > 0) we have that. (6.49) is a sequence of the form of (6.35) of Theorem 6.30.

Now Theorem 6.30, our induction assumption, and Lemma 6.17 show that the sequence (6.49) extends uniquely to a resolution (T7N,W1V,EN)

(To,WU,EU)

with functions yy and gt satisfying assumptions 1-5 of Lemma 6.33. By Lemma 6.33, the functions (gj, gz) extend to functions

F" - Id - ldX 1d-1 satisfying the conclusions of Theorem 6.34. Consider the values of the fiuictions fd which we constructed at a point q c F. fd(q) has d coordinates, and has one of the following three types:

1. fd(q) =

(td(q),td-1(q),...,td

2. fd(q) _ (td(q), td 1(q), ... , td-'*(q),1'(q), oo, ... , oo).

3. fd(q) =

(td(q),td-1(q),...,0(q))

6. Resolution of Singularities in Characteristic Zero

98

Each coordinate is a function defined for a GBO of the corresponding dimension. We have t' (q) = (w-ord`(q),77a(q)) with w-ord%(q) > 0.

In case 1, td-r is such that q c R.(1)(Maxt`l-' ); that is, Max t` -r has coclimension 1 in a d -r dirrrerrsional GBO, and dicn(Max fd) = d - r - 1. In case 2, w-ordr1-(''+1)(q) = 0 and r is the function defined in Proposition 6.29 for a monomial GBO and dim(Max f id) = d - r - I'1 - 1, where 1,1 is the first coordinate of max 17. In case 3, dim(Max fd) = 0. Thus Max fd is non-singular and equidimensional, and 4 c follows.

f

Theorem 6.35. Suppose that d is a positive integer.

Then there are a

totally ordered set Id and functions pd with the following properties: 1. For each pair (WO, En) with d = dim WO and each ideal sheaf Jo C Olwlo there is a function g)Od :V(.10)-,Id

with the property that Maxpp C V(J0) is permissible for (Wo, Eo). 2. If a sequence of transformations of pairs with centers Y;, C V (J; )

(Wr, Er) - .. - (Wo, Eo) Id, for 0 < i < r - 1, has been defined with the property that Yz = Max pd and V(Jk) # 0, then there is I,i such that Max p`;, is permissible for a function pd : V (Jr) and functions pd : V (J=)

>

(W,., h.,.).

3. For each pair (Wo, E, o) and JO c OW;, there is an index N such that the sequence of transformations (WN, EN) - ....__, (Wo, Eo)

constructed inductively by 1 and 2 is such that V( IN) = 0. The corresponding sequence will be called a "strong principalization" of Ju.

4. Property 4 of Theorem 6.34 holds.

Proof. Let Jo = JO, and set bo - max r-o. By Theorem 6.34, there exists a resolution (WN,, (JN1, bu), EN1) - ... -, (Wo, (Jo, bo), Eo)

of the simple basic object (Wo, (Jn,bo),En). Thus we have that .1N, (which is the weak transform of ./o on Ow,,) satisfies bl - rna.x VJN < bo. We now can construct by Theorem 6.34 a resolution (WNZ,(7

2

bi),FNa)-'...- (WN,,(JN,,b1),EN,)

6.8. Resolution of singularities in characteristic zero

99

of the simple basic object (WN, (J1vl,bi), EN,). A fter constructing a finite number of resolutions of simple basic objects in this way, we have a sequence of transformations of pairs (6.50)

(WN, EN) - ... - (Wo, Eo) such that the strict transform JN of .Io on WN is JN = OW,,. We have upper semi-continuous functions

fd:Maxvj -*Id satisfying the conclusions of Theorem 6.34 on the resolution sequences (WN4+1

,

(7n;+1, bi), EN,+,) - ... -r (WNi7 (JN,, bi), Ev,)

Now apply Leuinia. 6.33 to the sequence (6.50), with Fi = V(Ji), g-i v j,, g' = f id. We conclude that there exist functions pdi : V (Ji) -> Id = Z x Id with the desired properties. O

6.8. Resolution of singularities in characteristic zero Recall our conventions on varieties in the notation part of Section 1.1.

Theorem 6.36 (Principal 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 of monodial transforms which is an isoriorphisin. away from the closed locus of points where I is not locally principal, such that 10w1 is locally principal.

Proof. We can factor I =1112, where 11 is an invertible sheaf and V(12) is the set of points where I is not locally principal. Then we apply Theorem 6.35 to JO =12. 0 Theorem 6.37 (Embedded Resolution of Singularities). Suppose that. X is an algebraic variety over a field of characteristic zero which is embedded in a non-singular variety W. Then there exists a birational projective morphism

7r:WI -'W such that n is a sequence of monodial transforms, 7r is an embedded resolution of X, and it is an isomorphism away from the singular locus of W.

Proof. Set Xo = X, Wo = W, Jo =1x0 C own. Let (WN, 1'N) - (Wo, Eo)

6. Resolution of Singularities in Characteristic Zero

100

be the strong principalization of Jo constructed in Theorem 6.35, so that the weak transform of .Io on WN is J N = OwN. With the notation of the proofs of Theorem 6.34 and Theorem 6.35, if Xo has codimension r in the d-dimensional variety Wo, then for q E Reg(Xo),

_ (1, (1, 0), ... , (1, 0), 0, ... , 0) E Id, where there are r copies of (1, 0) followed by d - r zeros. The function pod is thus constant on the non-empty open set Reg(Xo) of non-singular points Pod(q)

of X0. Let this constant be c c Id. By property 4 of Theorem 6.34 there exists a unique index r < N - 1 such that max pd = c. Since W,. - Wo is an isomorphism over the dense open set Reg(Xn), the strict transform Xr of X0 must be the union of the irreducible components of the closed set Max pT of W. Since Max pd is a permissible center, Xr is non-singular and makes simple normal crossings with the exceptional divisor E.

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

7r: X1 -'X such that 7r is a projective morphism which is an isomorphism away from the singular locus of X. Proof. This is immediate from Theorem 6.37, after choosing an embedding of X into a projective space W.

Theorem 6.39 (Resolution of Indeterminacy). Suppose that K is a field of characteristic zero and (h : W - V is a rational map of projective Kvarieties. Then there exist a projective birational morphism 7r : W1 -+ W such that W1 is non-singular and a morphism A : W1 --4V such that \ _ 0 o 7r. If W is non-singular, then 7r is a product of m.onoidal transforms.

Proof. By Theorem 6.38, there exists a resolution of singularities 0 : W1 -' W. After replacing W with W1 and 0 with V) o 0, we may assume that W is non-singular. Let I'(fi be the graph of 0, with projections 7rl : I',f, W and 7r2 : I'O - V. As 7r1 is birational and projective, there exists an ideal

sheaf I C Ow such that ro = B(I) is the blow-up of I (Theorem 4.5). By Theorem 6.36, there exists a principalization in : W1 -4W of Z. The universal property of blowing up (Theorem 4.2) now shows that there is a morphism A : W1 - V such that A _ 0 o 7r.

Theorem 6.40. Suppose that in : Y -* X is a birational morphism of projective non-singular varieties over a field K of characteristic 0. Then

IIi(Y, OY) = H, (X, Ox) V i.

6.8. Resolution of singularities in characteristic zero

101

Proof. By Theorem 6.39, there exists a projective morphism f : Z - Y such that g - 7r o f is a product of blow-ups of non-singular subvarieties, 9n 9n-1 91 r .92 ...Z194Ze=X. We have ([65] or Lemma 2.1 [311)

i>0,

0,

Rzgj*Oz,

Oz,- i=0.

Thus, by the Leray spectral sequence, (651) .

R` g* Oz

=

ifi>0,

10,

ifi-0

Oa

,

and

9* :

II:(X,OX)

Ht(Z,Oz)

for all i. Since ,q* = f * o it*, we conclude that ir* is one-to-one. To show that lr* is an isomorphism we now only need to show that f* is also one-to-one.

Theorem 6.40 also gives a projective morphism y : W -a Z such that, ,0 = f o ry is a product of blow-ups of non-singular subvarieties, so we have

0*: Hi(y,Or)

H2(W,Ow) for all i. This implies that f * is one-to-one, and the theorem is proved.

Cl

For the following theorem, which is stronger than Theorem 6.38, we give a complete proof only the case of a hypersurface. In the embedded resolution constructed in Theorem 6.37, which induces the resolution of 6.38, if X is not a hypersurface, there may be monodial transforms Wt+1 -- Wz whose centers are not contained in the strict transform Xi of X, so that the induced

morphism X;+1 -> X% of strict transforms of X will i11 general not be a monoclial transform.

Theorem 6.41 (A Stronger Theorem of Resolution of Singularities). Suppose that X is an algebraic variety over a field of characteristic: zero. Then there is a resolution of singularities

ir:Xl ->X such that it is a sequence of monodial transforms centered in the closed sets of points of maximum multiplicity.

Proof. In the case of a hypersurface X = X0 embedded in a non-singular variety W = WO, the proof given for Theorem 6.37 actually produces a resolution of the kind asserted by this theorem, since the resolution sequence is constructed by patching together resolutions of the simple basic objects (Wi, (ii, b), Ei), where Ji is the weak transform of the ideal sheaf JO of Xo in Wo. Since Jo is a locally principal ideal, and each transformation in

102

6. Resolution of Singularities in Characteristic Zero

these sequences is centered at a non-singular subvariety of Maxa;, the weak transform Jt is actually the strict transform of J0. For the case of a general variety X, we choose an embedding of X in

a projective space W. We then modify the above proof by considering the Hilbert-Samuel function of X. It can be shown that there is a simple basic object (Wc, (Ko, b), Eo) such that Max UKp is the maximal locus of the Hilbert-Samuel function. In the construction of pd of Theorem 6.35 we use vjc, in place of rrti. Then the resolution of this basic object induces a sequence of monodial transformations of X such that the maximum of the Iilbert-Samuel function has dropped. More details about this process are given, for instance, in [55], [14] and [79].

Exercise 6.42. 1. Follow the algorithm of this chapter to construct an embedded resolution of singularities of: a. The singular curve y2 - x3 = 0 in A2. b. The singular surface z2 - x2;y3 = 0 in A3. c. The singular curve z - 0, y2 x3 = 0 in A3.

-

2. Give examples where the f d achieve the 3 cases of the proof of Theorem 6.34. 3. Suppose that K is a field of characteristic zero and X is an integral

finite type K-scheme. Prove that there exists a proper birational morphism 7r: Y -4 X such that Y is non-singular. 4. Suppose that K is a field of characteristic zero and X is a reduced finite type K-scheme. Prove that there exists a proper birational morphisrn 7r : Y X such that Y is non-singular. 5. Let notation be as in the statement of Theorem 6.35. a. Suppose that q E V(Jo) C Wo is a closed point. Let U spec(Owo,q), Dt = DinU for 1 < i < r and E = {Dl, ... , Dr} Consider the function pg of Theorem 6.35. Show that po(q) depends only on the local basic object (U, E) and the ideal (.In)q C OW,,,q. In particular, pp (q) is independent of the Kalgebra isomorphism of OX,q which takes JOOX,q to .IoOX,q

andDztoDi for1
6.8. Resolution of singularities in characteristic zero

103

0('rl (q)) = 7rI (O(q)) for all q E Wl and O(D) = D if D is the exceptional divisor of in. 6. Suppose that (Wo,En) is a pair and Jo C Oyyo is an ideal sheaf, with notation as in Theorem 6.35. Consider the sequence in 2 of Theorem 6.35. Suppose that G is a group of K-automorphisms of WO such that O(Dh) = D; for 1 < i < r and O*(Jo) = JO. Show that G extends to a group of K-automorphisms of each W;, in the sequence 2 of Theorem 6.35 such that O(Yi) - Yz for all O E G and each 7rn is G-equivarianl.. That is, it (E)(q)) = e(7ri(q))

for all q E W. Furthermore, the functions p4 of Theorem 6.35 satisfy pd(O(q)) = pd(q) for all q E W. 7. (Equivariant, resolution of singularities) Suppose that X is a variety over a field K of characteristic zero. Suppose that. G is a group of

K-automorphisms of X. Show that there exists an equivariant resolution of singularities in : XI - X of X. That is, G extends to a group of K-automorphisms of X so that 7r(O(q)) = E)(7r(q)) for all q c X1. More generally, deduce equivariant versions of all the other theorems of Section 6.8, and of parts 3 and 4 of Exercise 6.42).

Chapter 7

Resolution of Surfaces in Positive Characteristic

7.1. Resolution and some invariants In this chapter we prove resolution of singularities for surfaces over an algebraically closed field of characteristic p > 0. The characteristic U proof in Section 5.1 depended on the existence of hypersurfaces of maximal contact, which is false in positive characteristic (see Exercise 7.35 at the end of this section). The algorithm of this chapter is sketched by Hironaka in his notes [53]. We prove the following theorem: Theorem 7.1. Suppose that S is a surface over an algebraically closed field K of characteristic: p > 0. Then there exists a resolution of singularities

A:T-S. Theorem 7.1 in the case when S is a hypersurface follows from induction on r = max{t I Singt(S) 0} in Theorems 7.6, 7.7 and 7.8. We then obtain Theorem 7.1 in general from Theorem 4.9 and Theorem 4.11.

For the remainder of this chapter we will assume that K is an algebraically closed field of characteristic p > 0, V is a non-singular 3-dimensional

variety over K, and S is a surface in V. It follows from Theorem A.19 that for t F N, the set Singt(S) = {p E S I vp(S) > t} is Zariski closed. 105

7. Resolution of Surfaces in Positive Characteristic

106

Let

r - rnax{t I Singi(S) 0 0}. Suppose that p E Siiigr(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 6yp). There is an expansion

f=

aijkx!?l"zk

i+j+k>r

with oxjk E Kin Ov,p - K[[x, y, z]]. The leading form of f is defined to be L(x, y, z) =

37 ai jkxi yjzk.

i+j+k-r We define a new invariant, r(p), to be the dimension of the smallest linear subspace M of the K-subspace spanned by x, y and z in K[x, y, z] such that L E k[M]. This subspace is uniquely determined. This dimension is i n fact, independent of the choice of regular parameters (x, y, z) at p (or in o,,,,). If T, y, z are regular parameters in Ovp, we will call the subvariety N = V(M) of spec(Ov,p) an "approximate manifold" to S at p. If (x, y, z) are regular parameters in Ov,p, we call N = V (M) c spec(nv,p) a (formal) approximate manifold to S at p. M is dependent on our choice of regular parameters at p. Observe that 1 < T(q) < 3.

If there is a non-singular curve C C Singr(S) such that q E C, then r(q) < 2.

Example 7.2. Let f = x2 + y2 + z2 + a:5. 11 char(K) 74 2, then r = 3 and a: - y = z = 0 are local equations of the approximate manifold at the origin. However, if char(K) = 2, then T =1 and x -F y I- z = 0 is a local equation of an approximate inani10ltl at the origin.

Example 7.3. Let f = y2 + 2xy + x2 + z2 + z 5, char(K) > 2. Then

L=(y+x)2+z2, so that r = 2 and x + y = z = 0 are local equations of an approximate manifold at. the origin.

Lemma 7.4. Suppose that q c S is a closed point such that vq(S) = r, r(q) = n (with 1 < rt. < 3), and Ll = = L,, = 0 are local equations at q of a (possibly formal) approximate manifold N of S at q. Let 7r : V1 V be the blow-up of q, E _ 7r-1(q) the exceptional divisor of 7r, S1 the strict transform of S on V1, and N1 the strict transform of N on V1. Suppose that ql E it-1(q). Then vq, (Sl) < vq(S), and if vq, (S,) = r, then r(q) < 7-(q,)

and q, c N, n E. Moreover, N1 n E = 0 if r(q) = 3, Nl n E is a point if r(q) = 2, and N1 n E is a line if r(q) = 1.

7.1. Resolution and some invariants

107

Proof. Let f = 0 be a local equation of S at q, and let (x, y, z) be regular parameters at p such that

1. x=y=z=0 are local equations of N if r(q) = 3, 2. ,y = z = 0 are local equations of N if r(q) = 2, 3. z = 0 is a local equation of N if T(q) = 1. In dv,p = K[[x, ,y, z]], write

f=

aijkxiylzk =L+C,

where L is the leading forlrl of degree r, and C has order > r. qi has regular parameters (xl, yi, zi) such that one of the following cases holds:

1. x = xi, y = xi (yi + a), z = xi (zi +,(3) with a, /3 E K. 2. x = xiyi, y = yl, z _ yi(zi + f3) with f K.

3. x = xizi, y = ylzi, z = zl. Suppose that the first case holds. S1 has a local equation fi = 0 at q1 such that

fi = L(1,yi +rr,zl +0)+x1S1. Thus vq, (f i) < r, and vq, (fl) = r implies

L(1, yi + a, zi + A) _

aijk(y1 + (k)' (Z] + /3)k = > bjkyi zi i+j+k=r

j+k=r

for some bjk E k. Thus, the equality

L(1, x, x) = E (x - cx)j(. - /3)k j+k=r implies V (y-ax, z-(3x) C N. We conclude that r(q) < 2 if vq, (Sl)

vq(S).

Suppose that r(q) = 2 and vq, (Sl) = vq(S). Then we must have a = 13 = 0, so that q1 E N1. We further have

fi = L(yi, zi) + xi Ih, so that T(qi) > 7-(q)If r(q) = 1, then L = czr for some (non-zero) c E K, and we must have (3 = 0, so that qi E N1. We further have

fi = L(zi) + x11, so that T(ql)

T(q).

There is a similar analysis in cases 2 and 3.

Lemma 7.5. Suppose that C C Sing, (S) is a non-singular curve and q E C is a closed point. Let 7r : V1 -> V be the blow-up of C, E = 7r-'(C) the exceptional divisor, and S1 the strict transform of S on V1. Suppose that

7. Resolution of Surfaces in Positive Characteristic

108

(x. Y, z) are regular parameters in OV,q (or Ov,q) such that x = y = 0 are local equations of C at q, and x = 0 or x = y = 0 are local equations of a (possible formal) approximate manifold N of S at q. Let Ni be the strict transform of N on V1. Suppose that qi E it-i(q). Then vgl(Si) < r, and if vq, (Si) = r, then r(q) < r(ql) and qi c Ni n E.

Furthermore, N1 n E = 0 if r(q) = 2, and N1 n E is a curve which maps isomorphically onto spec(Or,q) (or slpec(Oc,q) in the case where N is

formal) if r(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 Sing,.(Sj),

V. - V.-1 where Si is the strict transform of S on are non-singular.

V,

so that all curves in Sing,.(S,,,)

Proof. This is possible by Corollary 4.4 and since (by Lemma 7.4) the blow-up of a point in Sing,.(S2) can introduce at most one new curve into Sing,.(Si+i), which must be non-singular.

Theorem 7.7. Suppose that all curves in Sing,.(S) are non-singular, and

... J,Vn -+Vn-i -...-V is a sequence of blow-ups of non-singular curves in Sing,.(Si) where Si is the strict transform of S on Vi. Then Sing,.(Si) is a union of non-singular curves and a finite number of points for all i. Further, this sequence is finite. That is, there exists an n such that Sing,.(S.) is a finite set. Proof. If rr : Vi --> V is the blow-up of the non-singular curve C in Sillg,.(S),

and C1 C Sing,.(Si) n 7r-1(C) is a curve, then by Lemma 7.5, Ci is nonsingular and maps isomorphically onto C. Furthermore, the strict transform of a non-singular curve must be non-singular, so Singr(.Si) nest, be a union of non-singular curves and a finite number of points for all i. Suppose that we can construct an infinite sequence

...-+ Vt -pV i by blowing up non-singular curves in Sing,.(S1). Then there exist curves Ci C Sing,.(Vi) for all i such that Ci+l -i Ci are dominant morphisms for all i, and

=r

vOve.al,

for all i. We have an infinite sequence (7.1)

Ovc-'Ov,.c,

--'...-Ov,,,c..-...

7.2. r(q) = 2

109

of local rings, where, after possibly reindexing to remove the maps which are the identity, each homomorphism is a localization of the blow-up of the maximal ideal of a 2-dimensional regular local ring. As explained in the characteristic zero proof of resolution of surface singularities (Lemma 5.10), we can view the OS,,c, as local rings of a closed point of a curve on a nonsingular surface defined over a field of transcendence degree 1 over K. By Theorem 3.15 (or Corollary 4.4), vpv

c,,

(Is ,Cn) < r

0

for large n, a contradiction. We can now state the main resolution theorem.

