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Algebraic resolution of formal ideals along a valuation

Dissertation
Author: Samar El Hitti
Abstract:
Let X be a possibly singular complete algebraic variety, defined over a field k of characteristic zero. X is nonsingular at p ∈ X if Ox,p is a regular local ring. The problem of resolution of singularities is to show that there exists a nonsingular complete variety X , which birationally dominates X . Resolution of singularities (in characteristic zero) was proved by Hironaka in 1964. The valuation theoretic analogue to resolution of singularities is local uniformization. Let ν be a valuation of the function field of X , ν dominates a unique point p , on any complete variety Y , which birationally dominates X . The problem of local uniformization is to show that, given a valuation ν of the function field of X , there exists a complete variety Y , which birationally dominates X such that the center of ν on Y , is a regular local ring. Zariski proved local uniformization (in characteristic zero) in 1944. His proof gives a very detailed analysis of rank 1 valuations, and produces a resolution which re ects invariants of the valuation. We extend Zariski's methods to higher rank to give a proof of local uniformization which reflects important properties of the valuation. We simultaneously resolve the centers of all the composite valuations, and resolve certain formal ideals associated to the valuation.

TABLE OF CONTENTS ACKNOWLEDGMENTS...................................................ii ABSTRACT..............................................................iii Chapter 1.INTRODUCTION.................................................1 History of the problem. Statement of the main result. 2.PRELIMINARIES.................................................7 Notations and Definitions 3.Example...........................................................9 4.Resolution in highest height......................................11 Perron Transforms. Etale Perron Transforms. Structure Theorems. 5.Resolution in all height............................................82 Perron Transforms. Extension of results to all height. 6.Local Uniformization.............................................94 Proof of the main result. BIBLIOGRAPHY........................................................97 VITA....................................................................103 iv

Chapter 1 Introduction 1.1 History of the problem One of the fundamental problems in Algebraic Geometry is the problem of Resolu- tion of Singularities.If Y is a singular algebraic variety,a resolution of singularities of Y is a proper mapping φ:X → Y which is an isomorphism on a dense open subset of Y,such that X is nonsingular.Hironaka [29] proved in 1964 that there exists a resolution of Y if Y is defined over a field of characteristic zero.The proof uses the existence of a hypersurface of maximal contact to reduce to an induction on the dimension of Y.There have been significant simplifications of this theo- rem in recent years,including Abramovich and de Jong [5],Bierstone and Milman [7],Bogomolov and Pantev [8],Bravo,Encinas and Villamayor [9],Encinas and Hauser [21],Encinas and Villamayor [22],Hauser [26],Koll´ar [35],Villamayor [44], and Wlodarczyk [45]. Since Zariski introduced general valuation theory into algebraic geometry,val- uations have been important in addressing resolution problems. 1

Suppose that K is an algebraic function field over a base field k,and V is a valuation ring of K.V determines a unique center on a proper variety X whose function field is K.The valuation gives a way of reducing a global problem on X, such as resolution,to a local problem,studying the local rings of centers of V on different varieties X whose function field is K. The valuation theoretic analogue of resolution of singularities is local uni- formization.A variety X is nonsingular at a point p if and only if the local ring O X,p is a regular local ring. The problemof local uniformization is to find a regular local ring Ressentially of finite type over k with quotient field K such that the valuation ring V dominates R. That is,R ⊂ V and m V ∩R = m R .In 1944,Zariski [48] proved local uniformization over fields of characteristic zero.To be precise,he proved: Theorem 1.1.1.(Zariski) Suppose that f ∈ R.Then there exists a birational extension of regular local rings R →R 1 such that R 1 is dominated by ν,and ord R 1 f ≤ 1 where f is the strict transform of f in R 1 .If ν has rank 1,then there exists R 1 such that f is a unit in R 1 . Zariski first proved local uniformization for two-dimensional function fields over an algebraically closed field of characteristic zero in [47].He later proved local uni- formization for algebraic function fields of characteristic 0 in [48]. Abhyankar has proven local uniformization in positive characteristic for two di- 2

mensional function fields,surfaces and three dimensional varieties [1],[2]. One of the most important techniques in studying resolution problems is to pass to the completion of a local ring (the germ of a singularity).This allows us to reduce local questions on singularities to problems on power series. The first question which arises on completion is if the following generalization of local uniformization is true: Question 1.1.2.Given f ∈ ˆ R,does there exist a birational extension R → R 1 where R 1 is a regular local ring dominated by V such that ord ˆ R 1 f ≤ 1,where f is the strict transform of f in ˆ R 1 ? The answer is surprisingly NO!We give a simple counter example to our ques- tion in Chapter 3,which comes from a discrete valuation. The example can be understood in terms of an extension of our valuation ring V to a valuation ring dominating ˆ R.It is a fact that the rank (page 7) of a valuation V dominating R often increases when extending the valuation to a valuation ring ˆ V dominating ˆ R.Some papers where this is studied are Spivakovsky [41],Heinzer and Sally [28],and Cutkosky and Ghezzi [19]. The first measure of this increase of rank is the prime ideal 3