Theorem 7.8. Suppose that Sing,.(S) is a finite set. Consider a sequence of blow-ups V,

(7.2)

where V+1

Vi is the blow-up of a curve in Sing,.(Si) if such a curve exists,

and V.i+i -' Vi is the blow-up of a point in Sing,.(Sj) otherwise. Let Si be the strict transform of S on Vi. Then Sing,.(Si) is a union of non-singular curves and a finite number of points for all i. Further, this sequence is finite. That is, there exists Vb such that 0. Theorem 7.8 is an immediate consequence of TheoreTri 7.9 below, and Theorems 7.6 and 7.7.

Theorem 7.9. Let the assumptions be as in the statement of Theorem 7.8, and suppose that q E Sing, (S) is a closed point. Then there exists an n such that all closed points q,, E S,4 such that q,4 maps to q and Vgn(Sn) = r satisfy

r(qn) > r(q)-

The proof of this theorem is very simple if r(q) = 3. By Lemma 7.4 the multiplicity must drop at all points above q in the blow-up of q. We will prove Theorem 7.9 in the case r(q) = 2 in Section 7.2 and in the case r(q) = I in Section 7.3. 7.2. -r(q) = 2

In this section we prove Theorem 7.9 in the case that r(q) = 2.

Suppose that r(q) = 2, and that there exists a sequence (7.2) of infinite length. We can then choose an infinite sequence of closed points qi f Sing,.(Si) such that qi+i maps to qj for all i. By Lemma 7.4, we have T(qj) = 2 l'ar all i, and each qi is isolated in Sing,.(Si).

Thus each map Vi+1 -> V is either the blow-up of the unique point qj E Sing,,(Sj) which maps to q, or it is an isomorphism over q. After

7. Resolution of Surfaces in Positive Characteristic

110

reindexing, we may assume that each map is the blow-up of a point qj such that qi+i maps to qj. for all i. Let Ri = OV,gti. We have a sequence

&

(7.3)

---

R1 - ... - Ri m ...

of infinite length, where each Ri+l is the completion of a local ring of a closed

point of the blow-up of the maximal ideal of spec(Rj), and vgi(Si) = r and T(qi) = 2 for all i. We will derive a contradiction. Since T(q) = 2, there exist regular parameters (x, y, z) in Ro such that a local equation f = 0 of S in R0 has the form

Vz k aijkX

f= i+j+k>r

and the leading form of f is aijoxsyj.

L(r, y)

For an arbitrary series

9.

bijkxY

xk

ERA,

define

- (k,+j) I bijk # 0 and i -F j < r} E T N U {oo}

7xyz(g) = mill{r

with the convention that, 7xyz(g) = oo if and only if biik = 0 whenever i + j < r. We have yxyz (,q) < 1 if and only if vim, (y) < r. l1i rt hermore, 7wyz(9) = oo if and only if g c (x, y)r. Set y = -ynz(g). Define bijkXiyjzk.

[9}xyz = (i+j )"Y+k-r7

't'here is an expansion (7.5)

bijkx2y3zk.

9 = [91xyz + (i+j )'Y+k >rry

Returning to our series f, which is a local equation of S, set y = 7xyz(f ), and

TY = {(i, j)

k

+j)

= 7, i + j < r, for some k such that aijk # 0}.

Then (7.6)

[fIxyz - 1L + E ai,j,'Y(r-.z j)x

y7zy(r-i-j)

(i,j)ET.

Definition 7.10. [ f ]xyz is solvable if 7 E N and there exists a, ,O E K such

that [f]xyz = L(x - ciz-1, y - fz-f ).

7.2. r(q) = 2

111

Lemma 7.11. There exists a change of variables n

n

1:1 - :r: -

OjZz,

13-i z

i=1

i=1

such that

i

yt _ y -

is not solvable.

Remark 7.12. We. can always choose (x, y, z) t hat are regular parameters in Ovo,go Then (x, y, z) are also regular parameters in Ovo,go However, since we are only blowing up maximal ideals in (7.2), we may work with any formal system oI' regular parameters (a:, y, z) in Rc (see Lernrna 5.3).

Proof. We prove 7.11. [f there doesn't, exist such a change of variables, then there exists an infinite sequence of changes of variables for n E N

xn = x -

aiz^f`,

i-i

yn -

> 13i z-11

i-1

such that yn E N for all n, yn+1 =

(f ), and yn+i > yn for all n. Let

00

xoo = x -1

aiz1

,

yoc = y - >2 biz

Thus NO = YX.,yxx,X(f) = oC and 00

f E (x -

00

ENizt)r aiz', V -

C RU.

But by construction of VO, p is isolated in Sing, (S). If I C Ovq is the reduced

ideal defining Sing,.(S) at q, then by Remark A.21, I = IRS is the reduced ideal whose support is the locus where f has multiplicity r in spec(Ro). Thus (x, y, z) = I C (x - > aizi, y - E,3izi), which is impossible. We may thus assume that [ f ] yy,r is not solvable. Since the leading form

of f is L(x, y) and r(p) = 2, Ri has regular parameters (xi, yi, zi) defined by

x=x121,

(7.7)

y=ytzt, z=zl.

Let f, = f = 0 be a local equation of the strict transform of f in R1.

Lemma 7.13. 1. 7xiy1 z(fi) = 'Yayz(f) - 1.

2. [ f i]x, y,

z = z [f 1:ryzf so [f1 .yz not solvable implies [ f i]xy,z not solv-

able.

3. If7x,y,,(f1) > 1, then the leading formof f1 is L(xi,yi) = z L(x,y).

7. Resolution of Surfaces in Positive Characteristic

112

Proof. The lemma follows from substitution of (7.7) in (7.4), (7.5) and (7.6).

We claim that the case yxl,y,,x(fl) = 1 cannot occur (with our assump-

tion that vq, (Sl) = r and r(qi) = 2). Suppose that it does. Then the leading form of fi is [fi]xiyiz, and there exist a form W of degree r and a, b, c, d, e, 7 E K such that

[fi]x-,ylz = W(axi + byi + cz, dxi + 41 + f z). We have an expression

L(xi, yi) + z!Q = W(axi + byi + rz, dx, + ey, 1- 7z)

so that L(x, y) = W(ax + by, dx + ey). We have ae - bd 0, since -r(q) = 2. By Lemma 7.14 below, solvable, a contradiction to 2 of Lenima 7.13.

is thus

Since y;,;lvlz(fi) < 1 implies vk,(fi) < r, it now follows from Lemma Ri, where Ri has regular

7.13 that after a finite sequence of blow-ups Rp parameters (xi,yi, zi) with

x - xizi, y =yizi, z=zi. By 3 of Lemma 7.13, we reach a reduction vq; (S,) < r ar vq, (Si) = r, r(qi) = 3, a contradiction to our assumption that (7.3) has infinite length. Thus Theorem 7.9 has been proven when r(q) = 2. Lemma 7.14. Let ob = -,D(x, y), T = W (u, v) be forms of degree r over K and suppose that 4) (x,y) = 4(ax+by,dx+ey) for some c E K with ae - bd # 0. Then for all a, f e K there exist a,,3 E K such that

(:x I az,y+Oz) _ %F (ax+by+cz,dx+ey+ fz).

Proof. Indeed, (x + az, y +.3z) _ T (a (x + cm) + b(y + /)z), d(x + az) + e(y +,8z)) 'l

x+by-I (aa+bf(3)z,dx+ey+(da+efi)z),

so we need only solve

da+e,3= f for a, fi.

113

7.3. T(q) = 1

7.3. r(q) = 1

In this section we prove Theorem 7.9 in the ease that T(q) = 1. We now assume that r(q) = 1. Suppose that there is a sequence (7.2) which does not terminate. We can then choose a sequence of points q E V, with qo = q, such that q for all it and r, T(q,,) = 1 for all it. After possibly reindexing, we may assume that for all it, qt is on

the subvariety of Vn which is blown up under Vn+i - V. Let R = 6,,,q for n > 0. We then have an infinite sequence (7.8)

Let. Ji C C')V,,g; be the reduced ideal defining Sing,,.(Si) at qj. Then by Lemmas 7.4 and 7.5, Ji is either the maximal ideal mi of or a regular

height 2 prime ideal pi in °y;,q;. Then the multiplicity r locus of 65,,q., (which is a quotient of Ri by a principal ideal) is defined by the reduced ideal .Ix = JjRj, by Remark A.21. We have Ji = 7/Li = rrtiRi or .. %i = p; = piRi, a regular prime in Ri. By the construction of the sequence (7.2), we see that. OV;+i,q,+i 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+l is the completion of a local ring of the blow-up of nti or p)i. We are thus free to work with formal parameters and equations (which

define the ideal Zs,,q; = TS,,g1I i) in the Ri, since the ideals ani and pi are determined in Ri by the i nltiplicy r locus of Singr(Os;,q,). Suppose that T = K[[x, y, z]j is a power series ring, i E N and q

= T, bijkxi.yazk

E T.

We can construct a polygon in the following way. Define

A(,q;X,y;Z)={(

t

3

r-k'r-k ) E

Q2 I k < r and bijk LO}.

Let JAI be the smallest convex set in 1W2 such that, A C IAI and (a, b) E JAI

implies (a + c, b + d) E IDI for all c, d > 0. For a E R, let S(a) be the line through (a, 0) with slope -1, and V (a) the vertical line through (a, 0). Suppose that JAI 4 0. We define nxy,z(g) to be the smallest a appearing in any (a., b) E IDI, and f3,,,,,,(g) to be the smallest b such that (cxyz(g), b) E IAI.

Let yxyz(9) be the smallest number y such that S(y)nIAI 4- 0 and let oxyz(g) be such that (yxyz(9)-oxyx(9), axyz(9)) is the lowest point on S(yxyz(9))nI AI Then (r.Xxya(9), J3Ty.(g)) and (-y--,, (g) - axyz(9), oxyz(9)) are vertices of IAI.

7. Resolution of Surfaces in Positive Characteristic

114

Define Fa,yy(g) to he the absolute value of the largest slope of a line through (axyx(g),,3xyz(g)) such that no points of IA(g; x, y, z) I lie below it.

Lemma 7.15. 1. The vertices of I A I are points of A, which lie on the lattice T,7Zx T,Z.

2. ,R(g) < r holds if and only if JAI contains a point on S(c) with c < 1, which in turn holds if and only if there is a verte:r: (a, 1)) with

a+b<1.

3. ayz(_q) < 1 if and only if g ¢ (:r., z)r. 4. A vertex of I A I lies below the line b = 1 if and only if g 9' (y, Proof. Lemma 7.15 follows directly from the definitions.

Suppose that g t- T is such that vT(g) - r. Let. bi jkxi

L=

zk

iIjIk=r be the leading form of g. (x, y, z) will be called good parameters for g if boor = 1.

We now suppose that vT(g) = r, and (:r, y, z) are good parameters for g. Let. 0 = 0(g; :r., y, z) and suppose that (a, b) is a vertex of JAI. Let

S(a,b)={kI

(r'k'r't k ) = (a, b) and bipc

0}.

Define tnh

zr +

ba(r-k),h(r-k),kxa

(r k) b(r - k)zk. ?/

kES(a,b)

We will say that {g},yz is solvable (or that the vertex (a,, h) is not prepared on JA (g; x, y, z)1) if a, b are integers and {g}ah2 = (z

- 7)xayb)r

K. We will say that, the vertex (a, b) is prepared on if {g}a y,z is not solvable. If {g}xyz is solvable, we can make an (a, b) preparation, which is the change of parameters for some 71

s`

i (g; x, y, z) l

zt = z - >)xa yh.

If all vertices (a, b) of JAI are prepared, then we say that (g; x, y, z) is well prepared.

Lemma 7.16. Suppose that 1' is a power se7 ies ring over K, g E T is such that YT(g) = r, T(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.

7.3. 'r(q) = 1

115

Proof. vr(,q) = r implies all vertices (a, b) of I A(g; x, y, z) I satisfy a + b > 1. The leading form of g is

L = E bijkxi y2 zk, i+j+k=r

where boor = 1. r(y) = 1 implies there exists a linear form ax + by + cz such

that (ax + by + cz)r = I..

As boa,. = 1, we must have c = 1. If a # 0, then (1,0) is a vertex of IL(g; x, y, z) 1, which is not prepared since {g} (1y'0) _ (z + ax)r.

Thus a = 0. If b # 0, then (0,1) is a vertex of JA(g; x, y, z) 1, which is not prepared since

(z+by)'

.

Thus L = zr, and z = 0 is an approximate manifold of g = 0.

Lemma 7.17. Consider the terms in the expansion k

(7.9)

h

A, -,\ (k) 1

77

a-0 obtained by substituting zl = z - i?x°yb into the monomial xiyizk. Define a projection for (a, b, c) E N3 such that c < r and a , b

ir(a,b,c) _ (

r-c r-c

Then:

1. Suppose that k < r. Then the exponents of monomials in (7.9) with non-zero coefficients project into the line segment joining (a, b) to i JL (r k' r -k then all these mono7niaGs project to a. If (a, b) , (a, b).

b. If (a, b) # (rk

then xiyJZi is the unique monomial in No monomial in (7.9) (7.9) which projects onto projects to (a,b). 2. Suppose that r < k, and (i, j, k) j4 (0, 0, r). Then all exponents in (7.9) with non-zero coefficients and z1 exponent less that r project into ((a,b) + Q 0) - {(a,b)}. 3. Suppose that (i, j, k:) = (0, 0, r). Then all exponents in (7.9) with non-zero coefficients and zl exponent less than r project to (a, b).

7. Resolution of Surfaces in Positive Characteristic

116

Proof. If k .L r, we have

_

T1 j+(k-A)b

r-

a + r) r ki - aJ1

= b + r-k

b

.

Suppose that. k < r. Since 0 < A < k, we have 0 < rk < 1. Thus

i+(k-A)a j+(k-A)b r-A

r-A

and 1 follows.

is on the line segment joining (a, b) to

Ifk>rand0
k:.

Suppose that k = r and 0 < A < r. Then

(i+(k_9if(k_A)by...( ir-Ar r -A

-A+n'r-A b

and thelast case of 2 and 3 follow. We deduce from this lemma that

Lemma 7.18. Suppose that zi = z - zlx" yb is an (a, b) preparation. Then: 1. JA(g; x, y, z1)) C I A(g; x, y, z)I - ((a, b)}. 2. If (a.', b') is another vertex of JA(y; x, y, z)I, then (a', 1/) is a verle:e {,q}.,j zr is obtained from {g}x,y by substiof IA(g; x, y, z1) I and tuting z1 for z.

Example 7.19. It is not always possible to well prepare after a finite number of vertex 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 I A(g; x, y, z) I are (0,1 + ), (1, r) and (m, 0). We can remove the vertex (zn, 0) by the (m, 0) preparation zl = z - x". 11

Thus

g = y(y - x)(y - 2x) ... (y - rx) -I- (z - 1yXr = y(y - x)(y - 2x)...(y - rx) + (z - x x3 can only be well prepared by the formal substitution W

z1 = z -

I(-1)ixi. i=2

Lemma 7.20. Suppose that ,q is reduced, vT(q) = r, r(q) = 1 and (x, y, z) are good parameters for g. Then there is a formal series O(x, y) E K[[x, y]] such that under the substitution z = z1 + O(x, y), (x, y, z1) are good parranaeters for 9 and (q; x, y, z1) is well prepared.

7.3. r(q) = 1

117

Proof. Let vL be the lowest vertex of IA(g; x, y, z)I. Let h(VL) be the second coordinate of VL. Set b = h(vt). If vL is not prepared, make a vT, preparation zl = z - yxayb (where (a, b) = vL) to remove VL in IA(g; x, y, zl)I. Let vL, be the lowest vertex of I A (g; x, y, zl) 1. If VL, is not prepared

and h(vL,) _ h(VL), we can again prepare 1L, by a vL1 preparation z2 = zl - yi xa' yb. We can iterate this procedure to either achieve I0(g; x, y, such that the lowest vertex vL is solvable or h(vL,,) > h(vL), or we can construct an infinite sequence of vL,, preparations (with b k r a.ll n) I

« b

where (an, b) = VL,,, such that the lowest vertex vLn of I A(q; x, y, zn)I is not prepared and h(vr,) for all n. Since an+1 > an, for all n, we can then make the formal substitution z' -- z - E°_°e r1zxa. yb, to get I A (g; x, y, z') I whose lowest vertex VL' satisfies h(VL') > h(VL).

such that if we set z' - z -

In summary, there exists a series

yh(VL)t'(x), and vL' is the lowest vertex of IA (g; x, y,z')I, then either VL' is prepared or h(UL') > h(vt,).

By iterating this procedure, we construct a series

y) such that if

x = z - 4'(x, y), then either the lowest vertex v, of Io(g; x, y, z) I is prepared, or IA(y;.xr,'y, )I - 0. This last. case only occurs if g = unit Zr, a11d thus cannot occur since g is assumed to be reduced. We can thus assume that vL is prepared. We can now apply the same procedure to vT, the highest vertex of Io(g; x, y, z)I, to redlu ce to the case. where v`1' and VL are both prepared. Then after a finite number of preparations we find a change of variables z' = z - -P (x, y) such that (g; x, y, z') is well prepared. We will also consider change of variables of the form

Yl =y - 7]xn for y E K, n a positive integer, which we will call translations.

Lemma 7.21. Consider the expansion (7.10)

h=E A_n

(.? 1 x:+(,i-A)ignzk W

obtained by substituting yi = y - nxn into the monomial xayjz". Consider the projection for (a, b, c) E N3 with c < r defined by

ir(a,b,c)=(

a

,

b

r-c r-c

).

7. Resolution of Surfaces in Positive Characteristic

118

Suppose that k < r. Set (a, b) = (,-

). Then xiylzk is the unique monomial in (7.10) whose exponents project onto (a, b). All other monomials ,

in (7.1 0) with non-zero coefficient project to points below (a, b) on the line through (a, b) with slope - n .

Proof. The slope of the line through the points (a, b) and ri+(j-A)n

A1

l r-k 'r - k/J is

j-A

1

i-(i+(j-.1)n) -n' Since i'

<

for A < j, the lemma follows.

Definition 7.22. Suppose that ,q E T is reduced, vT(g) = r, r(g) = 1 and (x, y, z) are good parameters for g. Let a = axyz(g), 13 = 13xyz(q), y = yzy.z(g), rS = 6.y. (g), e = exyz(g). Then (g; x, y, z) will be said to be very well prepared if it is well prepared and one of the following conditions holds.

1. (y - 6,6) ,-E (a,#), and if we make a translation yl = y - rix, with subsequent well preparation zl = z - -t(x, y), then axylzi (g) = a,

Qxy'zt (g) = /3,

yayizi(g) = y

and

bxyizi (g) < 6-

2. (y - 6,6)

and one of the following cases holds:

a. a=0. h. c 0 and is not an integer. c. c 0 and n = E is a (positive) integer and far any j E K, if yi = y - rIx'a is a translation, with subsequent well preparation zl = z - 4) (x, y), then exyjz(g) -- e. Further, if (c, d) is the lowest point on the line through (a, /3) with slope -c in IA(g; x, y, zl)I and (cl, dl) is the lowest point on this line in IA (g; x, yl, zi)I, then dl < d.

Lemma 7.23. Suppose that y E T is reduced, vp(g) = r, 'r(g) = 1 and (x, y, z) are good parameters for g. Then there are formal substitutions

zl = z - ¢(x, y), yl = y - O(x), where O(x, y), i(x) are series such that (g; x, yi , zl) is very well prepared. Proof. By Lemma 7.20, we may suppose that (g; x, y, z) is well prepared. Let a = ayz(g), N = /xyz(g), y = yayz(g), 8 = Sxyz(g), e = c (g). Let (y - t, t) be the highest vertex on I A x, y, z) I on the line a + b = y. Let

119

7.3. r(q) = 1

17 E K and consider the translation y1 = y - 27x. By Lemma 7.21, (ry - t, t) and (a, /J) are prepared vertices of J0(g; x, y1, z) 1. Thus if z1 = z - D(x, y) is a well preparation of (g; x, y1i z), then ('y - 1., 1.) and (a,;3) are vertices of Wg; x, y1, zl)I

We can thus choose ?1 E K so that after the translation yl = y - yx, and a subsequent well preparation z1 = z - 4)(x, y), we have cr., z, > b is a maximum for substitutions :?1 zl = ^l and y1 = y - 71x, z1 = z - fi(x, y) as above. il' ('y - 5xy1z1, 4ryiz1) (a, Q), then after replacing (x, y, z) with (x, yl, z1), we have achieved condition 1 of the lemma, so that (g; x, y, z) is very well prepared. We now assume that (g; x, y, z) is well prepared, and ((r, 1) - (y - 6.6). If c = 0 or is not an integer, then (g; x, y, z) is very well prepared. Suppose that n = is an integer. We then choose y E K such that with the translation y1 -,y - 71:0, and subsequent well preparation, we maximize d for points (c, d) of the line through (a, /3) with slope -e on the boundary of I0(f ; x, yi i zl)1. By Lemma 7.21 a, 3 and -y are not changed. If we now have d ,3, we. are very well prepared. If this process does not end after a finite number of iterations, then there exists an infinite sequence of changes of variables (translations followed by well preparations) yi = Yi+1 + r1i+1X " ',

zi = zi+1 + i+l (x, yi+1)

such that ni+1 > ni for all i. Given n E N, there exists a(n) E N such that u(n) > a(n - 1) for all n, and i > v(n) implies all vertices (a, b) of IA(g; x, y,, zi)I below ((Y, /3) have a > n. This last condition follows since all

vertices must lie in the lattice Z x T Z, and there are only finitely many points common to this lattice and the region 0 < a < n, 0 < b < 03. Thus xn I ii (x, yi) if i > a(n). Let

-

'Pi(x, Y) = E Oj j=1

71ky-,,.k);

(x, y -

k=1

so that zi. = z - tPi(x,, y) for all i. (Ti(x, y)} is a Cauchy sequence in K[[x, y]]. Thus z' = z - E°_1 ¢i(x, yY) is a well defined series in K[[x, y, z]]. Set y' = y 77i x'°. Then I A(g; x, y', z') I has the single vertex (a,,3), and (g; x, y', z') is thus very well prepared.

Definition 7.24. Suppose that g E T, g is reduced, vT(g) = r, T(g) = 1, and (x, y, z) are good parameters for g. We consider 4 types of monodial transforms T -* T1, where T is the completion of the local ring of a monoidal

transform of T, and T1 has regular parameters (xi, yl, z1) related to the regular parameters (x, y, z) of T by one of the following rules.

7. Resolution of Surfaces in Positive Characteristic

120

T1. Sing,.(g) =V(x,y,z):

x=x1, y=xi(y1+71),

z = x1 z1,

with rt E K. Then g1 =

is the strict transform of g in T1, and if vT1(g1) = r and r(g1) = 1, then (x1, y1 i z1) are good parameters for 91.

T2. Sing,. (g) = V (X' y, z), x = x1y1,

y - y1,

z = yiz1

Then g1 = jr is the strict transform of ,q in T1, and if v7) (g1) = r and r(y1) = 1, then (x1, y1, z1) are good parameters for g1. T3. Sing,.(q) = V(x, z),

x=x1, '/=y1, z=x171. Then 91 = is the strict transform of ,q in T1, and if VT, (g1) = r arid r(g1) = 1, then (x1, yl, z1) are good parameters for 91. T4. Sing, (g) = V (,y, z),

x=x1, y=y1, z=yIz1 Then y1 = y is the strict transform of g in T1, and if vT, (g1) = r and r(g1) = 1, then (xi, y1, z1) are good parameters for 41. Lemma 7.25. With the assumptions of Definition 7.24, suppose that g c: T is reduced, vT(g) = r, r(g) -- 1, (:r, y, z) are good parameters for g, (x., y, z) and (a:1, J1, z1) arc related by a transformation of one of the above types T1-T4, and vT, (g) -r(gj) = 1. Then there is a 1-1 correspondence

v : 0(y,x,y, z) - 0(91,x1,yt, z1) defined by

b - 1, b) if the transformation is a '1'1 with r) = 0, 2. o (a, b) = (a, a + b - 1) if the transformation is a T2, 3. a((L, h) = (a - 1, b) if the transformation is a T3. 4. o,(a, b) _ (a, b - 1) if the transformation is a T4. 1. rr(a.. h) = (a

I

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 0' (a, b) (al, b1) is a vertex of IO(91; x1, yl, z1)I, then (a, b) is a vertex of and if (g; x, y, z) is (a, b) prepared then (91; xt, y1, z1) is (a1, b1) prepared. In particular, (g1; x1, y1, z1) is well prepared if (g; x, y, z) is well prepared.

7.3. r(q)=1

121

Proof. In all cases, a (when extended to R2) takes line segments to line segments and interior points of JA(g; x, y, z)I to interior points of all A(9; x, y, z)I )

Thus the boundary of a(jA(g; x, y, z) I) is the union of the image by a of the line segments on the boundary of a(IA(g; x, y, z)I ). If (al, b1) is a vertex of a(IA(g; :r., y, z) I), it must then necessarily be a(a, b) with (a, b) a vertex of a(IA(g; x, y, z)I ) To sec that (gi; xii yli z1) is (a1, bl) prepared if (g; x, y, z) is (a, b) prepared, we observe that n1,h1

191 }x1 +yl , z1

{g}X,; i,z

-

yr {g} N,z

if we perform a T3 transformation or a T1 transformation with rl = 0, if we perform a T2 or T4 transformation.

Lemma 7.27. Suppose that assumptions are as in Definition 7.24 and (x, y, z) and (x1i y1, z1) are related by a T3 transformation. Suppose that v7) (gl) = r and r(g1) = 1. If (g; x, ,y, z) is very well prepared, then (91; X1, yi, z1) is very well prepared. Moreover, /32,.ly1z1(91)

= $xyz(9),

6x1ylz1(91) = dxyz(g).

Fx1?11z1(91) = Cxyz(9)

and

ax1y1z1(91) = Ckxyz(g) - 1,

y 111iz1 (91) = 7xyz(.9) - I.

We further have Singr(gj) C V (xi , z1).

Proof. Well preparation is preserved by Lemma 7.26. Suppose is very well prepared and

z)

(axyz, Oxyz) 7- (yxyz - bxyz, 6xyz)

Let y = -Yxyz (9), y1 = yxyz (g1) = y - 1. The terms in g which contribute to

the line S(y) are (7.11)

biykxtyJzk.

i+j+7k-'r7 k
k-r

(7.12)

(i + k - r) +j +ylk=ryl k < r

11

7. Resolution of Surfii.c;es in Positive Characteristic

122

which are the terns in gl which contribute to the line S(-y1). We have 6xyz(9) = 5x1,y1,z1(91) = min{r

k I b,jk # 0 in (7.11)}.

If a translation y = 7 + -qr. transforms (7.11) into

E

(7.13)

l.ijkxt J z k I

j.+j+yk=ry

k
then the translation yi = ill + rtx transforms (7.12) into (7.14)

Ciakxt VI zl

(i+k-r)+j+ylk=ryl k
We can make a similiar argument if (axyz03xyz) _ (yxyz - Jxyz,Jxyz)

The statement Singr(91) C V (xl, zl) follows since Sing,- (9) = V (X, y, z), so that Sing, (91) C V (xl ), the exceptional divisor of T3, and Sing,. (,91) C V(zi) since z = 0 is 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 (xi, yi, zj) are related by a TI transformation with 77 = 0, (g; x, y, z) is very well prepared and 1 1(gl) = r, r(gl) = 1. Let (gl; xl, y', z') be a very well preparation of (gl; x1, yi, z1). Let a = axyz(.9),

y = l'xyz(9),

'3 = $xyz(.q),

al = axly'z'(91),

31 = 1x,y'z'(91),

bi = aTty'21(91),

8 = oxyz(g),

E = Exyz(9),

yl = yxly'z'(.gi),

Cl = Cr.1 J'z' (91)

Then (fil,(51,

l,al) < (f,h, 1,a)

in the lexicographical order, with the convention that if e = 0, then If Q1= 0, a1 = S and e 0, then = F - 1. We further have Sing,. (g1) C V(x1i z').

oo.

7.3. r(q) = 1

123

Proof. Let a : 0(g; x, y, z) - A(g1; x1 i y1, z1) be the 1-1 correspondence of Lemma 7.25 1, which is defined by a(a, b) = (a+b-1, b). a transforms lines of slope m # -I to lines of slope +1+ , and transforms lines of slope -1 to vertical lines.

First suppose that -E < -1. Then (a, /3) # (ry - 5, b), and we are in case 1 of Definition 7.22. Then the line segment through (a, /3) and (ry - S, 5) (which has slope < -1) is transformed to a segment with positive slope, or a vertical line. Thus /3x1 yj,zj = 5 < /3. Since (g; x1 i y1 i z1) is well prepared, after very well preparation we have 131 < 0.

Suppose that -1 < -e < -2. We have ((x, /3) = (-y - 5,5). Then the line segment through (a, /3) and (c, d) (with the notation of Definition 7.22) is transformed to the line segment through (a + /3 - 1, /3) and (c+ d - 1, d), which has slope -E -cxlylzl = e -}- 1 < -1. Thus

(axtytzt,lixtytzt) - (a+0- 1,/i) and ('Yxiyjz1(91) - ax1y1z1(91),Sxjylzl(91)) 71 (ax1y1zt(91),I3x1y1z1 (gl))

Let (yxj j,1 zj (g1)

- t, t) be the highest point on the line S(yx, y, x, (.91)) in

ID(g1i x1, yi, zl)I. By Lenlnla 7.26, (gl; xi, yl, zl) is well prepared. Thus (by Lemma 7.21) very well preparation does not affect the vertices

/ (axlylzl(91),/3xlylzl(91)) and (`Yxjylz,(91) - t,t) of Io(gl;xl,yl,z1)I which are below the horizontal line of points whose second coordinate is d. Thus

Q1= /3 and 51<
-E -e + 1

so that -1 < -cx1y, z1 < 0.. Thus (aaxtytzt,l3xtytzt) = ('Yxtytzt -dxtytzt)axtylzl)

1113),

and (c1, d) = (c + d - 1, d) is the lowest point on the line through the point (ax,ylzj)f3xjy,zj) with slope -exl.yl,1 which is on IA(gl;xl,yl,zl)I By Lemma 7.26, (91i x1, y1i z1) is well prepared. We will now show that (91; xl, yl, z1) is very well prepared. If 1 ¢ N, then this is immediate, 1 so suppose that

n=

1 Exlylzl

EN.

124

7. Resolution of Surfaces in Positive Characteristic

We necessarily have n > 1, and n + 1. If we can remove the vertex (c1, d) from IA(gl; xi, yi, zl)I by a translation ill = yl - r):rl, then the translation y = y - ?Jx'°+1 will remove the vertex (c, d) from J0(.g; x, y, z)1, a contradiction to the assumption that (g; x, y, z) is very well prepared. Thus (gl; xl, yl, z1) is very well prepared, with 131 = 13, b1 = 6 and

1 -1-1. E1

E

The final case is when e = 0. Then /_i = S, (gl; xl, yl, zl) is very well prepared,

(a1,131)=('yi-61,bl)=(a+/3 -1,f) and el = 0. Thus /31 = 0, 61 = S and 1- = i = oc. Also, al < a, since El

E

Sing,(g) = V(x, y, z) (by assumption), so e = 0 implies 0 < 1. By Lemmas 7.4 and 7.16, Sing,. (g1) C V (:r.1, zl ). If Sirig,. (g1) = V (xl, zl ),

then all vertices of IA(gl; x1, yl, zl) I have the first cordinate > 1, and thus

V(xl,z')

O

Lemma 7.29. Suppose that assumptions are as in Definition 7.24, (x, y, z) and (x1, yl, zl) are related by a T2 or a T4 transformation, and (g; :r., y, z) is well prepared. Suppose that vT, (g1) = r and r-(gi) = 1. We have that (g1 i X1, y1 i z1) is well prepared and Sing,.(,gl) C V (yl, zl). Let (g1; xi, y', z') be a very well preparation. Then 0x1.y1,71 _ 13x1,y',z'(gl) < 0xyz(9)

Proof. First suppose that T -+ T1 is a T2 transformation. The correspondence a : A(g; x, y, z) - 0(gl; xi, yl, zl) of Lemma 7.25 2. is defined by a(a, b) _ (a, a + b 1). a takes lines of slope m to lines of slope rn.+1. Thus

-

(crx,y,z, (91),+3x1y1z1(91)) _ (axyz(.9), axyz(g) +,3xyz(9)

- 1).

Since Sing,.(g) = V (x, y, z) by assumption, we have axyz(g) < 1, and thus Oxlylx,(gl) < /3xyz(g).

(91; xl, in, z1) is well prepared by Lemma 7.26. Therefore, the vertex (axlylzr (91), 0.11Y,.1(91))

is not affected by very well preparation. We thus have Nx, y,z, (gl) < J&.1jz (g). Now suppose that T -+ T j is a T4 transformation. The 1-1 correspondence rr : y, z) - A(gl; xi, yl, zl) of Lemma 7.25 4 is defined by a(a, b) = (a, b - 1). Thus x17/lzl (91) < 13xyz(g). Also, (gl; xi, y1i zl) is well

prepared by Lemma 7.26, and the vertex (ax,ylzl(gl),,3xly,z,(gl)) is not affected by very well preparation. Thus /3xlu',x'(,gl) _ ,3xlylz, < 0xyz(9)

Lemma 7.30. Suppose that p is a prime, s is a non-negative integer and ro is a positive integer such that p f ro. Let r = rope. Then:

125

7.3. r(q) = 1

1. (a) = 0 mod p, if ps { A, 0 < A < r is an integer. mod p, if 0 < \ < ro is an integer. 2. Proof. Compare the expansions over Z of (x + y)'' = (xPB + yPB)TU.

0

Theorem 7.31. Suppose that assumptions are as in Definition 7.24, (x, y, z) and (xj, yl, z1) are related by a T1 transfol7nation with rl # 0, and (g; x, y, z) is very well prepared. Suppose that rrl; (gl) = r, r(gl) = 1 and /3xyz(g) > 0. zi) is Then there exist good parameters (x, yi, zi) in 7 such that. (g1 i very well prepared, with ,13xl,yi,xi(gl) < fl .y,z(g) and Sing,.(gi) C V(xi, z').

Proof. Let cr - a,:yz(9), Ii = /ixyz(9), 'y = 7 yz(9), S = Sxyz(9). Lemma 7.16 implies z = 0 is an approximate manifold of g = U. Thus

a+ 13>1.

(7.15)

Apply the translation y' = y - rlx and well prepare by some substitution

z' = z -

y'). This does not change (cr, j8) or y. Set 6' = 6x.y,,z' (.9)

First assume that b' < 3. We have regular parameters (xi, y1, z 1) in Tsuch

that x=xr,y' =xiy1,z'=x1-i1 by Lemma 7.16, so we have a 1-1 correspondence (by Lemma 7.25 1)

o : A(9; x, y; z) -t A

x1, y1, z1)

defined by a(a, b) = (a+b-1, b). (91i xl,,yl, z1) is well prepared (by Lemma 7.26). We are essentially in case 1 (-c < --1) of the proof of Lemma 7.28

(this case only requires well preparation), so that (al, /31) = o(y - 8', 8') and 01 = 6' < ,13. After very well preparation, we thus have regular parameters xl, yl, zi in T1 such that (g1; xj, y 'l, z'j) is very well prepared and Oxl,yi,zi(91) < flxyz(9)

We have thus reduced the proof to showing that 8' < ;13. If (a,3) (y - S, 6), we have S' < 5 < 13 since (,q; x, ,y, z) is very well prepared.

Let g = E aijkxiyjzk. Suppose that (a., /3) _ (y - 6, 6). Set

W={(i,j,k)EN3I k
3

r-kr-

) = (CZ, /3) },

aijkx'yzk. (i,j,z)EW

By assumption, zT + F is not solvable. Moreover, aijk?7A (A) xi+A(y )j-Azk

(7.16) F(x, y, z) = F(x, y' + i7x, z) _ (i,j,k)EW A=O

(\

7. Resolution of Surfaces in Positive Characteristic

126

By Lemma 7.21, the terms in the expansion of g(x, y' + rlx, z) contributing to (y, 0) in IA(g; x, ,y', z) 1, where y = a +,3, are 0.

F(7,o) (i,j,k)E W

If (g; x, y', z) is (y, 0) prepared, then 8' = 0 < 3. Suppose that {g} 1',z = zr + F(7,o)

is solvable. Then y E N, and there exists r/i E K such that (z - , 'x'y)? = zr + F(y,a),

(7.17)

so that, with al=4EK,for0
0 (in K), then i = a(r - k), j = /.3('r - k) E N and WV

aijk = G

k

2. If i = a(r - k), j = 3(r - k) E N and (r1 k) = 0 (in K), then aijk=0. Thus (by Lemma 7.30) ai.Ik = 0 if p" } k.

Suppose that K has characteristic 0. By Remark 7.32, we obtain a is solvable. Thus 0 = d' < /..1 contradiction to our assumption that if K has characteristic 0. Now we consider the case where K has characteristic p > 0, an id r = p'ro with p { ro, ro > 1. By (7.18) and Lemma 7.30, we have i = aps, j = ,3p3 E N and ai,j,(ru-1)p° 0. We have an expression lip' = ept, where p } c. Suppose that t > s. Then Q E N, which implies a = y - Q E N, so that (z + Wxay/3)r = zr + F,

a contradiction, since zr + F is by assumption not solvable. Thus t < s. Suppose that e = 1. Then 03 = pt-" < 1 and a < 1 (since we must have Singr(g) = V (x, y, z)) imply -y = a +,(3 = 1, a contradiction to (7.15). Thus c > 1. Also, Zr + F = (zP8 + WP" xaP" y$P" )ro (zP3 +WP5XCP9(yl (zP3

+

)r0

+

WPsxr,psY[(yI)Pt

le.

+

?7P,xp,]

ru

7.3. r(q) = 1

127

Now make the (-y, 0) preparation z = z' - rIywx'r (from (7.17)) so that (g; x, y', z') is (y, 0) prepared. Let G zr + F. Then G

= ((z )P' +

ewP'nPt(e-1) (y )ptxrp +p'(e-1)

ro I

(y1)2ptS2) )

(z')P'ro + ro rewp"77P°(e-1)(y')p'Xap"+pt(e-1) + (y,)2PtfZJ (z')P`(ro-1) y')(z'll)Ps(ro-2)

+ ... + Aro(x, y')

+A2(:r,

for some polynomials f (x, y'), A2i ... , A,.,,, where (y') iP' I Ai for all i. Since

all contributions to S(y) n I A(,q; x, y', z') must come from this polynomial (recall that we are assuming (a, f3) = (y S, 5)), the term of lowest second coordinate on S(y) n I A x, y', z')I is (n, b)

I\apt+p1(F-1) Pt/ ph , py

which is not in N2 since t < s, and is not (a, 0) since e > 1. Thus (g; x, y', z') is not (a, b) solvable, and t

p8 s

< - =/3. 0

Remark 7.32. In Theorem 7.31 suppose Lhat K has characteristic 0. From (7.18) we see that if {g}10,1, is solvable, then for 0 < k < r - 1 there exist i., j E N with i = a(r - k), j = Q(r - k) so that a,,3 E N. Comparing (7.18) with the binomial expansion (z (z +

i-o

r (wx G-)

ay41)r-izi = zr

+

we sec that zr 4 F is solvable, a contradiction. Thus (y, 0) is not solvable on IA(g; x, y', z) I if K has characteristic 0, and after well preparation, (,y, 0) is a vertex of I (g; x, y', z') 1.

Example 7.33. With the notation of Theorem 7.31, it is possible to have (y, 0) solvable in JA(g; x, y', z)l if K has positive characteristic, as is shown by the following simple example. Suppose that the characteristic p of K is greater than 5. Let g = zP + xyP2 -1 +

xP2-1

y2.

The origin is isolated in Sing (g = 0), and (g; x, y, z) is very well prepared. The vertices of IA(g; x, y, z) I are (a, /3) = (y - 6, 5) = (p, p and p)

7. Resolution of Surlkes in Positive Characteristic

128

p-1, 2). Sety=y'+rlxwith 0#r)E K. Then zp + x(.y' +

9

zp +

P2

+

rlx)p2-1

1(y/ + ?7x)2

:c,p2

(P2-1)77p2-1-ixp2-i(y)i)

1

l i +i72xp2+1 + 2yy/Xp2 +;rp2-1(y/)2 z=o

x, y', z) I are (a 3) _

The vertices of I A

However, (ry, 0) is solvable. Set z' = z + P 2-1

g = (z,)1,+

2

(p i

Since

(, p - 1) and (y, 0) - (1), 0).

rlp-px1'.

llyps-l-ixp2-i(y/)i +77

/J1

(7)21)

Then 2x'2+I+2rlyxn2+xp2-1(y+)2.