Q ˆ R = {f ∈ ˆ R | ν(f) ≥ n for all n ∈ N}. This prime has been previously defined and studied by Teissier [43],Cutkosky [16] and Spivakovsky. If the rank V = 1,then there is a unique extension of V to the quotient field of ˆ R/Q ˆ R dominating ˆ R/Q ˆ R .Thus the rank of the extension does not increase,and the prime ideal Q ˆ R led to this obstruction. In spite of the fact that we cannot resolve the singularity of f = 0 by a birational extension of R,we can resolve the formal singularity,whose local ring is ˆ R/Q ˆ R ,by a birational extension of R.This is proven for valuations of rank 1 by Cutkosky and Ghezzi [19]. Theorem 1.1.3.(Cutkosky,Ghezzi) Suppose that rank V = 1.Then there exists a birational extension R →R 1 where R 1 is a regular local ring dominated by ν such that Q ˆ R 1 is a regular prime Zariski proves local uniformization by constructing special birational extensions R → R 1 dominated by rank 1 valuation rings which he calls Perron transforms. Cutkosky and Ghezzi make essential use of Perron transforms and Zariski’s reso- lution algorithm in their proof of Theorem 1.1.3. We extend Perron transforms to arbitrary rank in Chapter 5,and prove a strong form of local uniformization,which generalizes both Theorem 1.1.1 and Theorem 1.1.3. 4

1.2 Statement of the main result First we introduce some notation which is necessary for the statement of out the- orem.Let (0) = P t V ⊂ · · · ⊂ P 1 V ⊂ P 0 V be the chain of prime ideals in V.Let (0) = P t R ⊂ · · · ⊂ P 1 R ⊂ P 0 R be the induced chain of prime ideals in R,where P i R = P i V ∩R.Let ν i be a valuation whose valuation ring is V i = V P i V . Consider the following condition (1.1) on a Cauchy sequence f = {f n } in

R P i R . For all l ∈ N,there exists n l ∈ N such that ν i (f n ) ≥ lν(m

R P i R ) if n ≥ n l .(1.1) Define the prime ideal Q

R P i R ⊂

R P i R for the valuation ring V P i V by Q

R P i R =

f ∈

R P i R | f satisfies (1.1).

Theorem 1.2.1.Suppose that R is a local domain which is essentially of finite type over a field k of characteristic zero,and V is a valuation ring of the quotient field of R which dominates R.Let 5

(0) = P t V ⊂ · · · ⊂ P 0 V be the chain of prime ideals of V.Then there exists a birational extension R →R 1 such that R 1 is a regular local ring and V dominates R 1 .Further,P i R 1 = P i V ∩ R 1 are regular primes for all i,and Q

R 1 P i R 1 ⊂

R 1 P i R 1 are regular primes for all i. 6

Chapter 2 Preliminaries In this section,we introduce some notations and assumptions that will hold in chapters 4 and 5. Suppose that T is a regular local ring of dimension q,which is essentially of finite type over a field k of characteristic zero,with maximal ideal m T ,such that T/m T is an algebraic extension of k.Suppose that ν is a valuation of the quotient field of T which dominates T.Let QF(T) be the quotient field of T. By definition ν is a homomorphism ν:QF(T) ∗ →Γ from the multiplicative group of QF(T) onto an ordered abelian group Γ such that: 1.ν(ab) = ν(a) +ν(b) for a,b ∈ QF(T) ∗ , 2.ν(a +b) ≥ min {ν(a),ν(b)} for a,b ∈ QF(T) ∗ , 3.ν(c) = 0 for 0 = c ∈ k. We extend ν to QF(T) by setting ν(0) = ∞. Let V = {a ∈ QF(T)|ν(a) ≥ 0}.Then V is a ring and is called the valuation ring of ν. Let m V = {a ∈ QF(T)|ν(a) > 0}.Then m V is the unique maximal ideal of V. The rank of a valuation ring V is the number of proper prime ideals in V,which are necessarily ordered by inclusion. 7