- -1 mod p,

it follows that 61 = 6xy,zt(g) = p, and the vertices of IA(q; x, y', z')l are and (p +y, 0). (a, Q)=( ,p-E),(-1,-61,61)-(p-p,

p)

Theorem 7.34. For all n E N, there are f,,, f: Rn, such that f is a generator of for f.,, in R,,, such that if and good parr22ameter/sp(xn, y,,, N,,,,

ri = Nxnynzn (fit), LY7t -

(fn)+

En. = E2n?/n .,i (fn),

6n - 6XnynZn (fn ),

a(n) _ (Qn, 6n,

1 En

, Ckn ),

then a(n + 1) < a(n) for all n in the lexicographical order in. IN x r,N x (Q+ U {oc}) x rL N. If n+r = On, 6n+1 = 6n and Fl # oo, then, 1

En41

_

-En 1

As an immediate corollary, we find a contradiction to the assumption that (7.8) has inl'rnit.e length. We have now verified Theorem 7.9 for the final case r(q) = 1.

Proof. We inductively construct regular parameters (xn, yn, zn) in Rn, a generator f of q Rn such that (xn, yn, z t) are good parameters for In, is well C V (xri., z,,L) or Singr(fn) C V(yn, zn) and (fn; xri, yri, prepared. If Singr(fn) C V (xn, z,,), we will further have that (fn; xn, Yn; zn) is very well prepared. Let f be a generator of ZS,gRo. We first choose (possibly formal) regular parameters (x, y, z) in R which are good for f = 0, such that I A (f ; x, y, z) I

7.3. z(q)=1

129

is very well prepared. By Lemma 7.16, z = 0 is an approximate manifold for f = 0. Let a = a;,,,y,, (f ), /I = By assumption, q is isolated in Sing,.(f). Suppose that fi, and regular parameters (xi,yi, zi) in Ri have been defined for i < n as specified above. If Sing, (fn) = V(yn, zn), then Rn - Rn+1 must be a T4 transformation, Xn, = xn+1,

Thus (fn 1; x,+t,

Yn = yn+1,

-n = yn+I zy'?.+1

-

zn+1) is prepared by Lemma 7.29. Further, Singr(fn+1) C V(yn+l, z,

If Singe. (fn 11) - V (xn+1, 'Jn+1, zn+1), make a change of variables to very well prepare. Otherwise, set Yn I 1 - Jrx+1, zn+1 We have On+l < 13,, by Lerluna 7.29. If Sing,.(fn) = V(xn, zn), then R7, R,,+1 must be a T3 transformation xn = xn 1 1,

Yn = Yn+1,

zn = xn,+lzn+l,

and (fn; xn, yn, zn) is very well prepared. Thus (fn+1; xn+i, yn+1, zn-F 1) is also very well prepared. Moreover, 1

(19n+1, sn+1,

En+1

,

(k71+1) < 03., 6,,

1n

f,

an)

by Lemma 7.27. Further, Sing,.(f,,+I) c V(xn+1, zn+1) Now suppose that Sing ,.(fn) = V (xn, yn, zn). Then (f ; xn, yn, zn) is very well prepared, and R,,.+1 must be a Ti or T2 transformation of If Rn+i is a T2 transformation of R,,, xn = xn+1Yi.+1 ;

yaar+i,

Yn --

2n =

yn+1zni+l,

then (fn+l; xn+1, Z/n+l, z, l) is well prepared with j3n+l < /3n by Lemur a 7.29. Further, Sing,(fn+1) C V(yn+1, -'+1) If Sing,(fn+1) = V (xn+l, y, 11, zn+I ),

then make a further change of variables to very well prepare. (Allerwise, set Yn+1 = yn+1, zn+1 = Z", i 1 If-

is a T1 transformation of Rn, Xn. _ Xn+1,

I

'4n = xn+1(yn+-1 + i ),

zn = xn+l zn+1'

then Sing4,,,+1) C V(xn+l,zn+l) by Lemma 7.5. If Qn 0 0 and rl 0 in the TI transformation relating Rn and R,,+1, we can change variables to ?/n+i, zn+1 to very well prepare, with ,3n+1 < /3n and Sirlgr (fn+1) C V zn+i) by Theorem 7.31.

130

7. Resolution of Surfaces in Positive Characteristic

If ri = 0 in the Ti transformation of Rn, and (xn+l,.?/n+1, zn+1) are z;,+i) by very well regular parameters of R,,,,+i obtained from (x,,,+l, preparation, then (fn 11; xn 11, yn+l, zn+1) is very well prepared, (r+1, Sn+1,

1

En+1

,

an+1) < (Nn, dn;

1 En

,

an),

and Sing,.(f,,+1) C V(x,,, t 1, z. i 1) by Lemma 7.28.

If Ran = 0, we can make a translation yn = yn + 77xn, and have that are, unchanged. (gnx,,,, yn, z,,.) is very well prepared, and O Thus we are in the case of ii = 0.

7.4. Remarks and further discussion The first proof of resolution of singularities of surfaces in positive characteristic was given by Abhyankar ([1], [3]). The proof used valuation theoretic methods to prove local uniformization (defined in Section 2.5). The most general proof of resolution of surfaces, which includes excellent surfaces, was given by Lipman [63]. This lucid article gives a self-contained geometric proof. This proof of resolution of surfaces which are hypersurfaces over an algebraically closed field (Theorem 7.1) extends to give a proof of resolution of excellent surfaces. Sonie general remarks about the proof are given in the final 3 pages of Hironaka's notes [53]. The first part of our proof of Theorem 7.1 generalizes to the case of an excellent surface S, and we reduce to proving

a generalization of Theorem 7.9, when V = 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 Sing,.(R./I) = V(m). There are two plain new points which must be considered. The first point is to find a generalization of the polygon I A (f ; x, y, z) I to the case where 6s,q ^_' R/I is as above. This is accomplished by the general theory of characteristic polyhedra [54]. The second point is to extend the theory to the case of a non-closed residue field. This is accomplish ed in [23]. We will now restrict to the case when our surface S is a hypersurface over

an arbitrary field K. The proof of Theorem 7.9 in the case when T(q) = 3 is the same, and if r(q) = 2, the proof is really the same, since we must then have equality of residue fields K(qi) = K(qi+1) for all i in the sequence (7.3), under the assumption that. T(qi) = 2; vq, (Si) - 7 for all i. This is the analogue of our proof of resolution of curves embedded in a non-singular surface, over an arbitrary field. In the case when r(q) = I., some extra care is required. It is possible to have non-trivial extensions of residue fields k(qi) --> k(qa+1) in the sequence

7.4. Remarks and further discussion

131

(7.8). However, if k(qi.+,) # k(qi), we must have fi,+t < flu. (This is proven in the theorem on page 26 of [23]). Thus (,3i, bi, 1, ai) drops after every transformation in (7.8), and we see that we must eventually have ri drop or T increase. The essential difficulty in resolution in positive characteristic, as opposed to characteristic zero, is the lack of existence of a hypersurface of maximal contact (see Exercise 7.35 at the end of this section). Abhyankar is able to finesse this difficulty to prove resolution for 3dimensional varieties X, over an algebraically closed field K of characteristic p > 5 (13.1, [4]). We reduce 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 argument (12.4.3, 12.4.4 [4]) to reduce the problem to the case of a point on a normal variety of multiplicity < 3! = 6. A simple exposition of this projection method is given on page 200 of [62]. We will sketch the proof of resolution, in the case of a hypersurface singularity of multiplicity < 6. We then have that a local equation of this singularity is

+....+ao(x,y,z) =0.

(7.19)

Since r < 6 < char(K), it follows that T E K, and we can perform a Tschirnhausen transformation to reduce to the case w' + a,.-2(x, y, z)W'.-2 + ....}.. ao(x, y, z) = 0.

Now w = 0 is a hypcrsurface of maximal contact, and we have a good theoremn for embedded resolution of 2-dimensional hypersurfaces, so we reduce to the case where each ai is a monomial in x, y, z times a unit. Now we have reduced to a combinatorial problem, which can be solved in a characteristic free way.

A resolution in the case where (7.19) has degree p = char K is found by Cossart [24]. The proof is extraordinarily difficult. Suppose that X is a variety over a field K. In [36], de Jong has shown that there is a dominant proper morphism of K-varieties X' X such that

dim X' = dim X and X' is non-singular. If K is perfect, then the finite extension of function fields K(X) -> K(X') is separable. This is weaker than a resolution of singularities, since the map is in general not hirational (the extension of function fields K(X') - K(X) is finite). This proof relies on sophisticated methods in the theory of moduli spaces of curves.

Exercise 7.35. 1. (Narasimhan [691) Let K be an algebraically closed field of characterisitc 2, and let X be the 3-fold in AK with equation

f =wz {-xy3+yz3+zx7=0.

132

7. Resolution of Surfaces in Positive Characteristic

a. Show that the maximal multiplicity of a singular point on X is 2, and the locus of singular points on X is the monomial curve C with local equations y3 + zx6 = 0,

xy2 + T3 = 0,

yz2 + x7 = 0, w2 4 zx7 = 0, which has the parameterization t - (t7, t19, t15, t32). Thus C has embedding 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 [501) Let K he an algebraically closed field of characteristic

2, and let S be the surface in 4K with equation f = x2 + y4z + y2z4 + z7 = 0. a. Show that the maximal multiplicity of points on S is 2, and the. singular locus of S is defined by.the singular curve y2 +z 3 = 0,

x+yz2=0. b. Suppose that X is a hypersurface on a non-singular variety W,

and p E X is a point in the locus of maximal multiplicity r of X. A hypersurface H through p is said to have permanent contact with X if under any sequence of blow-ups it : W1 W of W, with non-singular centers, contained in the locus of points of multiplicity r on the strict transform of X, the strict transform of H contains all points of the intersection of the strict transform of X with multiplicity r which are in the fiber 7r -1(p). Show that there does not exist a non-singular hypersurface of perrnanerrl, contact at the origin for the above surface S. Conclude that a hypersurface of maximal contact does not exist for S. c. Resolve the singularities of S.

Chapter 8

Local Uniformization and Resolution of Surfaces

In this chapter we present a proof of resolution of surface singularities through local uniformization of valuations. Our presentation is a modernization of Zariski's original proof in [86] and [88]. An introduction to this approach was given earlier in Section 2.5

8.1. Classification of valuations in function fields of dimension 2 Let L be an algebraic function field of dimension two over an algebraically

closed field K of characteristic zero. That is, L is a finite extension of a rational function field in two variables over K. A valuation of the function field L is a valuation v of L which is trivial on K. v is a hoinomnorphisui v : L* -+ r from the multiplicative group of L onto an ordered abelian group

r such that 1. v(ab) = v(a) + v(b) for a, b E L*, 2. v(a + b) > min{v(a), v(b)} for a, b E L*,

3. v(c)=0 for0

cEK.

We formally extend v to L by setting v(O) = oo. We will assume throughout

this section that v is non-trivial, that is, r # 0. Some basic references to the valuation theory of algebraic function fields are (92] and [3]. 133

8. Local Uniformization and Resolution of Surfaces

134

A basic invariant of a valuation is its rank. A subgroup r' of r is said to

be isolated if it has the property that if 0 < a E r' and Q E r is such that 0 < [3 < a, then 13 E V. The isolated subroups of r form a single chain of subgroups. The length of this chain is the rank of the valuation. Since L is a two-dimensional algebraic function field, we have that rank(v) < 2 (cf. the corollary and the note to the len-irna of Section 10, Chapter VI of [92]). If r is of rank 1 then r is embeddable as a subgroup of R (cf. page 45 of Section

10, Chapter VI [92]). If r is of rank 2, then there is a non-trivial isolated subgroup r1 c r such that r1 and r/rl are embeddable as subgroups of R. If these subgroups of R are discrete, then v is called a discrete valuation. In particular, a discrete valuation (in an algebraic function field of dimension two) can be of rank 1 or of rank 2. The ordered chain of prime ideals p of V corresponds to the isolated subgroups r' of r by

P={fEvjv(f)er-r'}u{o} (cf. Theorem 15, Section 10, Chapter VI [92]). In our analysis, we will find a rational function field of two variables L' over which L is finite, and consider the restricted valuation v' = v I L'. We will consider L' = K(x, y), where x, y E L are algebraically independent elements such that v(x) and v(y) are non-negative. Let [L' : L] = r. Suppose that w E L, and suppose that (8.1)

F(x, y, w) = Au(x, y)w' + ... + A,. (x, y) _ 0

is the irreducible relation satisfied by w, with Ai E K[x, y] for all i. Since v(F) - -xa, there must be two distinct terms Aiw''-i, AAw"-2 of the same value, where we can assume that i < j. Thus (j - i)v(w) = v(4-), and we have that r!r' C F. Thus r and r' have the same rank, and r is discrete if and only if r' is discrete. Associated to the valuation v is the valuation ring

V={f ELIv(f)>0}. V has a unique maximal ideal

mv={fEVIv(f)>0}. The residue field K(V) = V/mv of V contains K and has transcendence degree r* <,r- over K. v is called an r*-dimensional valuation. In our case (v non-trivial) we have r' = 0 or 1. If R is local ring contained in L, we will say that v dominates R if R C V and my fl R is the maximal ideal of R. If S is a subring of V, we will say that the center of v on S is the prime ideal my fl S. We will say that R is an algebraic local ring of L if R is essentially of finite type over K (R is a localization of a finite type K-algebra) and R has quotient field L.

8.1. Classification of valuations in function fields of dimension 2

135

We again consider our algebraic extension L' = K (x, y) -> L. The valuation ring of v' is V' = V f1 L, with maximal ideal my? = my fl L. The residue field K(V') is thus a subfield of K(V). Suppose that w* E K(V), and let uw be a lift of w* to V. We have an irreducible equation of algebraic dependence (8.1) of w over K[x, V]. Let Ah be such that v(Ah) = min{v(A1)...... v(Ar)}.

We may assume that h r, because if v(A,.) < v(Ai) for i < r, then we would have v(Ar) < v(Aiw''-i) for 1 < i < r - 1, since v(w) > 0, so that v(F) = v(Ar) < oo, which is impossible. Thus v(A) > 0 for 0 < i < r, so that A e V' for all i. Let a.? be the corresponding residues in K(V'). We have a relation (w*)r-h +

... + a, = 0. Since r - It > 0, w* is algebraic over K(V'). Thus v and v' - v I L' have (1 (W*)? + ... + a1

1(w*)7 h 1 +

ah+l('n)*)r-h-1 +

the same dimension. We will now give a classification of the various types of valuations which occur on L.

8.1.1. One-dimensional valuations. Let x E L be the lift of an element of K(V) which is transcendental over K. Suppose that every y c- L which is transcendental over K(x) satisfies v(y) = 0. Then v must be trivial when restricted to K(x, y) for any y which is transcendental over K(x). Since L is finite over K (x, y), we would have that v has rank 0 on L, so that v is trivial on L, a contradiction. We may thus choose y E L such that x, y are algebraically independent over K, v(x) = 0, v(y) > 0 and the residue of x in K(V) is transcendental over K. Let L' - K(x,y), and let v' = v I L' with valuation ring V' - V fl L'. By construction, if rnvi is the maximal ideal of V', there exists c E K such that mv' fl K[x, y] C (x - c, y). Suppose (if possible) that p - rray? fl K[x, ,y] _ (x - c, y). Then the residue of x in K(V) is c, a contradiction. Thus p is a height one prune in K[x, y] and p = (y). It follows that the valuation ring V is R?, (cf. Theorem 9, Section 5, Chapter VI [921) and the value group of v' is Z. The valuation ring V is a localization of the integral closure of the local Dedekind domain V' in L. Thus V is itself a local l)edekirid domain which is a localization of a ring of finite type over K. In particular, V is itself a regular algebraic local ring of L.

8.1.2. Zero-dimensional valuations of rank 2. Let rl be the nontrivial isolated subgroup of r, with corresponding prime ideal p C V. The composition of v with the order-preserving houiomorphism r - r/rl defines a valuation vl of 1., of rank 1. Let V1 be the valuation ring of vi. V, consists of the elements of K whose value by v is in 1'1 or is positive. Hence

136

8. Local Uniformization and Resolution of Surfaces

V1 = V. v induces a valuation 1.2 of K(Vp) with value group F1, where v2 (f) = v(f *) if f * E Vp is a lift of f to Vp. Since K(Vp) has a non-trivial valuation, which vanishes on K, K(Up) cannot he an algebraic extension of K. Thus K(Vp) has positive transcendence degree over K, so it must have transcendence degree 1. Let x* E K(Vp) be such that x* is transcendental over K. l,et :r, he a lift of :t:* to V,.,. Then K(x) C Vp, so that 111 is trivial on K (x), and v1 is a valuation of the one-dimensional algebraic function field L over K(x). Let V2 C K(Vp) be the valuation ring of v2i and let 7r : Vp -> K(Vp) be the residue map. Then V = 7r-1(V2). v1 is a 1-dimensional valuation of L, so Vp is an algebraic local ring which is a local Dedekind domain. The residue field K(Vp) is an algebraic function field of dimension 1 over K, so V2 is also a local Dedekind domain. Thus IF is a discrete group. Let w be a regular parameter in V,. Let w be a regular parameter in Vp. Given any non-zero element f E L, we have a unique expression f = w"1g, where m = vi (f) E Z, and g is a unit, in Vp. Let. g* be the residue of g in K(1' ). Let n, = 112(9*) E Z. Then, up to an order-preserving isomorphism of t, v(f) _ (m, n) E t = Z2. where Z2 has the lexicographic order.

8.1.3. Discrete zero-dimensional valuations of rank 1. Let x E L be such that v(x) = 1. If y E L and v(y) = nl, then there exists a unique cl E K such that, n2 - v(y - cl:t:"') > 11(9). If we. iterate this construction infinitely many times, we have a uniquely determined Laurent series expansion C1x'E' + C2xf2 + .. .

with v(y - C1xnl -

- CmX ,,,) = nm+1 > nn for all m. This construction

defines an embedding of K-algebras 1, C K((a:)), where K((:1:)) is the field of Laurent series in x. The order valuation on K((x)) restricts to the valuation v on L.

8.1.4. Non-discrete zero-dimensional valuations of rank 1. We subdivide this case by the rational rank of v. The rational rank is the dimension of l' O Q as a rational vector space. Since v has rank 1, we clay assume that

r C R. If 1 ¢ t, we can choose 0 < A E t and define an order-preserving group isomorphism F -> ar by i(c) of 1. with the same valuation ring

Then v = 0 o v is a. valuation

If ELjv(f)>O}-{f ELI v(f)>O}=V as v. We may thus normalize v (by replacing v with the equivalent valuation

v) so that r is an ordered subgroup of R and I E r.

8.2. Local uniforrnization of algebraic function fields of surfaces

137

Suppose that v has rational rank larger that 1. Choose x, y E L such that v(x) = 1 and v(y) = r is a positive irrational number. Suppose that F(x, y) - ai.jx'yj is a polynomial in x and y with coefficients in K. Then v(aijxzyi) = i + jr for any term with ate 0 0, so all the monomials in the expansion of F have distinct values. It follows that v(F) = min{i + Tj I aid # 0}. We conclude that x and y are algebraically independent, since a relation F(x, y) = 0 has infinite value. Furthermore, if L1 = K(x, y) and vl = v I L1, we have that the value group I'1 of vi is Z + Zr. Since r/I'1 is a finite group, we have that r is a group of the same type. In particular, r has rational rank 2. Now suppose that v has rational rank 1. We can normalize r so that it is an ordered subgroup of Q, whose denominators are not bounded, as r is not discrete. In Example 3, Section 15, Chapter VI [92], examples are given of two-dimensional algebraic 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 (cf. MaeLane and Shilling [64]) where the denominators of the rational exponents in the Puiseux series expansion are not bounded.

Exercise 8.1. Let K be a field, R = K[x, y] a polynomial ring and L the quotient field of R.

1. Show that the rule v(x) = 1, v(y) = 7r defines a K-valuation on L which satisfies

v(f) = min{i +.jir I ail 0 0}

arjx'yj c K[x,y]. 2. Compute the rank, rational rank and value group r of v. 3. Let V be the valuation ring of v. Suppose that r E r. Show that

if f =

L7={f EV I v(f)>r} is an ideal of V. 4. Show that V is not a Noetherian ring.

8.2. Local uniformization of algebraic function fields of surfaces In this section, we prove the following local uniformization theorem.

Theorem 8.2. Suppose that K is an algebraically closed field of characteristic zero, L is a two-dimensional algebraic function field over K and v is a valuation of L. Then there exists a regular local ring A which is essentially of finite type over K with quotient field L such that v dominates A.

8. Local Uniformization and Resolution of Surfaces

138

Our proof is by a case by case analysis of the different types of valuations of I, which were classified in the previous section. If v is 1-dimensional, we

have that R = V satisfies the conclusions of the theorem. We must prove the theorem in the case when v has dimension zero, and one of the following three cases hold: v has rational rank 1, or v has rank 1 and rational rank 1,

or v has rank 2. The case when v has rank 1 includes the cases when v is discrete and v is not discrete. We will follow the notation of the proceeding section. Suppose that K [xi , ... , a:,,,] is a polynomial ring, and m = (xl, ... , x,,,).

For f E (cf.

ord(f) will denote the order of f in K[xj .... ,

Section A.4). In particular, ord(f) = r if and only if f E mr and

f V M"+1

.

8.2.1. Valuations of rational rank 1 and dimension 0. Lemma 8.3. Suppose that S = K[xi, ... , x,,,] is a polynomial zing, v is a rank 1 valuation with valuation ring V and maximal ideal m v which contains

S such that K(V) = K and my fl S = (xl, ... , xn). Suppose that there exist fi E S for i E N such that v(fl) < v(f2) < ... < v(fn) < ...

is a strictly increasing sequence. Then limi_,,,0 v(fi) = oo.

Proof. Suppose that p > 0 is a positive real number. We will show that there are only finitely many real numbers less than p which are the values of elements of S. Let r = min{v(f) I f c r is well defined since S is Noctherian. In fact,, if there were not a minimum, we would have an infinite descending sequence of positive real numbers.

rl > 7-2 > ...ri > ... and elements gi E (:r.1, ... , x) such that v(gi) = Ti. Let

Ir, - {f E SI'(f)>ri}. The Ir, are distinct ideals of S which force a strictly ascending chain

Irl CIr2C...CIr,C...

I

which is impossible.

Choose a positive integer r such that r > 2. If f E S is such that. v(f) < p, then write f = g + h with h E x,,)T and deg(g) < r. v(h) > p implies v(g) = v(f ). Thus there exist a finite number of elements 9t, . , g7,, E S (the monomials of degree < r) such that if 0 < A < p is the value of an element of S, then there exist cl, ... , c:.. E K such that (8.2)

v(cig1 + ... + cingm) _ X

8.2. Local uniformization of algebraic function fields of surfaces

139

We will replace the gi with appropriate linear combinations of the gi so that, 11(91) < 11(92) < ... < V(9m).

Then A = v(gk) in (8.2), where ck is the first non-zero coefficient. By induction, it suffices to make a linear change in the gi so that (8.3)

11(91) < v(9i) if i > 1.

After reindexing the gi, we can suppose that there exists an integer l > 1 such that 11(91) = 11(92) = ... = 11(91)

and

v(gi) > v(gi) if t < i.

The equality v(2L) = 0 for 2 < i < l implies 2i E V and there exist ci E K(V) = K such that y has residue ci in K(V). Then s - ci E my implies u(g - ci) > 0, so that v(gi - cigl) > v(91) for 2 < i < 1. After replacing gi with gi - cig1 for 2 < i < 1, we have that (8.3) is satisfied.

O

Theorem 8.4. Suppose that S = K[x, y] is a polynomial ring over an algebraically closed field K of characteristic zero, v is a rational rank 1 valuation of K(x, y) with valuation ring V and maximal ideal'rav, K(V) = V/mv = K such that S C V and the center of V on S is the maximal ideal (x, y). Further suppose that f c K[x, y] is given. Then there exists a birational extension K[x, y] - K[a:', y'] (K[x, y] and K[x', y'] have a common quotient field) such that K[x', y'] C V, the center of v on K[x',,y'] is (x', y'), and f = (x')15, where 1 is a non-negative integer and b E K[x', y'] is not in (x,, y').

Proof. Set r = ord f (0, y). We have 0 < r < oo. If r = 0 we are done, so suppose that r > 0. We have an expansion x«`yi(x)y'st +1: xa''Yi(x)ys',

1=

(8.4)

i>d

i=1

where the first sum is over terms with minimal value p = v(x"Yi(x)yy`) for 1 < i < d, 'yj(x) E K[x] are polynomials with non-zero constant term, 01

<...<8d,$d+1

pif i>d.

-

a with a, b E N, (a, b) = 1. There exist non-negative integers bet V(X) v(u b a', U' such that ab' - ba' I. Then v(xT) = 0 implies there exists 0 0 c E

K(V) = K such that v(

- c) > 0. Set

x = x' 1(Y1 + C)"'

y = x1(y1 + c)b".

8. Local Uniformization and Resolution of Surfaces

140

We have that Si =x

b' y-a'

yt +C = x-bya,

so that v(yi) > 0 and

v(xi) =

v(by) [ab'

- Al =

v(by)

> 0.

Set a = ala + J31b. We have that aia +,31b = a for 1 < i < d. Thus there exist ei c:: Z such that n.'

b') (13i-Q1

ei 0

)

for I < i < d. By Cramer's rule, we have

Qi-,d1 - Uet

0 ej

\aa' l

Thus ei = 111- as'.

We have a factorization f - x,"fi (xi, yi), where d

h (xi, YO

_

i (y1 + c) i°

(y1 + C)a'a'+b'131(

) + xi S2

i=1

is the strict transform of f in the birational extension K[xi, yi] of K[x, y]. We have r1 < oo. ord f (0, y) = r implies there exists an i such that ai = 0, Qi = r. Thus v(yr) > p, and,3i < r for 1 < i < d. We have 7.1 = ord f i (0, yi) <

13d - 01 <

a If ri < 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 that r1 = r. Then /3d = r, 01 = 0, a = 1 and ad = 0. Further, there is an expression d

(0) (y,

+ cryd(0)yi.

i=1

Let u be an indeterminate, and set d

9('u) _

(0)u,0,. ='Yd(0)(u- c)'.

i-1 The binomial theorem implies that 3d-1 = r - 1, and thus

v(y) = v(xad-'). Let 0 A e k(V) = k be the residue of i . Then v(y - ax"d-1) > v(y). Set y' = y - Axad-' and f'(x, y') = f (x, y). We have ord f'(0, y') _ ord f (0, y) = r.

8.2. Local uniformization of algebraic function fields of surfaces

141

V is a minimal value term of f (x, y) in the expansion (8.4), so v(y) <

v(f) Replacing y with y' and f with f and iterating the above procedure after (8.4), either we achieve a reduction ord f, (0, yl) < r, or we find that there exists a'(x) E K[x] such that v(y' - a'(x)) > v(y'). We further have that v(y') < v(f ). Set y" = y' - a'(x) and repeat the algorithm following (8.4) until we either reach a reduction ord f, (0, yi) < r or show that there exist polynomials a(i> (x) E K[x] for i E N such that we have an infinite increasing hounded sequence v(a(x)) < v(a'(x)) < .

Since each v(a(')(x)) is an integral multiple of v(x), this is impossible. Thus

-

there is a polynomial q(x) E K[x] such that after replacing y with y q(x), and following the algorithm after (8.4), we achieve a reduction rl < r in K[xl, yl]. By induction on r, we can achieve the conclusions of the theorem.

We now give the proof of Theorem 8.2 in the case of this subsection, that is, v has dimension 0 and rational rank 1. Let x, y E L be algebraically independent over K and of positive value. Let w be a primitive element of L over K(x, y). We can assume that w has positive value, for if v(w) = 0, there exists c E K such that the residue of in - c is uou-zero in K(V) = K, and we can replace w with w - c. Let R = K[x, y, w]. Then R C V, the center of v on R is the maximal ideal (x, y, w), and R has quotient field L. Let z he algebraically independent over K(x, y), and let f (x, y, z) in the polynomial ring K[x, y, z] be the irreducible polynomial such that R K[x, y, z]/(f) (with w mapping to z + (f )). After possibly making a change of variables, replacing x with x + liz and y with y + yz for suitable ii, 'y E K, we may assume that

r=ord f(0,0,z) <no. If r = 1, then R,,,,t,nq, is a regular local ring which is dominated by v (by Exercise 3.4), so the conclusions of the theorem are satisfied. We thus suppose that r > 1. Let it : K[x, y, z] K[x, y, w] be the natural surjcction. For g E K[x, y, z], we will denote v(ir(g)) by v(g). Write d

(8.5)

f (x, y, z)

ai(x,

_ d=1

y)z°i

+ E ai(x,

y)z°i,

i>d

where ai(x, y) E K[x, y] for all i, the ai(x, y)z°1 are the terms of minimal

value p = v(ai(x, y)z°i) for 1 < i < d, v(aj(x, y)z°*) > p for i > d and

vi«2<...
8. Local Uniformization a,ud Resolution of Surfaces

142

By Theorem 8.4, there exists a hirational extension K[x, y] with K[xi, yl] C V and my fl K[x1, yi] = (xi, y1) such that

K[a.1, yl]

d

f-

xlai(x1, y1)z°q + E xl`u'i(xi,

yi).z

i>d

i=1

where ai(0, 0) # 0 for all i. Ri = K[xi, yl, zi]/(f) is a domain with quotient

field L, R - R1 is birational, R1 C V and my fl R1 = (xi, yi, z). Write v(X; = i with s, t relatively prime positive integers. Let s', t' be positive integers such that s't - st' = 1. Set X1 = x2(z2 + (:2)t',

z = a;2(z2 + c2)",

y1 = Y2,

where 0 -1 c2 E K is determined by the condition v(z2) > 0. Let a = AI It + a, s. We have a = Ait + ai,s for I < i < d. We have (as in the proof of Theorem 8.4) that t°i is a non-negative integer for 1 < i < d and f = x2 f2 (x2, y2, Z2), where d

f2(x2, y2, z2) _ (x2 +

c2)t'A1+s'°i

of ol) (z2 + C2)

+ x212

i-1 is the strict transform off in k:[x2i y2: z2]. Set R2 = K[a:2, y2, z2]/(f2). Then R1 --> R2 is birational, R2 C V and my n R2 = (x2, y2, z2). Set

ri = ord f2(0, 0, z2) <

od - Q1

t

< r.

If r1 < r, we have a reduction, so we may assume that r1 = r. Then we have ad = r, Qi = 0, t = 1 and ad(0, 0) 0. We further have a relation d

T,ai(0,0)(z2 + c2)°` = ad(0,0)z?. i=1

Set d

0)u°`

9(u)

i-1

= ad(0, 0)(u - c2)r.

The binomial theorem shows that ad-1 = r - 1 and v(z) = v(ad_i(x, y)). There exists c E k such that v(z - cad_1(x, y)) > v(z). Then radz''-1 is a minimal value term of , so that v(Y) > v(z). Set z' = z - cad- I (X, Y) f'(x, y, z') = f (x, y, z). We have ord(f'(0, 0, z')) = r and ez' = We repeat the analysis following (8.5), with f replaced by f and z replaced with z'. Either we get a reduction ri = ord(f2(0, 0, z2)) < r, or there exists a'(x, y) E K[x, y] such that v(z' - a'(.x, y)) > v(z') and v(z')

< v(OF') = v(8z

8.2. Local unitorrrritation.of algebraic function fields of surfaces

143

Set z" = z'-a'(x, y) and repeat the algorithm following (8.5). If we do reach a reduction r1 < r after a finite number of iterations, there exist polynomials ai') (x, y) E K[x, y] for i c N such that v(a(x, y)) < v(a'(x, y)) < ...

is an increasing bounded sequence of real numbers. By Lemma 8.3 this is impossible. Thus there exists a polynomial a(x, y) E K[x, y] such that after replacing

z with z -a(x, y), the algorithm following (8.5) results in a reduction r1 < r.

We then repeat the algorithm, so that by induction on r, we construct a birational extension of local rings R. - R1 so that R1 is a regular local ring which is dominated by V.

8.2.2. Valuations of rank 1, rational rank 2 and dimension 0. After normalizing v, we have that its value group is Z + Z r, where r is a positive

irrational number. Choose x, y E L such that v(x) = 1 and v(y) = r. By the analysis of Subsection 8.1.4, x and y are algebraically independent over K. Let w be a primitive element of L over K(x, y) which has positive value. There exists an irreducible polynomial f (x, y, z) in the polynomial ring K[x, y, z] such that T = KM, y, w] ^_' K[:r., y, z]/(f) (where w maps to z + (f)) and my flT = (x, y, v)). After possibly replacing x with x + 0z" and y with y I ryz"', where in is sufficiently large that mv(w) > max{v(.r), v(y)}, and y E K are suitably chosen, we may assume that r = ord(f (0, 0, z)) < oa.

If r = 1, is a regular local ring (by Exercise 3.4), so we assume that r > 1. Let it : K[x, y, z] K[:r, y, w] be the natural surjection. For g E K[x, y, z], we will denote v(7r(g)) by v(g). Write d

f = T aixai yiji F7r i=1

ajxCki yA 27i, i.>d

where all ai E K are non-zero, aix`Yi yr3i z7i for 1 < i < d are the minimal value terms, and y1 < y2 < < yd. There exist integers s, t such that v(z) = q+rri, and there exist integers M, N such that v(.X.ai-YAZ79

= M + Nr

for 1 < i < d. We thus have equalities (8.6)

M = a1+sy1 =... =ad+ryd, N = 13, +ty1 =...=Qd+tyd.

We further have

('i+syi) +(/3i+tyi)rr > M+Nr

8. Local Uniforrnization and Resolution of Surfaces

144

if i > d. Since ord(f (0, 0, z)) = r and adxad'''3dzyd is a minimal value term, we have ryd < r. We expand T into a continued fraction:

hl+

1 .

1

h2 +

Let y be the convergent fractions of r (cf. Section 10.2 [46]). Since lint 9 = T, we have

(ai+sryi)+Wi+tyi)f' >M+Nf 93

9a

for j - p - 1 and j = p, whenever i > d and p is sufficiently large. We further have that sgp+tfp > 0 and sgp 1+t fp_1 > 0 For p sufficiently large. - c) > 0. Since v(xRyt) = v(z), there exists c E K = K(V) such that v( Consider the extension

K[x, y, z] - K[x1, yl, zl],

(8.7)

where (8.8)

xY=

r9' y1

x yt(zl + c) =

1 tfP-'

(z1 + c).

We have fp-19p - fpgp-1 = E = ±1 (cf. Theorem 150 [46]), and E, -'r+ f -' and r - 9 have the same signs (cf. Theorem 154 [46]). From (8.8), we see

that

- xfP-1 x1 p

y9P-1

.Y9P

:

yl

rfP ,

from which we deduce that (8.7) is birational, and Er/(xl) = gp- 1( fp-1 qp-1

-T), w(yl)

gp(T - ). 9p

Hence v(x1), v(yi), v(zl) are all > 0. We have A'.I9P+Nfp Mg,-, FNfr-t

f(x,y,z) = x1

fl(xl,yl,Nl),

y1

where fi = (zt + c)y1(E ai(zl + c)yi y,) + xiy1 H i-1

is the strict transform of f in K[x1i y1, z1]. We thus have constructed a birational extension T --y Tl = K[xii yli z1]/(f1) such that. TI C V and my (1 Ti = (xl, yl, z1). We have f 1(0, 0, 0) - 0, so 'rr, = e is a root of d

O(U) = r ii=1

CLi11yi-y1 = 0.

8.2. Local unilbrmization of algebraic function fields of surfaces

145

Let rl = ord f1(0, 0, z1). Then r1 < yd - yl r. If r1 < r, we have obtained a reduction, so we may assume that r1 = r. Thus y1 = 0, yd = r, ad =,3d = 0 (since yi < r for I < i < d) and

0(u) = ad(u - )r. Hence yd-1 = r - I. and ad-1 # 0, by the binomial theorem. We have v(z) _ ad_1+/d_1T. Our conclusion is that there exist non-negative integers

m and n such that v(z) -in +'n,7. There exists c E K such that v(z - nx yn) > v(z). We make a change of variables in K [x, y, z], replacing z with z' = z-cxmyn, and let f'(x, y, z') = f (x, y, z). We have v(z') > v(z) and ord f'(0, 0, z') = r. Since radzr-1 is a minimal value term of , we have v(z) < v(f). We further have _ We now apply a transformation of the kind (8.8) with respect to the new variables x, y, z'. If we do not achieve a reduction r1 < r, we have a relation v(z') = -ml +n17 with mi ni non-negative integers, and we have

v(z) < v(z') <

of

i9z ).

We have that there exists c2 E K such that v(z' - c2xmr ynl) > v(z').

Set z" = z' - c2x"" y" ". We now iterate the procedure starting with the transformation (8.8). If we do not. achieve a reduction r1 < r after a finite number of iterations, we construct an infinite bounded sequence (since 5z 7 0 in T)

m+n7 T1 such that T1 C V and (T ).,,,,vr,Ti is a regular local ring.

8.2.3. Valuations of rank 2 and dimension 0. Suppose that v has rank 2 and dimension 0. Let 0 C p C my be the distinct prime ideals of V. Let, x, y E L be such that x, y are a transcendence basis of L over K, v(x) and v(y) are positive and the residue of x in K(Vp) is a transcendence basis of K(Vp) over K. Let w be a primitive clement of L over K(x,y) such that v(w) > 0. Let T _ K[x, y, w] C V. Let f (:c:, y, z) he an irreducible element in the polynomial ring K[x, y, z] such that (8.9)

K[x, y, z]/(f)

where z + (f) is mapped to w.

K[x, y, w],

8. Local Uniforrnization and Resolution of Surfaces

146

By our construction, the quotient field of T is L and trdegK(T/p n T) 1. Thus p n 1' is a prime ideal of a curve on the surface spec(T). Set Ro = R,,,,nTT. Let Co be the curve in spec(T) with ideal p n Ro in Ro. Corollary 4.4 implies there exists a proper birational morphism it X --, spec(T) which is a sequence of blow-ups of points such that the strict transform Co of Co on X is non-singular. Let X 1 - X be the blow-up of Co. Since X1 1, spec(T) is proper, there exists a unique point al E X1 such that V dominates R1 = °Xi,al. Rl is a quotient of a 3-dimensional regular :

local ring (as explained in Lemma 5.4). Let C1 be the curve in X, with ideal p n RI in R.I. (R,)pnR, dominates RpnR and (R1)pnR1 is a local ring of a point on the blow-up of the maximal ideal of RpnR. We can iterate this procedure to construct a sequence of local rings (with quotient field L) Ru--+

such that V dominates.Rj for all i, R, is a quotient of a regular local ring of dimension 3 and (R;,_t1)pnRi+1 is a local ring of the blow-up of the maximal ideal of (Rz)pnR; for all i. Furthermore, each Ri is a localization of a quotient of a polynomial ring in 3 variables. Since (Ro)pnRo can be considered as a local ring of a point on a plane curve over the field K(x) (as is explained in the proof' of Lemma 5.10), by Theorem 3.15 (or Corollary 4.4) there exists an i such that (R,)pnR; is a regular local ring. After replacing the T in (8.9) with a suitable affine ring, we can now assume that TpnT is a regular local ring. If Tm,,ni is a regular local ring we are done, so suppose T,,,,vni- is not a regular local ring. Let 1) _ 7r 1(p f1 T), where 7r : K[x, y, z] T is the natural surjection. If g E K[x, ,y, zj is a polynomial, we will denote v(7r(y)) by v(g). Let I C K[x, y, z] be the ideal I = (f, , , W0 ) defining the singular locus of T. Since I (as TpnT is a regular local ring), one of 7r(), 7r(), 7r( ) ¢ p n T. Thus

min{v(Of ), v(Oayf ), v(Of )} _ (0,'n) az for some it E N. Write

f=F,.+F,.il+...+Fm, where Fj is a form of degree i in x, y, z and r = ord(f). After possibly reindexing the x, y, z, we may assume that 0 < v(x) < v(y) < v(z). Since K (V) - K, there exist c, d E k (which could be 0) such that

v(y - cx) > v(x) and v(z - dx) > v(x).