The rational rank of ν is the maximal number of rationally independent ele- ments in Γ,which is bounded above by the dimension of T. Let (0) ⊂ · · · ⊂ P V ⊂ m V be the chain of prime ideals in V. Let Γ 1 ⊂ Γ V be the rank 1 isolated subgroup of the valuation ring V/P V .We have an embedding Γ 1 ⊂ R of ordered groups,such that Z ⊂ Γ 1 .In this way we identify the integers with a subgroup of Γ 1 .We will sometimes say that an element γ ∈ Γ V is ∞ if γ ∈ Γ 1 ,and γ < ∞ if γ ∈ Γ 1 .If γ ∈ Γ 1 ,then there exists n ∈ N such that γ ≤ n. The maximal ideal of T is m T = m V ∩T. In chapters 4 and 5,we assume that trdeg T/m T V/m V = 0.(2.1) Let P T = P V ∩T. Suppose that V/P V has rational rank s. In chapters 4 and 5,we assume that trdeg T P T /P T T P T V P V /P V V P V = trdeg QF(T/P T ) QF(V/P V ) = 0.(2.2) 8

Chapter 3 Example Example 3.0.2.There exists a discrete valuation ring V dominating a regular local ring R of dimension 3 such that for all r ≥ 2,there exists f ∈ ˆ R such that for all birational extensions R →R 1 of regular local rings dominated by V,the strict transform of f has order ≥ r in ˆ R 1 . Proof.Let t,φ(t),ψ(t) ∈ k[[t]] be algebraically independent elements of positive order. We have the inclusion k(x,y,z) →k((t)) defined by x = t,y = φ(t) and z = ψ(t). The order valuation on k((t)) (with valuation ring k[[t]]) restricts to a discrete rank 1 valuation on k(x,y,z),dominating R = k[x,y,z] (x,y,z) . Q ˆ R = (y −φ(x),z −ψ(x)) ⊂ ˆ R is a regular prime of height 2,and it defines a curve γ ⊂ Spec ( ˆ R). Q ˆ R ∩R = (0). 9

Suppose that r ∈ N(r ≥ 2). Let f = (y −φ(x)) r +(z −ψ(x)) r+1 ∈ ˆ R. ord ˆ R f = r and ord γ f = ord (y−φ(x),z−ψ(x)) f = r. Suppose that R → R 1 is a birational extension where R 1 is a regular local ring dominated by ν. In ˆ R 1 ,write y −φ(x) = h 1 g 1 and z −ψ(x) = h 2 g 2 where g 1 is the strict transform of y −φ(x) in ˆ R 1 ,h 1 ∈ R 1 and g 2 is the strict transform of z −ψ(x) in ˆ R 1 ,h 2 ∈ R 1 . ∞= ν(y −φ(x)) = ν(h 1 ) +ν(g 1 ) and ν(h 1 ) < ∞thus ν(g 1 ) = ∞. Similarly,ν(g 2 ) = ∞. Let γ be the strict transform of γ in Spec( ˆ R 1 ),I γ = (g 1 ,g 2 ) = Q ˆ R 1 ⊂ m ˆ R 1 ord ˆ R 1 f ≥ ord γ f = ord γ f = r In the previous example,we have f ∈ Q ˆ R ,and has value larger than any element in the value group. This means that the rank of the valuation ring V must increase when passing to the completion ˆ R. 10

Chapter 4 Resolution in highest height 4.1 Perron Transforms Throughout this chapter we assume that the assumptions of chapter 2 hold.We will define two types of Peron Transforms. Perron Transforms of type (1,0): Suppose that x 1 ,...,x s ,...,x p ,...,x q is a regular system of parameters in T,such that s ≤ p,x 1 ,...,x p ∈ P T , x p+1 ,...,x q ∈ P T ,and ν(x 1 ),...,ν(x s ) are rationally independent. Let τ i = ν(x i ) for 1 ≤ i ≤ s. We define two types of transforms of type (1,0): • Transforms of type I. Set τ i (0) = τ i for 1 ≤ i ≤ s.For each positive integer h define s positive, rationally independent real numbers τ 1 (h),...,τ s (h) by the ”Algorithm of 11

Perron” [48]. τ 1 (h −1) = τ s (h) τ 2 (h −1) = τ 1 (h) +a 2 (h −1)τ s (h) . . . τ s (h −1) = τ s−1 (h) +a s (h −1)τ s (h) Where a j (h −1) =

τ j (h −1) τ 1 (h −1)

,2 ≤ j ≤ s the greatest integer in τ j (h) τ 1 (h) .There are nonnegative integers A i (h) such that τ i = A i (h)τ 1 (h) +A i (h +1)τ 2 (h) +· · · +A i (h +s −1)τ s (h) for 1 ≤ i ≤ s. Then Det(A i (h +j)) = (−1) h(s−1) (See [48] page 385). These numbers have the important property that lim h→∞ A i (h) A 1 (h) = ν(x i ) ν(x 1 ) (4.1) we refer to [48] page 385. Let A i (h +j) = a ij+1 ,and define: x 1 = x 1 (1) a 11 ...x s (1) a 1s . . . x s = x 1 (1) a s1 ...x s (1) a ss x s+1 = x s+1 (1) . . . x q = x q (1). (4.2) Then Det(a ij ) = ±1 and ν(x 1 (1)) = τ 1 (h),...,ν(x s (1)) = τ s (h) are rationally independent.We necessarily have that x 1 (1),...,x s (1) ∈ P T 1 . Define a transformation T →T 1 of type I along ν by T 1 = T[x 1 (1),...,x s (1)] T[x 1 (1),...,x s (1)]∩m V . 12

T 1 is a regular local ring,QF(T) = QF(T 1 ) and ν dominates T 1 . • Transforms of type II r . Now we define T →T 1 of type II r along ν,we refer to [48] (with the restric- tion that s +1 ≤ r ≤ p),as follows: Set ν(x r ) = τ r .τ r is rationally dependent on τ 1 ,...,τ s since the ratio- nal rank of ν is s.There are thus integers λ,λ 1 ,...,λ s such that λ > 0,(λ,λ 1 ,...,λ s ) = 1 and λτ r = λ 1 τ 1 +· · · +λ s τ s .(4.3) We first perform a transform T → ˜ T(1) of type I along ν where ˜ T(1) has regular parameters ˜x 1 (1),...,˜x s (1) defined as in (4.2).Then ν(˜x i (1)) = τ i (h) for 1 ≤ i ≤ s,v(˜x r (1)) = τ r .Set λ i (h) = λ 1 A 1 (h +i −1) +λ 2 A 2 (h +i −1) +· · · +λ s A s (h +i −1) for 1 ≤ i ≤ s.Then λτ r = λ 1 (h)τ 1 (h) +· · · +λ s (h)τ s (h). Take h sufficiently large that all λ i (h) > 0.This is possible by (4.1),since λ 1 τ 1 +· · · +λ s τ s > 0.We still have (λ,λ 1 (h),...,λ s (h)) = 1 since Det(A i (h+ j − 1) = ±1.After re-indexing the ˜x i (1),we may suppose that λ 1 (h) is not divisible by λ.Let λ 1 (h) = λµ + λ

,with 0 < λ

< λ.Now perform the following transform ˜ T(1) → ˜ T(2) along ν defined by ˜x 1 (1) = ˜x r (2), 13

˜x r (1) = ˜x 1 (2)˜x r (2) µ ,and ˜x i (1) = ˜x i (2) otherwise.Set τ

i = ν(˜x i (2)) for all i. τ

1 ,...,τ

s ,τ

r are positive and λ

τ

r = λ

1 τ

1 +· · · +λ

s τ

s where λ

1 = λ,λ

i = −λ i (h) for 2 ≤ i ≤ s.Thus we have achieved a reduction on λ,and by repeating the above procedure,we reduce to the case τ r = λ 1 τ 1 +· · · +λ s τ s In this case we define x 1 = N 1 . . . x s = N s x r = N λ 1 1 ...N λ s s N r . Thus in the general case,there exists a ij ∈ N,1 ≤ i,j ≤ s +1 such that x 1 = N a 11 1 ...N a 1s s N a 1s+1 r . . . x s = N a s1 1 ...N a ss s N a ss+1 r x r = N a s+11 1 ...N a s+1s s N a s+1s+1 r . where Det(a ij ) = ±1 and ν(N 1 ),...,ν(N s ) are positive and rationally inde- pendent,and ν(N r ) = 0. Define a transformation T →T 1 of type II r along ν by T 1 = T[N 1 ,...,N s ,N r ] T[N 1 ,...,N s ,N r ]∩m V . We necessarily have that N 1 ,...,N s ,N r ∈ P T 1 ,QF(T) = QF(T 1 ) and ν dom- inates T 1 . 14