8.2. Local uniformizationi of algebraic function fields of surfaces

147

Thus we have a birational transformation K[x, y, z] -> K[xi, yi, zi] (a quadratic transformation), where :r.1 - x,

y - CZ;

Y1 =

Z1 =

x

z - dx x

with inverse

x=xi, y=xl(yl+c),

z = xi(zl + d).

In K[xi, yi, zi] we have f (x, y, z) = xj fl(xl, yl, zl), where

fi(xi,yi,zi) = x'&-rFm(l, yl + c, Zi + d)

is the strict transform of f. Let T1 = K[xi, yl, zi]/(fl). Then T -- Ti is birational, Tl C V and m fl T1 = (x1, yl, z1). We calculate

xl''-i (XI

(yi+c)

11

-(yi+d)

Zi

+rfl),

= x17-1M xl zii

ri

Hence

thin{v(Ux ),

v(j ), v(jz )} > (r - 1)v(xi) + min{v(dxl ), v(

), i

Since we have assumed that Tmvni- is not regular, we have r > 1. We further have that v(xl) > 0, so we conclude that

minfv("fl

8x1), v( f ii ), v(af i )} _ (0, ni) < (0, n).

Let ri = ord(fl) < r. (T1),nvnT, is a regular local ring if and only if rl = 1. Thus we may assume that rl > 1. We now apply to T j a new quadratic transform, to get min{v(19x2 ), v( yf

2 ),

v(az1)} _ (0, n2) < (0,710-

Thus after a finite number of quadratic transforms T -+ T1, we construct a Tl such that (Ti),,,vnTj is a regular local ring which is dominated by V. Remark 8.5. Zariski proves local uniformization for arbitrary characteristic zero algebraic function fields in [87].

148

8. Local Uniformization and Resolution of Surfaces

8.3. Resolving systems and the Zariski-Riemann manifold Let L/K be an algebraic function field. A projective model of L is a projective K-variety whose function field is L. Let E(L) he the set of valuation rings of L. If X is a projective model of L, then there is a natural morphism lrX : E(L) -+ X defined by 7rx (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 V on X. There is a topology on E(L), where a basis for the topology are the open sets 7rX1(U) for an open set U of a projective model X of L. Zariski shows that E(L) is quasi-compact (every open cover has a finite subcover) in [89] and Section 17, Chapter VI [92]. E(L) is called the Zariski-R.iemann manifold of L.

Let N he a set of zero-dimensional valution rings of L. A resolving system of N is a finite collection {S1,. .. , S} of projective models of L such that for each V E N, the center of V on at least one of the Si is a non-singular point.

Theorem 8.6. Suppose that L is a two-dimensional algebraic function field over an algebraically closed field K of characteristic zero. Then there exists a resolving system for the set of all zero-dirrtensional valuation rings of L.

Proof. By local uniformization (Theorem 8.2), for each valuation W of L there exists a projective model Xw of L such that the center of W on Xw is non-singuar. Let Uw be an open neighborhood of the center of W in X1,1, which is non-singular, and let AW = irw- (UVV), an open neighborhood

of W in E(L). The set {Aw I W E E(L)) is an open cover of E(L). Since

E(L) is quasi-compact., there is a linite subcover {Aw} of E(L). Then Xw1, ..., XWr is a resolving system for the set of all zero-dimensional valuation rings of L.

Lemma 8.7. Suppose that I, is a f.'wo-dirn.ensioraal algebraic fartction field over an algebraically closed field K of characteristic zero, and R0cR1c...cR=c...

is a sequence of distinct normal two-dimensional algebraic local rings of L. Let, 1 - U R;.. If every one-dimensional valuation ring V of L which contains 1 intersects Ri in a height one prime for some i, then S2 is the valuation ring of a zero-dimensional valuation of L. Proof. 5l is by construction integrally closed in L. Thus ft is the intersection of the valuation rings of L containing fl (cf. the corollary to Theorem 8, Section 5, Chapter VI [92]). Observe that St # L. We see this as follows.

If rni is the maximal ideal of R, then m = U mi is a non-zero ideal of 92. Suppose that 1 E m. Then 1 E mi for some i, which is impossible.

8.3. Resolving systems and the Zariski-Riemann manifold

149

We will show that SZ is an intersection of zero-dimensional valuation To prove this, it suffices to show that if V is a one-dimensional valuation ring containing Sl, then there exists a zero-dimensional valuation

rings.

ring W such that V is a localization of W and Sl c W. Since V is onedimensional, V is a local Dedekind domain and is an algebraic local ring of L. Let K(V) be the residue field of V. Let my be the maximal ideal of V, and let pi = Ri fl mv. By assumption, there exists an index j such that i > j implies that pi, is a height one prime in R,. (R,.)p; is a normal local ring which is dominated by V. Since both (Ri)p; and V are local Dedekind domains, and they have the common quotient, field L, we have

that (Ri)p; - V for i > j (cf. Theorem 9, Section 5, Chapter VI [92]). Thus the residue field Ki of (Ri)p, is K(V) for i > j. The one-dimensional algebraic local rings Si = Ri./pi for i > j have quotient field K(V), and S,+1 dominates Si. U Si is not K(V) by the same argument we used to show that Sl is not L. Thus there exists a valuation ring V' of K(V) which dominates U Si. Let 7r : V -> K(V) be the residue map. Then W = 7r-'(V') is a zero-dimensional valuation ring of V and W contains 0. Since we have established that fl is the intersection of zero-dimensional valuation rings, we need only show that fZ is not contained in two distinct zero-dimensional valuation rings. Suppose that fl is contained in two distinct. zero-dimensional valuations, V1 and V-2. We will first construct a projective surface Y with function field L such that V1 and V2 dominate (unique) distinct points qi and q2 of Y. Let. a, y

be a transcendence basis of L over K. Let vi, v2 be valuations of L whose respective valuation rings are V1 and V2. If vi(x) and v2(x) are both < 0, replace x with .1. Suppose that v1(x) > 0 and v2(x) < 0. Let c E K he such that the residue of x in the residue field K(V1) ^- K of V1 is not c. After replacing x with x - c, we have that vi(x) = 0 and v2(x) < 0. We can now replace x with . to get that vl(x) > 0 and v2(x) > 0. In this way, after possibly changing x and y, we can assume that x and y are contained in the valuation rings V1 and V2. Since Vl and V2 are distinct zero-dimensional

valuation rings, there exist f E V1 such that f V V2i and g E V2 such that g' V (cf. Theorem 3, Section 3, Chapter VI [921). As above, we can

assume that v, (f) = 0, v2 (f) < 0, vi (g) < 0 and v2 (g) = 0. Let a = f, b=

9,

and let A be the integral closure of K[x, y, a, b] in L. By construction,

A c V fl V2. Let ml = A fl mrty,, nt2 - A fl mnv2. We have that a E m2 Thus ml #m2. We nowletY but be the 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 generated K-algebra Hi which has quotient field L. Let X, be the

150

8. Local Uniformization and Resolution of Surfaces

normalization of a projective closure of spec(Bi). Since Ri is integrally closed, the projective surface Xi has a point ai with associated local ring Ox.,ai = R,.-

There is a natural birational map Ti : Xi -> Y. We will now establish that Ti cannot be a morphism at ai. If Ti were a morphism at ai, there would be a unique point bi on Y such that Ri = Ox;,.; dominates OY{,b Since V1 and V2 both dominate V1 and V2 both dominate OY;,b,, a contradiction. By Zariski's main theorem (cf. Theorem V.5.2 [47]) there exists a onedimensional valuation ring W of L such that W dominates and W

dominates the local ring of the generic point of a curve on Y. Let I'll; be the set of one-dimensional valuation rings W of K such that W dominates Ox,,., and W dominates the local ring of the generic point of a curve on Y. l'i is a finite set, since the valuation rings W of I'i are in 1-1 correspondence with the curves contained in 1ri 1(ai), where irl I'1:, - Xi is the projection of the graph of Ti onto Xi. Since R.i+1 = 04, l,at, 1 dominates Ri = we have I`i+1 C ri for all i. Since each Fi is a non-empty finite set, there exists a one-dimensional valuation ring W of L such that W dominates Ri for all i, a contradiction. :

Theorem 8.8. Suppose that L is a two-dimensional algebraic function field over an algebraically closed field K of characteristic zero, R is a two-dimensional normal algebraic local ring with quotient field L, and

Ri--+ ...

is a sequence of normal local rings such that Ri+1 is obtained from Ri by blowing up the maximal ideal, normalizing, and localizing at a maximal ideal so that Ri+1 dominates Ri. Then SZ = U Ri is a zero-dimensional valuation ring of L.

Proof. By Lemma 8.7, we are reduced to showing that there does not exist a one-dimensional valuation ring V of L such that Sl is contained in V and my n Ri is the maximal ideal for all i. Suppose that V is a one-dimensional valuation ring of L which contains Sl. Let v be a valuation of L whose associated valuation ring is V. Since V is one-dimensional, there exists a E V such that the residue of a in K(V) is transcendental over K. Let mi be the maximal ideal of Ri. Assume that my fl R is the maximal ideal mo of R. Write a = n with , it c R. Then a R, since R/mo = K. Hence 71 E mo, and also E mo since v(a) > 0. Since R1 is a local ring of the normalization of the blow-up of mo, mOR1 is a principal ideal. In fact if 1

8.3. Resolving systems and the Zariski-Riemann manifold

151

(C mo is such that v(() = nun{v(f) I f E rno}, we have that (R1 = m0Rl. Thus i = S E Ri and 771 = S c Ry. If we assume that my fl Ri = mi, we have that 1, i7i C ml, a = 2, 7l2 E R2 and

with m

v() > v(1) > 0,

v(Si) > v(6),

v(ii) > v(771) > 0,

v(771) > V(772)-

More generally, if we assume that mi = my fl v for i = 1, ... , n, we can find elements ri, 77; E Ri for 1 < i < n such that a = and 77,

v(r7)>v(r71)>...>v(r7,z)>0. Since the value group of V is Z, it follows that n < v(r7)). Hence, for all i sufficiently large, my fl R2 is not the rnaxirnal ideal. rrty fl R, must then be a height one prime in Ri, for otherwise we would have my f1Ri = (0), which would imply that V = L, against our assumption.

Remark 8.9. The conclusions of this theorem are false in dimension 3 (Shannon [76]).

Theorem 8.10..quppose that S, S' are normal projective surfaces over an algebraically closed field K of characteristic zero and T is a birational map from S to S'. Let St -> S be the projective morphism defined by taking the normalization of the blow-up of the finitely many points of T, where T is not a morphism. If the induced birational map T1 from Si to S' is not a morphism, let S2 -> S1 be the normalization of the blow-up of the finitely many fundamental points of T1. We then iterate to produce a sequence of birational morphisms of surfaces

S-S1 l-S2<-...*-- S,, +- ..., This sequence must be of finite length, so that the induced rational map Si - S' is a morphism for all i sufficiently large.

Proof. Since Si is a normal surface, the set of points where Ti is not a morphism is finite (cf. Lemma 5.1 [47]). Suppose that no Ti : Si -- S' is a morphism. Then there exists a sequence of points pi E Si for i e N such that pi+1 maps to pi and Ti is not a morphism at pi for all i. Let R4 = OSi,pi. The birational maps Ti give an identification of the function fields of the Si and S' with a common field L. By Theorem 8.8, 11 = U Ri is a zerodimensional valuation ring. Hence there exists a point q E S' such that Il dominates OS',q. OS',q is a localization of a K-algebra B = K[ fl, ... , fr] for some f l ,- .. , fr E L. Since B C St, there exists some i such that 13 C R. Since 12 dominates Ri, we necessarily have that Ri dominates OSI,q. Thus Ti

152

8. Local Uniformization and Resolution of Surfaces

is a morphism al. pi., a contradiction to our assumption. Thus our sequence must be finite.

Theorem 8.11. Suppose that S and S' are projective s rfaces over an algebraically closed 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 resolving system for N.

Proof. By Theorem 8.1.0, there exists a birational morphism 34 - i S obtained by a sequence of normalization of the blow-ups of all points where the birational map to S' is not. defined. Now we construct a birational morphism S' - S', applying the algorithm of Theorem 8.10 by only blowing up the points where the hirational map to S is not defined and which are nonsingular points. The algorithm produces a birational map 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 in : 5* - S and 7r2 : S* - S'. We will show that S* is a

resolving system for N. Suppose that V c N. Let r, p', 13' and p* be the centers of V on the respective projective surfaces S, S', S and S*. First suppose that p' is a singular point,. Then 73 must be non-singular, as {S, S'} is a resolving system for N. The birational map from S to .4'' is a morphism at p, and irl is an isomorphism at 15. Thus the center of V on S* is a non-singular point. Now suppose that p' is a non-singular point. Then gi is a non-singular point of S' and the birational map from S' to S is a morphism at p'. 't'hus 72 is an isomorphism at p', and the center of V on S* is a non-singular point. Theorem 8.12. Suppose that 1, is a two-dimensional algebraic function field over an algebraically closed field of characteristic zero. Then there exist.,.( a non-singular projective surface S with function field L.

Proof. The theorem follows from Theorem 8.6 and Theorem 8.10, by induction on r applied to a resolving system {Sl,... , S,.) for the set of zerodimensional valuation rings of L.

Remark 8.13. Zariski proves the generalization of Theorem 8.11 to dimension 3 in [90], and deduces resolution of singularities for characteristic zero 3-folds from his proof of local uniformization for characteristic zero algebraic function fields [87]. Ahhyankar proves local uniformization in dimension three and characteristic p > 5, from which he deduces resolution of singularities for 3-folds of characteristic p > 5 [4].

8.3. Resolving systems and the Zariski-Riemann manifold

153

Exercise 8.14. 1. Prove that, all the birational extensions constructed in Section 8.2 are products of blow-ups of points and non-singular 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 characteristic p > 0. Some care is required to ensure L is separable. In [1], Abhyankar accomplishes this that K (x. y) and gives a very different proof in the remaining case to prove local uniformization in characteristic p > 0 for algebraic surfaces.

Chapter 9

Ramification of Valuations and Simultaneous Resolution

Suppose that f is an algebraic function field over a field K, and trdegK L < 0U is arbitrary. We will use the notation for valuation rings of Section 8.1.

Suppose that R is a local ring contained in L. We will say that R is algebraic (or an algebraic local ring of L) if L is the quotient field of R and R is essentially of finite type over K. Let K(S) denote the residue field of a ring S containing K with a unique maximal ideal. We will also denote the maximal ideal of a ring S containing a unique maximal ideal by mg. Suppose that L - L* is a finite separable extension of algebraic function fields over K, and V* is a valuation ring of L* /K associated to a valuation

v* with value group 1'*. Then the restriction v = v* I L of v* to 1, is a valuation of L/K with valuation ring V = L n V*. Let 1' be the value group of v.

There is a commutative diagram

L T

V=LnV*

L T

V*.

155

9. Ramification of Valuations and Simultaneous Resolution

156

There are associated invariants (cf. page 53, Section 11, Chapter VI [92]) namely, the reduced ramification index of v* relative to v,

e = [r* : r] < oo, and the relative degree of v* with respect to v, f = [K(V*) : K(V)]. The classical case is when V* is an algebraic local Dedekind domain. In

this case V is also an algebraic local Dedekind domain. If my = (y) and my = (x), then there is a unit ry c V* such that

x=ryye. Since we thus have

r*/r

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 the induced homomorphism on completions with respect to the respective maximal ideals V

-} V

is given by the natural inclusion of K algebras K[[x]] - K[[?!]],

where y = P yy and x = y . We thus have a natural action of r*/r on (a generator multiplies -y by a primitive e-th root of unity), and the ring of invariants by this action is

f/

(V*)I"/r.

In this chapter we find analogs of these results for general valuation rings. The basic approach is to find algebraic local rings R of L and S of L* such that V* dorninates S (S is contained in V* and my fl s = mg) and V

dominates R, and such that the ramification theory of V - V* is captured in (9.1)

H. --, .5',

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

[87] (also Section 2.5 and Chapter 8 of this book). It is proven in [87] that if char(K) = 0 and V is a valuation ring of an algebraic function field L/K, then V dominates an algebraic regular local ring of L. The case when trdegK L = 2 is proven in Section 8.2.

9. Ramification of Valuations and Simultaneous Resolution

157

We first consider the following diagram:

L -> (9.2)

L* T

S where L*/L is a finite separable extension of algebraic function fields over L, and S is a normal algebraic local ring of L*. We say that S lies above an algebraic local ring R 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 (9.2) where there does not exist a local ring R of L such that S lies above R (cf. [35]). To construct an extension of the kind (9.1), 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 the concept of a monoidal transform. Suppose that V is a valuation ring of the algebraic function field L/K, and suppose that R is an algebraic local ring of L such that. V dominates R. Suppose that p C R is a regular prime; that is, Rip is a regular local ring. If f E p is an element of minimal value, then is contained in V, and is an algebraic local ring of L which if Q = rnv n R[5], then R1 = is dominated by V. We say that R - R1 is a monoidal transform along V. R1 is the local ring of the blow-up of spec(R) at the non-singular subscheme

If R is a regular local ring, then Rr is a regular local ring. In this case there exists a regular system of parameters (xi, ... , xn) in R such that V (p).

if ht(p) - r, then R1 = R[?; ..

, s71

]q.

The main tool we use for our analysis is the Local Monomialization Theorem 2.18.

With the notation of (9.1) and (9.2), consider a diagram L

L*

I

T

V=V*nL

V* T

S*

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 (9.3))

V

V*

T

T

R -4 S T

S*

9. Ramification of Valuations and Simultaneous Resolution

158

such that S* -1, S is a sequence of inonoidal transforms, S is a regular local ring, and there 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 by Abhyankar in [2]. However, it follows from a refinement in Theorem 4.8 [35] (strong monomialization) of Theorem 2.18 that if K has char-

acteristic zero, trdegK(L*) is arbitrary and V* has rational rank 1, then local simultaneous resolution is true, since in this case (2.8) becomes

xl = 5lyI x2 = y2, ... xn = Yn, and S lies over R13.

The natural next question to consider is weak simultaneous local resolution; that. is, does there exist a diagram (constructed from (9.3))

V

V*

T

I

R

S T

S*

such that S* - S is a sequence of monoidal transforms, S is a regular local ring, and there exists a normal local ring R such that S lies above R? This was conjectured by Abhyankar on page 144 of [8] (and is implicit, in [1]). If trdegK(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 answer to this question is yes, as we prove in [29] and [35]. This is a simple corollary of local rnonornialization. In fact, weak simultaneous local resolution follows from the following theorem, which will be of use in our analysis of ramification.

Theorem 9.1 (Theorem 4.2 [35]). Let K be a field of characteristic zero, L an algebraic function field over K, L* a finite algebraic extension of L, and v* a valuation of L'/K, with valuation ring V*. Suppose that S* is an algebraic local ring of L* which is dominated by v*, and R* is an algebraic local ring of L which is dominated by S*. Then there exists a commutative diagram

Ro - R - S R

CV*

T

T

->

where S* - S and R* jW are sequences of monoidal transforms along v* such that Rp -' S have regular parameters of the form of the conclusions of Theorem 2.18, R is a normal algebraic local ring of L with tonic singularities

9. Ramification of Valuations and Simultaneous Resolution

159

which is the localization of the blow-up of an ideal in RU, and the regular local ring S is the localization at a maximal ideal of the integral closure of R in L*. Proof. By resolution of singularities [52] (cf. Theorems 2.6 and 2.9 in [26]), we first reduce to the case where R* and S* are regular, and then construct, by the Local Monomialization Theorem 2.18, a sequence of monoidal transforms along v S

Ra T

C V*

T

R*

->

S*

so that R e is a regular local ring with regular parameters (xi, ... , xn), S is a regular local ring with regular parameters (yi, . . . , y1,), and there are units in S, and a matrix of natural numbers A = (ai5) with non-zero determinant d such that

xi = biy1aft ... Unn for 1 < i _< n. After possibly reindexing the yi, we may assume that d > 0. Let (bit) be the adjoint matrix of A. Set n

79

fi=Hxai'bb'')y1 ?-1

7=l

for 1 < i < n. Let R be the integral closure of R0[fi,... , f] in L, localized at the center of v*. Since m.k = ms, Zariski's Main Theorem (10.9 in [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 (9.4). In the theory of resolution of singularities a modified version of the weak simultaneous resolution conjecture is important. This is. called global weak simultaneous resolution.