Perron Transforms of type (2,0). Suppose that x 1 ,...,x s ,...,x p ,...,x q is a regular system of parameters in T such that ν(x 1 ),...,ν(x s ) are rationally independent. Suppose that d 1 ,...,d s ∈ N and that x j ∈ P T Define a transformation T →T 1 of type (2,0) along ν by T 1 = T[ x j x d 1 1 ...x d s s ] T[ x j x d 1 1 ...x d s s ]∩m V . In all cases,by (2.1),we have that dim(T 1 ) = dim(T) and T 1 /m T 1 is a finite extension of T/m T . For an ideal I ⊂ T,let ν(I) = min{ν(f) | f ∈ I}. Consider the following condition (4.4) on a Cauchy sequence {f n } in T. For all l ∈ N,there exists n l ∈ N such that ν(f n ) ≥ lν(m) if n ≥ n l .(4.4) Lemma 4.1.1.1.Suppose that {f n } and {g n } are two Cauchy sequences in T converging to f ∈ ˆ T and {f n } satisfies (4.4).Then {g n } satisfies (4.4). 2.Let Q ˆ T =

f ∈ ˆ T | A Cauchy sequence {f n } in T which converges to f satisfies (4.4)

. Then Q ˆ T is a prime ideal in ˆ T. 3.Q ˆ T ∩T = P T . 15

Suppose that f ∈ ˆ T is not in Q ˆ T .Let {f n } be a Cauchy sequence in T which converges to f.There exists an l ∈ N such that there are arbitrarily large n with ν(f n ) < lν(m).We can thus choose n 0 such that f n − f n 0 ∈ m l if n ≥ n 0 and ν(f n 0 ) < lν(m).For n ≥ n 0 ,we have f n = f n 0 +h with h ∈ m l .ν(h) ≥ lν(m) > ν(f n 0 ) implies ν(f n ) = ν(f n 0 ) for n ≥ n 0 . A similar calculation shows that the above value ν(f n 0 ) is independent of choice of n 0 satisfying the above conditions,and is independent of choice of Cauchy se- quence converging to f. We may thus define ν(f) = ν(f n ) for sufficiently large n,if f ∈ ˆ T −Q ˆ T .We will sometimes write ν(f) = ∞ if f ∈ Q ˆ T .Observe that an extension of ν to a valuation of QF( ˆ T) which dominates ˆ T is uniquely determined on elements of ˆ T −Q ˆ T .We now identify ν with an extension of ν to QF( ˆ T) which dominates ˆ T. Let σ(T) = dim( ˆ T/Q ˆ T ) and τ(T) = dim(T/P T ). We have σ(T) ≤ τ(T) with equality if Q ˆ T = P T ˆ T. Let ω(T) = dim(T) −dim T/m T (P T /m 2 T ∩P T ). We have dim T/m T (P T /m 2 T ∩P T ) ≤ height P T with equality if and only if T/P T is a regular local ring. Thus w(T) ≥ dim(T) − height P T = dim(T/P T ) and w(T) = dim(T/P T ) if and only if T/P T is a regular local ring. 16

4.2 Etale Perron Transforms Suppose that x 1 ,...,x s ,...,x p ,...,x q is a system of regular parameters in T,such that x 1 ,...,x s ,...,x p /∈ P T ,x p+1 ,...,x q ∈ P T and ν(x 1 ),...,ν(x s ) are rationally independent. Let β 0 ∈ ˆ T be a primitive element of T/m T over k.Let U = T[β 0 ] m ˆ T ∩T [β 0 ] ⊂ ˆ T. We thus have that the subfield k[β 0 ] ⊂ U is a coefficient field of U.We will identify k[β 0 ] with U/m U . ν extends to a valuation of QF(U) which dominates U by restricting our exten- sion of ν to QF( ˆ T) to QF(U).ν is uniquely determined on elements of U which are not in Q ˆ U = Q ˆ T . For the moment,let us call our extension of ν to QF( ˆ T) ˆν and let ˆ V be the valuation ring of ˆν.Let ν