Suppose that f : X -- Y is a proper, generically finite morphism of k-varieties. Does there exist a commutative diagram XI

fr

x

1'i 1

1

f -4

Y

such that fi is finite, X1 and Y1 are complete k-varieties with X1 nonsingular, Y1 normal, and the vertical arrows are birational?

160

9. Ramification of Valuations and Simultaneous Resolution

This question has been posed by Abhyankar (with the further conditions that Yl --> Y is a sequence of blow-ups of non-singular subvarieties and Y, X are projective) explicitly on page 144 of [8] and implicitly in the paper [2].

We answer this question in the negative, in an example constructed In Theorem 3.1 of [30]. The example is of a generically finite morphism X -* Y of non-singular, projective surfaces, defined over an algebraically closed field K of characteristic not equal to 2. This example also gives a counterexample to the principle that a geometric statement that is trice 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 [35]). Suppose that char(K) = 0 and we are given a diagram of the form of (9.3)

- L*

L

T

T

V=V*nL

V* T

s*. Let K be an algebraic closure of K(V*). Then there exists a regular algebraic local ring RI of L such that if Ite is a regular algebraic local ring of L which contains R1 such that Ro

--*

It -* S

CV*

T

S*

is a diagram satisfying the conclusions of Theorem 9.1, then:

1. There is an isomorphism Z /A' Z" '" r*/I' given by

(bl,...,bn)' blv*(yl)

+... + bn11*(yn)

2. r * /r acts on S(§K(S) K, and the invariant ring is

RtK(R)K

06K(S)K)'" ".

3. The reduced ramification index is

e = lr*/ri = I Det(A)I. 4. The relative degree is

f = [K(V*) : K(V)] = [K(S) : K(R.)]. It is shown in [35] and [32] that we can realize the valuation rings V and V* as directed unions of local rings of the form of /3 and S in the above

9. Ramification of Valuations and Simultaneous Resolution

161

theorem, and the union of the valuation rings is a Galois extension with Galois group r*/r We now give an overview of the proof of Theorem 9.2, referring to [35] for details. v* is our valuation of L* with valuation ring V*, and v = v* I L

is the restiction of v* to K. In the statement of the theorem r is the value group of v* and r is the value group of v. By assumption, we have regular parameters (xl, . . . , x,) in Ro and (yl, . . . , y,,) in S which satisfy the equations

xi=b&Hy°

for1
j-1 of (2.8). We have. the relations 14

v(.rj,)

= E ajjv*(yi) E r

f'or 1 < i < ii. Thus there is a group homomorphism

Zn/AtZ --> t*/I' defined by 1. To show that this homomorphism is an isomorphism we need an extension of local uionoinialiration (proven in Theorem 4.9 [35]). 3 is immediate from 1 H.0 is monomial, and Ro -> R is a weighted blow-up. Lernrrra 4.4 of [35] shows that if M* is the quotient field of S6K(S)K and M is the quotient field of R®K(R)K, then M*/M is Galois with Galois group Gal(M*/M) ^-'

Z /Atz'. Furthermore, R&I
counterexample to global weak simultaneous resolution stated earlier in this

section. This example is written up in detail in [30]. We will give a very brief outline of the construction. With the notation of (9.6), if a valuation v of K(Y) has at least two distinct extensions vi and v2 to K(X) which have non-isomorphic quotients rl/r and r2 /I', then this presents an obstruction to the existence of a map Xr Y1 (with the notation of (9.6)) which is finite, and X1 is non-singular. If there were such a map, we would have points pr and p2 on X1, whose local rings R1 and R2 are regular an(-] axe dominated by vl and v2 respectively, and a point p on Yl whose local ring R is dominated by p. If we choose our valuation v carefully, we can ensure that h is a quotient of Al by rl/I' and R is also a quotient of R2 by P2/r. We can in fact choose the extended valutions so that we can conclude that not only are these quotient groups not isomorphic, but the invariant rings are not isomorphic, which is a contradiction.

162

9. Ramification of Valuations and Simultaneous Resolution

An interesting question is whether the conclusions of local monomialization (Theorem 2.18) hold if the ground field K has positive characteristic. This is certainly true if trdegK(L*) = 1. Now suppose that trdegK (L*) - 2 and K is algebraically closed of positive characteristic. Strong monomialization (a refinement, in Theorem 4.8 [351 of Theorem 2.18) is true if the value group F* of V* is a finitely generated group (Theorem 7.3 [351) or if V*/V is defectless (Theorem 7.35 [35]). The defect, is an invariant which is trivial in characteristic zero, but is a power of p in an extension of characteristic p function fields. Theorem 7.38 [351 gives an example showing that strong monorrrialization fails if trdegK(L*) = 2 and char(K) > 0. Of course the value group is not finitely generated, and V*lV has a, defect. This example does satisfy the less restrictive conclusions; of local monomialization (Theorem 2.18). It follows from strong monomialization in characteristic zero (a refinement of Theorem 2.18) that, local simultaneous resolution is true (for arbitrary I:rdegK(L*)) if K has characteristic zero and rat.icmal rank v* = 1. We consider this condition when trdegK (L*) - 2 and K is algebraically closed of positive characteristic. In Theorem 7.33 [35] it is shown that in many cases local simultaneous resolution does hold if trdegK(L*) - 2, char(K) > 0 and the value group is not finitely generated. For instance, local simultaneous resolution holds if I' is not, p-divisible. The example of Theorem 7.38 [351 discussed above does in fact satisfy local simultaneous resolution, although it does not satisfy strong monomializat.ion.

Appendix. Smoothness and Non-singularity II

In this appendix, we prove the theorems on the singular locus stat.ed in Section 2.2, and prove theorems on upper semi-continuity of order needed in our proofs of resolution. The proofs in Section A.1 are based on Zariski's original proofs in [85].

A.I. Proofs of the basic theorems Suppose that P E A7 = spec(K[xl,... , xn]) has height n - r. Let, rnp C O,tn,p be the ideal of P. The differential d : 0A --+ Stan = 0A11 dxl $j .

e OAn dx,,

is defined by

d(u) =

ar1.

Ox,

dx1 + ... + au

Ox ,

dxv,.

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

StAn

p ® K(P).

Define D(P) to be the image of dp. Since OA,,.p is a regular local ring, dimK(p) mp/m, = n-r. Thus dirK(p) D(P) < n-r, and dimKlpi D(P) _ n - r if and only if dp : mp/(mP)2 -, D(P) is a K(P)-linear

Theorem A.1. Suppose that P E Aj = spec(K[xl,... , xn]) is a closed point. Then dimK(p) D(P) = n if and only if K(P) is separable over K. 163

Appendix. Smoothness and Non-singularity II

164

Proof. Suppose that P E spec(K [x 1, . . . , x, ]) is a closed point. Let m p C OAA,P be the ideal of P. Let ai be the image of xi in K(P) for 1 < i < n. Let fl (z1) E K[zi] be the minimal polynomial of al over K. We can then inductively define fi(zl, ... , zi) in the polynomial ring K[zi, ... , zi] so that, fi (a1, ... , ai_1, zi) is the minimal polynomial of ai over K(a1 i ... , ai_1) for

2
By construction I C mp and K[xi,...,X,J/I

K(P), so we have I = , fn(xi, .... xn) is a regular system of parameters in OAn,,p, and the images of f1 (x1), f2(x1, x2), ... , fn(x1 i ... , xn) ,trip. Thus fl(xi), f2(xi, x2),

in mp/4 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 v. at P, which in turn holds if and only if the determinant i=1

8xi

is not zero in K(P). This condition holds if and only if K(a1,... , ai) is separable over K(ai, .... ai_r) for I < i < n, which is true if and only if K(P) is separable over K.

Corollary A.2. Suppose that P E AK = spec(K[xi,... , xn]) is a closed point such that K(P) is separable over K. Let mp c OA-,p be the ideal of P, and suppose that Eli, ... , un E mp. Then ui,... 0171 is a regular system of parameters in %,p if and only if I (a; ;e) has rank n at P. Proof. This follows since dp : mp/mr -+ D(P) is an isomorphism if and only if K(P) is separable over K. Lemma A.3. Suppose that A is a regular local ring of dimension b with maximal ideal m, and p C A is an equidimensional ideal such that dim(A/p) = a. Then dirA/m(1) + rn2/m2) < b - a,

and dimA/,n(1) + rn2/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 sequence of k-vector spaces (A.1) 0 -> p/pflnm2 = p+rn2/m2 __+ m/m2 - m'/(na`)2 = ml(p+m2) __+ 0. The lemma now follows, since

dimk m`/(m')2 > a and A' is regular if and only if dimk 7n'/(rn')2 = a (Corollary 11.5 [13]).

A.1. Proofs of the basic theorems

165

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

Lemma A.4. Suppose that A is a regular local ring of dimension n with maximal ideal m, p C A is a prime ideal such that A/p is regular, and dim(A/p) = a - r. Then there exist regular parameters u1, ... , un in A such that p = (111, ... , ur) and ur+1i ... , un map to regular parameters in A/p. Proof. Consider the exact sequence (A.1). Since A and A' are regular, there exist regular parameters u1 i ... , un in A such that u,. } 1, ... , It,, reap to regular parameters in A', and 111, ... , ur are in p. (ul,... , ur) is thus a prime ideal of height r contained in p. Thus p = (u1, ... , ur). Corollary A.5. Suppose that Y is a subvariety of dimension t of a variety X of dimension n. Suppose that q E Y is a non-singular closed point of both Y and X. Then there exist regular parameters U1 i ... , u in OX,q such that 1y,, = (u1, ... , un-t), and un_t+l, ... , u, map to regular parameters in Oy,q. If vn_t_j_1i ... , v are regular parameters in OY,q, there exist regular parameters ul, ... , un as above such that u2 maps to v; for n - t + 1 < i < n.

Lemma A.6. Let I = (f1i ... , f,,) C K

... , xn] be an ideal. Suppose that P E V(I) and J(f; x) has rank n at P. Then K(P) is separable algebraic over K, and n of the polynomials fl, ... , fm form a system of regular parameters in OAIC,p.

Proof. Let m be the ideal of P in K[xl, ... , xn]. After possibly reindexing the fi, we may assume that I J(fl i ... , f,,,; x) I ¢ rn. By the Nullstellensatz, m is the intersection of all maximal ideals n of K[xl, ... , xn] containing art. Thus there exists a closed point Q E Al. with maximal ideal n containing

m such that IJ(fl,..., fn;x)I ¢Ti, so that. J(fli..., f,,; x) has rank n at n. We thus have dimK(Q) D(Q) = n, so that K(Q) is separable over K and (f1, ... , is a regular system of parameters in OAn,Q by Theorem A.1 and Corollary A.2. Since I C rra C it, we have P = Q. Lemma A.7. Suppose that P E AK = spec(K[xl,... , xn]) with ideal

m C K[xl,...,xn] is such that T r i has height n - r and t 1 1 ,---,t ,E

C OAn, p.

Let

ai be the image of xi in K(P) for 1 < i < r. Then the dete77ninunt , ... , tn_r; xr+l i ... , xn) I is non-zero at P if and only if a1, ... , ar is is a rega separating transcendence basis of K(P) over K and I J (t1

ular system of parameters in OAn,1p. xr+1i ... , xn) I is non-zero at P. There Proof. Suppose that I J(t1i ... ) exist 3o, Si, ... , sn-r E K[xl,... , xn] such that si E K[xl,... , xn] for all i,

so ¢ T i and o = ti for 1 < i < n -r. Then I.1(s1,

... , sn-r; xr+1, ... , xn) I

Appendix. Smoothness and Non-singularity II

166

is non-zero at. P. Let K* = K(al,... , ar), let n be the kernel of K* [xr+1, ... , ..y,,] _* K (P),

and let s* for I < i < n - r be the image of si under I1*[xr11,...,xn n is the ideal of a point Q E AK.r. We have K(P) = K*(Q), and at Q, w

w

rixr+l,...;xn

#0.

By Lemma A.6, K*(Q) is separable algebraic over K* and is a regular system of parameters in OA"-" ,Q = K* [xr+l, ... , xn]7z. Since x trdegK K(P) = r, al, ... , ar is a separating transcendence basis of K(P) over K. Thus K* c K[x1, ... , x,,] y,-, and K[x1, ... , xnI7n = K*[xr+1, ... , xn]n,

so that t, ... ,

is a. regular system of parameters in

OA- p = K[x1,... , xn] ii Now suppose that al, .

. .

, ar is a separating transcendence basis of K(P)

is a regular system of parameters in 0A!k,p = over K and t1 .... K[xl,... , xn]1r. Let K* _ K(al,... , ar), and let n be the kernel of K*[xri 1,.... xnJ - K(P). n is the ideal of a point Q E AZ-.r. Then K*(Q) = K(P) and K*(Q) is separable over K*. We have a natural inclusion K* C K[xl, ... , xnJm. Thus K[:r.1....,x.,,]ne = k*['firI1,...,:d:n]vt, , . . . , x , . ]],,. . . By and t1, ... , tn_r is a regular system of parameters in K* t.,,._,.; x7.+ 1, .. , a.n) # 0 at Q, and thus it is also Corollary A.2,

0atP.

Theorem A.8. Suppose that P E An = spec(K[xl,... x,,]) and tl,... , tn_r are regular parameters in °A,,,p. Then rank(J(t; x)) = n - r at P if and ,

only if K(P) is separable generated over K.

Proof. Let a17 I < i < n, be the images of xi in K(P), and lot m be the ideal of P in K[x1...... Suppose that rank(J(t; x)) = n - r at P. After possibly reindexing the xi and tj, we have IJ(tl,.. , to-r; xr+1, , ..%n)I 0 at P. Now K(P) is separably generated over K by Lemrrla A.7.

Suppose that K(P) is separably generated over K. Let (1, ... , Sr be a separating transcendence basis of K(P) over K. There exists q'j(x) E IC[x1:..., xn] for 0 < j < r with 4rp (x) 0 Mn such that o(Q} . Let

-

A. 1. Proofs of the basic theorems

167

K(P) n be the kernel of the K-algebra homomorphism K[xi, ... , ai for I < i < n, xi '- (_ ri if n < i. is is the ideal of defined by xi a point. Q E AK r. Let 'Do(W),'Pi (x), ... , n-r(x) E K[xi, ... , xn] be such

that fio¢mand t,

`Pz(x)

for 1 < i < n - r.

4 o(x)

Let ,1_,.+,i (x) = xn+.i Wo (x) - T j (x) for 1 <.j < r. Let I C K[xl, . , xn i ] thus I (_ n. We will show that be the ideal generated by are in fact a regular system of parameters in 4)1, ... , K[xl, ... , xn+r]n. QAn.1 r,Q

.

It. suffices to show that if F(x) E n, then there exists

A(i) E K[xi;..,:rn+r] such that AF E I. Given such an F, we have an expansion r

j(x) + G(xt, x2....... Cu.),

Wo(x)9F(x) _ 1: j=1

where H j (x) F K[a: t , ... , xn+r] and s is a non-negative integer. Now F E n

implies G E m. Thus there exists an expansion 71

r

G =

fiti

i-1

with fi E K [xl, .. , x,L]71b, and there exist h c_ K[xl , .. , x,,] - {m} and Ci E K[xi,.... xn] such that

hG -

Ci-ti(x) i-1

and hTo(w)2F(x) E I. Since -t1, ... , (Dn is a regular system of parameters hi K [x i , ... , x,n+,.] n. is a separating transcendence basis of K(Q) over K, it follows and

that IJ(`F1...... .,;?.t....... ,,..)I is of is nori-zero at. Q by Lemma A.7. Thus J ( 1 ) 1 rank n - r at P, and by Lemma A.7, CYl, ... , CYr, must, contain a separating transcendence basis of K(P) over K.

Proof that smoothness at a point is well defined Suppose that X is a variety of dimension s over a field K and P E X. In Definition 2.6 we gave a definition for smoothness of X over K at P which

Appendix. Smoothness and Non-singularity II

168

depended on choices of an afline neighborhood U = spec(R) of P and a presentation R K[xl, ... , x,z]/I with I = (f', .... f,,,). We now show that this definition is well defined, so that it is independent

of all choices U, f and x. Consider the sheaf of differentials S2X/h . We have, by the "second fundamental exact sequence" (Theorem 25.2 [66]), a presentation (A.2)

K xl, ... , xa x J(f;x) SZi

Tensoring (A.2) with K(P), we see that

0 R -4 QRIK -> 0.

0 K(P) has rank s if and

only if J(f ; x) has rank n -sat P.

Proof of Theorem 2.8

Let s = dim X . There exists an afliue neighborhood U = spec(R) I = (f r, .. , f1,). Let. rn p E spec(K[x j, ... , xn]) be the ideal of P in A7.. Let

of P in X such that R - K[xl,... ,

t = dim(K[:r;l,... , x,,]/mp).

Suppose that K(P) is separably generated over K and P is a nonsingular point of X. By Theorem A.8, dp :grip/rriy -- D(P) is an isomorphism. By Lemma A.3, dimK(pp) dp(I) = n-s (take A = OA",P, p = 111,,p iii the statement of the lemma, so that b = n - t, a = s - t), which is equivalent

to the condition that J(f ; x) has rank n - .r at P. Thus X is smooth over

K at P. Suppose that X is smooth at P, so that .1(f; :r) has rank n - s at P. Then dimK(p) dp(I) = n - s, so that diumlr(p)(I+ mp)/mp > n - s. Thus P is a non-singular point of X by Lemma A.3.

Remark A.9. With the notation of Theorem 2.8 and it's proof, for any point P E V(I), we have dimx (p) (I + mp)/mp < n - s by Lemma A.3, so that J(f ; x) has rank < n - s at P.

Proof of Theorem 2.7 It suffices to prove that the locus of smooth points of X lying on an open affine subset U = spec(R) of X of the form of Definition 2.6 is open.

Let A = In_s(J(f;x)). Then P E U - V(A) if and only if J(f;x) has rank

A.2. Non-singularity and uniformiziug parameters

169

greater than or equal to n - s at P, which in turn holds if and only if J(f ; x) has rank n - s at P by Remark A.9.

Proof of Theorem 2.10 when K is perfect Suppose that K is perfect. The openness of the set of non-singular points of X follows from Theorem 2.7 and Corollary 2.9. Let rt E X be the generic point of an irreducible component of X. Then OX,,1 is a field which is a regular local ring. Thus 71 is a non-singular point. We conclude that the non-singular points of X are dense in X.

A.2. Non-singularity and uniformizing parameters Theorem A.10. Suppose that X is a variety of dimension r over a field K, and P E X is a closed point such that X is non-singular at P and K(P) is separable over K.

1. Suppose that y,. .. , y,. are regular parameters in Ox,p. Then SZX/K, p

= dyiOX,p ®... Ed dy,.OX p

2. Let fn be the ideal of P in Ox,p. Suppose that

Emc OX,p and dfl, ... , df, generate SZY/K p as an OX,p module. Then fl, ... , f, are regular parameters in Ox,p.

Proof. We can restrict to an affine neighborhood of X, and assume that

X = spcc(K[x1,... , xn]/I). The conclusions of part 1 of the theorem follow from Theorem A.8 when

X = Ak'. In the general case of 1, by Corollary A.5 there exist regular parameters (I/1i ... , y,,) in A - k"[a: , ... , xn] (where m is the ideal of P in spec(K[xi, ... , xn]) such that IA = (yr+1, ... , yn) and %, ... ,1/r) map to the regular parameters (yj, ... , y,.) in B = A/IA. By the second fundamental exact sequence (Theorem 25.2 [66]) we have an exact sequence of B-modules

IA/1"A -+ QA'IK 'A B - na1K 0' This sequence is thus a split short exact sequence of free B-modules, with 122, /1t dyi BW ... EAdyrB, since IA/I2A = yr+i B 0-9,B and 01

-

L7 N dyIB (D

(D

nB.

Suppose that the assumptions of 2 hold. Let R = OX,p. Let yj,... , yr be regular parameters in R. Then dyl, ... , dyr is a basis of the R-module

Appendix. Smoothness and Non-singularity II

170

SZX/K.P by part 1 of this theorem. There are aj1 E R such that r

fiaijyj, 1<j
We have expressions T

?'

dfi -

aijdyi

I

daijyj)

j-1 Er

.,=1 aijdyj for 1 < i < r. By Nakayama's Lemma we have that the w2 generate SZXIK P as an R-module. Thus I((k?)I ¢ fn- and (aij) is i nvertible over R, so that f j.... , fr generate T71. Let 7 vi

Lemma A.11. Suppose that X is a variety of dimension r over a perfect field K. Suppose that P E X is a closed point such that X is non-singular at P, and 'yi i ... , y,. are. regular parameters in Ox,p. Then there e:rists an acne neighborhood U = spec(R) o f P in X such that, y ' , . . . , yr E R, the natural inclusion S = K[yi ..... y,.] -+ R. induces a morphism it : U - AK such that (A.3)

S2RIK = dye I.H. E ... W dyrR,

DerK (R, R)

- IIomp(IIR,/K, R) _

y ii

R W ... EB Ayr R

(With v, (dyj) = Sij), and the following condition holds. Suppose that a E 1 is a closed point, b = 7r(a), and (x1, ... , xr) are regular parameters in OArt b. Then (x1, ... , xr) are regular parameters in O ,a

yi,... , ,y, are called uniformizing parameters on U. Proof. By Theorem A.10, there exists an affine neighborhood U = spec(R) of P in X such that y1, ... , yT E R and dy1,... dy?. is a free basis of S21R,K. Consider the natural inclusion S - K[yi,... , yr] -+ R, with induced mor,

phism 7r : spec(R) -y spec(S). Suppose that a E U is a closed point with

maximal ideal m in R, and b = ir(a), with maximal ideal n in S. Suppose that (x11. .. , xr) are regular parameters in B = S. Let A = Rte,. By construction, ils/K is generated by dy1,... , dy, as an S-module. Thus SZA/K is generated by S1B/K as an A-module. By 1 of Theorem A.10, dxl,... , dxr generate RB/Ft as a B-module, and hence they generate illAIK as an Amodule. By 2 of Theorem A.10, x1, ... , xr is a regular system of parameters in A.

A.3. Higher derivations

171

A.3. Higher derivations Suppose that K is a field and A is a K-algebra.. A higher derivation of A over K of length m is a sequence

D = (Do. Di....1 Dm) if m < oo, and a sequence

D = (Do, Di.... ) if m. - or-, of K-linear maps Di : A - A such that Do = id and Di(xy) =

(A.4)

i-i

D.i(x)D,i-i(y)

A[t]/(tm+1) defined by Et (f) - Em O Di(f )t' is a K-algebra homomorphism with D0(f) = f.

Example A.12. Suppose that A = K[yl,... , y] is a polynomial ring over a field K. For I _< i < r, let tj he the vector in N" with a I in the jyrr E A and th coefficient and 0 everywhere else. For f = E ai...... i,.yi' m c N. define DmE; (f) _

yj'

ai(2)y1

I

... yr7 .

Then Dl = (Do, D?,

is a higher derivation of A over K of length oo for 1 < j < r. Observe that, the L are iterative; that is, DtE, o DkE, = (

i+k ';

D(i+k),,

For all i, k:. Define l/11r....ir = DZr Er o Dt, _lEr _1 O ... O Di,EI

for (iI....,i,.) C N. The D! r,,riT commute with each other. The Dt,,...,ir are called Hasse-Schuiidt derivations [56]. If K has characteristic zero, then w

1

(3i1

it li2! ... ir! Oyi1

... vy'

Appendix. Smoothness and Non-singularity II

172

Lemma A.13. Suppose that X is a variety of dimension r over a perfect field K. Suppose that P E X is a closed point such that X is non-singular at P, yi, ... , yr are regular parameters in Ox,p, and U = spec(R) is an affine neighborhood of P in X such that the conclusions of Lemma A.11

fir

hold. Then there are differential operators Dil..... ir on R, which are uniquely determined on R by the differential operators D4j ... jr on S of Example A. 12, such that if f F OX, p and

=

all,....iryl

. yrr

in Ox,p K(P)[[yi, ... , y,]], where we identify K(P) with the coefficient field of Ox,p containing K, then ai,,...,ir = l i,,...,ir(f)(P) is the residue of Di,..... ir (f) in K(P) for all indices i1i ... , ir. Proof. If K has characteristic zero, we can take. A..r)

Dil, ,ir

ail+ +ir

1

ill

i,.! V11 ..

ayrrir

Suppose that K has characteristic p > 0. Consider the inclusion of K-algebras

S = K[yi,

,

yr] - R

of Lemma A.11, with induced morphism

it : U = spec(R) - V = spec(S). Let Dj* for I < j < r be the higher order derivation of S over K defined in Example A.12. We have n1R/.9 = 0 by 1 of Theorem A.10, (A.3) and the "first fundamental exact sequence" (Theorem 25.1 [66]). Thus rr is non-ramified

by Corollary IV.17.4.2 [45]. By Theorem W.18.10.1 [45] and Proposition IV.6.15.6 [45], rr is etale. We will now prove that Ua extends uniquely to a higher derivation Dj of R over K for 1 < j < r. It suffices to prove that (Do*, DES , ... , D ,,, j) extends to a higher derivation of R over K for all m. The extension of Do* = id to Do = id is immediate. Suppose that an extension (DO, D1, ... , Dm) of (Do*, DES , ... , D,,*,,,,) has been constructed. Then we have a commutative diagram of K-algebra homomorphisms

R 9' T

S

where

f'

R[t]/(tm+')

T

R[t]/(tm+2) m

g(a) _ 1: Di,.,(a)ti, i=o

A.3. Higher derivations

173

71L+1

(a)t-

f (a) i=0

Since (t"t+t)R[t]/(t",+2) is an ideal of square zero, and S -> H is formally etale (Definitions P1.17.3.1 and 17.1.1 [45]), there exists a unique K-algebra homomorphism h making a commutative diagram

R 9'h

R[t]/(t7n+1)

I

? \ S

t H[t]/(I,--12)

Thus there is a unique extension of (Dc, DEf, ... ,

to a higher

derivation of R. Set Dil,...,i,. = Di,.F, o ... p

DiIF1.

Let A = Ox,p. Since K(P) is separable over K, and by Exercise 3.9 of Section 3.2, we can identify the residue field K(P) of A with the coefficient field of A which contains K. Thus we have natural inclusions

K[yi, ... , y,.] C A c A

K(P)[[yi,

We will now show that D , extends uniquely to a higher derivation of A over K for 1 < j < r. Let m he the maximal ideal of A. Let. I. be an indeterminate. Then A[[t]] is a complete local ring with maximal ideal Ta = mA[[t]] +

Define Et : A --> A[[t]] by Et(f) = E°°_o Df,Fj (f )t Et is a K-algebra honloinorphism, since D is a higher derivation. Et is continuous in the m adic topology, as Et(m") C n' for all c,. Thus Ft extends uniquely to a K-algebra homomorphsim A A[[t]]. Thus D,, extends uniquely to a higher derivation of A over K. We can thus extend Di1..... ir uniquely to A. Since K(P) is algebraic over K, and Di,,...,i,. extends D ,,,,j" it follows that Di...... j,. acts on A as stated in the theorem. O

Remark A.A. Suppose that the assumptions of Lemma A.13 hold, and that K is an algebraically closed field. Then the following properties hold.

1. Suppose that Q E U is a closed point and a is the corresponding maximal ideal in R. Then a = ( j , . . , y,.), where yi = yt - ai with .

ai=yi(Q)EKfor 1
fit

2. The differential operators Di...... i. on R. are such that if f c H, and a is a maximal ideal in R, then, with the notation of 1,

=

ai.....,i,.y ... y7

Appendix. Smoothness and Non-singularity II

174

in K [ [

1.

.

. .

..

} ] = Ra, where

ai...... ir =

Di...... ir

(f)(Q) is the

residue of Di,,...,ir (f) in K(Q) for all indices i1, ... , ir.

Proof. The ideal of rr(Q) in S is (y1-a1, ... , yr ar) with ai = yi(Q) E K. Let yi = yi - ai for I < i < r. We have a = (yl.... , yr) by Theorem A.10 2, and 1 follows. The derivations b , ... , are exactly the derivations ... , on S of Example A.12, so they extend to the differential operators nz, ..,ir on S of Example A.12. Now 2 follows from Lemma A.13 applied to Q and y1, ... , /r. O

.r

Lemma A.15. Let notation be as in Lemmas A.11 and A.13. In particular, we have a fixed closed point P E U = spec(R) and differential operators on R. Let I c_ R be an ideal. Define an. ideal C R by

irn,[i = {I)il, .ir(f) I f E I and i1 + -

+ it < m}.

Suppose that Q E U is a closed point, with corresponding ideal mQ in R, and suppose that (z1, ... , zr) is a regular system of parainet.er,s irn niQ. Le.t. be the differential operators constructed on OwQ by the construction of Lemmas A.11 and A.M. Then

(Jm.u),nq = (Di...... ir(f) I f E I and it + ... + it < m}'. Proof. When K has characteristic zero, this follows directly from difl'erentiation and (A.5). Assume that K has positive characteristic. By Lemma A.13 and Theo-" rem W.16.11.2 [45], U is differentiably smooth over spec(K) and {Di,, ..,qr i1, ... , i.,. < m,} is an R-basis of the differential operators of order less than m on R. Since {dzl,... , dzr} is a basis of SZ' /K (by Lemma A.11), I

{D,,..... ir ii, ... , i,. < m} is an R,,,Q-basis of the differential operators of order less than m. on R,,,,,, by Theorem IV.16.11.2 [45]. The lemma folI

0

lows.

A.4. Upper semi-continuity of v,2(Z) Lemma A.M. Suppose that W is a non-singular variety over a perfect field K wad K' is an algebraic field extension of K. Let W' = W X K K', with projection rrl : W' -+ W. Then W' is a non-singular variety over K'. Suppose that q E W is a closed point. Then there exists a closed point q' E W' such that rr1(q') = q. Furthermore,

1. Suppose that (fl, ... , fr) is a reqular system of parameters in OW,vq. Then (fi, . , f,.) is a regular system of parnnaelers inn OWI,gl. . .

A.4. Upper semi-continuity of vq(I)

175

2. If I C OWq is an ideal, then (IOw",q') (l Ow;q = I.

Proof. The fact that W' is non-singular follows from Theorem 2.8 and the definition of smoothness. Let q' E W' be a closed point such that 7ri (q') = q. We first prove 1. Suppose that U = spec(R) is an affine neighborhood of q in W. There is a presentation R = K[xi, ... , xn]/I. We further have that 7r, 1(U) = spec(R'), with R'= Let T7-7,

be the ideal of q in K[a,,... , xn], and n the ideal of q' in K'[x1,... , xn]. By the exact sequence of LeInina A.3; there exists a regular system of parameters f 1, ... , f n in K[x1i... , xn] ,, such that Irt. - (fr+1! ... , f n)ni

and f i maps to fi for I < i < r. By Corollary A.2, the determinant

J(f r , .... f n: x 1; ... , x,,,) I is non-zero at q, so it is non-zero at q'. By Corollary A.2, f 1i ... , f., is a regular system of parameters in K'[x1,... , xn]n. 'T'hus fr, ... , fr is a regular system of parameters in R;, - OW,gr. We will now prove 2. Since 7a maps to a maximal ideal of OW,q 0K K',

6w, q, = (OWq OK K');7 denotes completion with respect to nOW,q OK K'). Thus 6W",q is faithfully flat, over OW,ry, and (IOw,,q') flOW,q - I (by Theorem 7.5 (ii) [661). fl

Suppose that (R, m) is a regular local ring containing a field K of characteristic zero. Let K' be the residue field of R. Suppose that J C H. is an ideal. We define the order of J in N to be vn(J) = max{b I J C rrrh}.

If J is a locally principal ideal, then the order of J is its multiplicity. However, order and multiplicity are difl'erent in general.

Definition A.17. Suppose that q is a point on a variety W and J C 01,V is an ideal sheaf. We denote 1"q(J) ='UW.q (JOWq).

If X C W is a subva.riety, we denote

vq(X) = lIq(2X)

Remark A.18. Let notation be as in Lemma A.16 (except we allow K to be an arbitrary, not necessarily perfect field). Observe that if q E W and q' E W' are such that 7r(q') = q and J C Ow is an ideal sheaf, then VUyv,q (Jq) = UOw.a (Jg(')u.q) = VOw'.q'

vdw'.v' (JgOw',gr ).

176

Appendix. Smoothness and Non-singularity 11

Suppose that X is a noctherian topological space and I is a totally ordered set. f : X - I is said to be an upper semi-continuous function if for any a E I the set {q E X I f (q) > a} is a closed subset of X. Suppose that X is a subvariety of a non-singular variety W. Define

vX:W -ly by vx(q) = vq(X) for q E W. The order vq(X) is defined in Definition A.17.

Theorem A.19. Suppose that K is a pet feel field, W is a non-singular variety over K and I is an ideal sheaf on W. Then liq(I) : W -- N is an upper semi-continuous function.

Proof. Suppose that m E N. Suppose that U = spec(R) is an open affine subset of W such that the conclusions of Lemma A.11 hold on W. Set 1 = 1'(i1, I). With the notation of Lemmas A.11 and A.13, define an ideal Jm,U C R by

Jm,U = {Dii,...,ir(9) I g E I and it + . i,. < m}. By Lemma A. 15, this definition is independent of the choice of yl, ... , yr satisfying the conclusions of Lemma A.11, so our definition of determines a sheaf of ideals Jm on W. We will show that V(Jm) _ {77 E W I vn(I) > m}, from which upper semi-continuity of vq(I) follows. Suppose that. 77 E W is a point:. Let Y be the closure of {c/} in W, and

suppose that P E Y is a closed point such that Y is non-singular at P. By Corollary A.5, Lemma A.11 and Lemma A.15, there exists an affine neighborhood U - spec(R) of P in W such that there are yl, ... , yr E R

with the properties thatyl = ... _yt = 0 (with t < r) are local equations of Y in U and V 1 ,.. . , yr satisfy the conclusions of Lemma A.11. For g E Ip, consider the expansion

g _ Eaii..... iryll ...yr in OW,p L K(P)[[yl, ... , yr]], where we have identified K(P) with the coefficient field of Ow, p containing K and (with the notation of Lemma. A.13)

= Dil,...,ir(g)(P)

A.4. Upper semi-continuity of vq(Z)

177

Then y E V (J,,,.) is equivalent to Dil,...,;,.(g) E (yi,... , yi) whenever + i,. < m., which is equivalent to g E Zp and i1 + ai,,...,i,. = Dil,...,i,(9)(P) = 0

if g E Zp and it +

+ it < ni, which in turn holds if and only if

gEI pflOw,p=ZyP for all g E Zp, where the last equality is by Lemma A.16. Localizing Ow p at y, we see that vn(Z) > m if and only if y E V(J,,,).

Definition A.20. Suppose that W is a non-singular variety over a perfect field K and X C W is a subvariety. For b E N, define

Singb(X) = {q E W I vq(W) ? b}.

Singb(X) is a closed subset of W by Theorem A.19. Let

r= max{vx (q) I q C W}. Suppose that q E Sing, (X). A subvariety H of an affine neighborhood U of q in W is called a hypersurface of maximal contact for X at q i f

1. Sing,.(X) n U C H, and 2. if

Wn1 -W1]4 W

is a sequence of monoidal transforms such that for all i, 7ri is centered at a non-singular subvariety Yi C Sing,.(Xi), where Xz is the strict transform of x on Wi, then the strict transform Hn of H on H,,. such that With the above assumptions, a. non-singular 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 A.21. Suppose that W is a non-singular variety over a perfect field K and X C W is a subvariety, q E W. Let T= Owq, and define Singb(Ox,q) _ {P E spec(T) I v(Ix,gTp) > b).

Then Singb(Ox,q) is a closed set. If J C Ow is a reduced ideal sheaf whose support is 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 conclusions of Lemma A.13 (with p = q), and let notation be as in Lemma A.13 (with p = q). With the further notation of Theorem A.19 (and I = Ix) we see that r(U, j) = Jb,iJ. JqT is a reduced ideal by Theorem 3.6.

178

Appendix. Smoothness and Non-singularity II

Suppose that P E Singb(Ox,q). Then vrP(g) > b for all 9 E TX,q. Thus q E P(b) C T. and there exists h E T - P such that gh. E Pb. Now and induction in the formula (A.4) shows that Di,..... i,(yh) E ii -i*) C P if ii + . + it < b. Thus JyT C P. Dil.... ,i,. (g) E P(b Now suppose that P E spec(T) and JqT C P. Let S = Ow,q. If Jq = P1fl f1P,,,, is a primary decomposition, then all primes Pi are minimal since.lq is reduced. By Theorem 3.6 there are primes Pi1 in T and a function v(i) such that PT = Pii fl fl Pj,(i) is a minimal primary decomposition. Thus JgTP;j = PiTP;f = Pi9Tp;1 for all i, j. We have fl P1 j c P, so Pig C P For some i, j. Moreover, Tx,q C PbSp, implies ZX,q C PPfTp3, which in turn implies 2X,q C PbTP3,. Thus vrp (Zx yTP) > b and P E Singb(OX,q).

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Index Abhyankar, 12, 28, 42, 45, 130, 131, 152, 153, 157, 158, 160 Abramovich, 9 Albanese, 55 approximate manifold, 106 Atiyah, viii

Bierstone, 61 Bravo, 61 Cano, 13 Ohevalley, 24 Cohen, 22 completion of the square, 24 Cossart, 131 curve, 2

de Jong, 61, 131 derivations Hasse-Schmidt, 171 higher, 171 Eisenbud, viii Encinas, 61 exceptional divisor, 20, 39, 40 exceptional locus, 39, 40

Giraud, 70 Harris, viii Hartshorne, viii, 2 Hauser, 13, 61, 132 Heinzer, 13, 157 Hensel's Lemma, 23 Hironaka, 2, 15, 61, 62, 130 hypersurface, 2

intersection reduced set-theoretic, 62 scheme-theoretic, 2, 62 isolated subgroup, 134 Jung, 55 Karu, 9 Kawasaki, 11 Kuhlmann, 13 Levi, 55 Lipman, 130 local uniformization, 12, 130, 13R, 147, 156

Macaulayfcation, 11 Macdonald, viii MacLanc, 137 Matsuki, 9 maximal caut.art formal curve, 28 formal hypersurface, 177 hypersurface, 56, 67, 70, 131, 177 Milmau, 61 Moll, 61

monoial transform, 16, 39, 40 monouiialization, 13, 157 morphism birutiunal, 9 monomial, 14 proper, 9 multiplicity, 3, 19, 25, 175

Nagata, 11 Narasimhan, 131 Newton, 1, 3, 6, 12 185

Index

186

Newton Polygon, 34 normalization, 3, 10, 12, 26 ord, 22, 138

order, 3, 19, 25, 138 vj(R), 63 1R, 175

vX, 63, 176 vv(J), 175 vv(X), 175

upper semi-continuous, 176

l iltant, 13 principalization of ideals, 40, 43, 99 l uiseux, 6 Puiseux series, 6, 11, 137 resolution of differential forms, 13 resolution of indeterminacy, 41, 100 resolution of singularities, 9, 40, 41, 61, 100 102

curve, 10, 12, 26, 27, 30, 38, 39, 42 embedded, 41, 56, 99

cquivariant, 103 excellent surface, 130 positive characteristic, 10, 12, 13, 30, 42, 105, 152

surface, 12, 451 105, 152 three-fold, 12, 152 resolution of vector fields, 13 ring affine, 2

regular, 7 Rotthaus, 9, 13 scheme Cohen-Macaulay, 11 excellent, 9 non-singular, 7 quasi-excellent, 9

smooth, 7 Schilling, 137 Seidenberg, 13 Serre, 10 Shannon, 151 singularity

Sing(J,b), 66 Singb(X), 45, 105, 177 Singl,((7X,y), 177 SNCs, 29, 40, 49 Spivakovsky, 13 surface, 2

tangent space, 8 Teissier, 13 three fold, 2

transform strict, 20, 25, 38, 65 total, 21, 25, 37 weak, 65

Tschirnhausen transformation, 24, 28, 51, 56, 61, 67, 70

uniformizing parameters, 170 upper semi-continuous, 176 valuation, 12, 133 dimension, 134 discrete, 134 rank, 134 rational rank, 136 ring, 134 variety, 2 integral, 2 quasi-complete, 15 subvaricty, 2 Villamayor, 61 Walker, 55 Weierstrass preparation theorem, 24 Wiegand, 13 Wlodarczyk, 9, 61 Zariski, 1, 12, 55, 133, 147, 148, 152, 1 56

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