be the restriction of ˆν to QF(U) with valuation ring V

= ˆ V ∩QF(U). Since QF(U) is finite over QF(T),ν and ν

both have the same rank and rational rank. Thus the chain of prime ideals in V

is (0) ⊂ · · · ⊂ p V

⊂ m V

where p V

∩QF(T) = p V ,and the rank 1 valuation rings V

/p V

and V/p V have the same rational rank s. Let p U = p V

∩U. 17

We have τ(U) = τ(T) by (2.2) and w(U) ≤ w(T) with equality if P U = P T U. Let p = ω(T) and p 0 = ω(U),we have p 0 ≤ p. We also define two types of etale Perron transforms: • Etale Perron transform of type I: We have that x 1 ,...,x s /∈ P U . Let U →U 1 = U[x 1 (1),...,x s (1)] m V ∩U[x 1 (1),...,x s (1)] be a transformation of type I along ν.Define x 1 (1),...,x s (1) by: x 1 = x 1 (1) a 11 ...x s (1) a 1s . . . x s = x 1 (1) a s1 ...x s (1) a ss x s+1 = x s+1 (1) . . . x q = x q (1) Where Det(a ij ) = ±1. We have that the field k[β 0 ] ⊂ U 1 has the property that k[β 0 ] = U/m U = U 1 /m U 1 .We further have that U →U 1 is a birational extension. We identify ν with an extension of ν to the Quotient field of ˆ U 1 which domi- nates ˆ U 1 .Let p 1 = ω(U 1 ).We have that s ≤ p 1 = ω(U 1 ) ≤ ω(U) = p 0 ≤ w(T) = p. We have that x 1 (1),...,x s (1),x s+1 ,...,x q 18

a regular system of parameters in U 1 ,such that ν(x 1 (1)),...,ν(x s (1)) are rationally independent. T →U →U 1 (with the regular parameters x 1 ,...,x s ,...,x q and x 1 (1),...,x s (1),x s+1 ,...,x q ) is called an etale Perron transform of type I along ν. • Etale Perron transform of type II r :(s +1 ≤ r ≤ p) Let λ(t 1 ,...,t r−1 ) be a polynomial in the polynomial ring U/m U [t 1 ,...,t r−1 ],in the variables t 1 ,...,t r−1 . We have x 1 ,...,x s ∈ P U and we assume that x

r = x r −λ(x 1 ,...,x r−1 ) ∈ Q ˆ U . Let U →U

= U[N 1 ,...,N s ,N r ] m V ∩U[N 1 ,...,N s ,N r ] be a transformation of type II r along ν.Define N 1 ,...,N s ,N r by x 1 = N a 11 1 ...N a 1s s N a 1s+1 r . . . x s = N a s1 1 ...N a ss s N a ss+1 r x

r = N a s+11 1 ...N a s+1s s N a s+1s+1 r 19

where Det(a ij ) = ±1,ν(N 1 ),...,ν(N s ) are rationally independent and ν(N r ) = 0.Define x 1 (1) = N 1 ,...,x s (1) = N s , x r (1) = N r . Let β 1 ∈ ˆ U

be a primitive element of U

/m U

over k.Let U 1 = U

[β 1 ] m ˆ U

∩U

[β 1 ] . We identify ν with an extension of ν to the Quotient field of ˆ U 1 which domi- nates ˆ U 1 . Let α be the residue of x r (1) in U 1 /m U 1 .Let ˜x i (1) = x i if s < i ≤ p,i = r and ˜x r (1) = x r (1) − α. x 1 (1),...,x s (1),˜x s+1 (1),...,˜x r (1),...,˜x p (1),x p+1 ,...,x q are regular param- eters in U 1 .We necessarily have that ν(x 1 (1)),...,ν(x s (1)) are rationally independent and x 1 (1),...,x s (1) ∈ P U 1 . Let p 1 = ω(U 1 ).We have that s ≤ p 1 = ω(U 1 ) ≤ ω(U) = p 0 ≤ ω(T) = p. We choose a regular system of parameters x 1 (1),...,x p 1 (1),...,x p (1),x p+1 ,...,x q (4.5) in U 1 such that x 1 (1),...,x p 1 (1)/∈ P U 1 ,x p 1 +1 (1),...,x p (1),x p+1 ,...,x q ∈ P U 1 , and there exists a 1-1 map σ:{s + 1,...,p 1 } → {s + 1,...,p} such that x i (1) = ˜x σ(i) (1) for s +1 ≤ i ≤ p 1 . 20

Full document contains 109 pages
Abstract: Let X be a possibly singular complete algebraic variety, defined over a field k of characteristic zero. X is nonsingular at p ∈ X if Ox,p is a regular local ring. The problem of resolution of singularities is to show that there exists a nonsingular complete variety X , which birationally dominates X . Resolution of singularities (in characteristic zero) was proved by Hironaka in 1964. The valuation theoretic analogue to resolution of singularities is local uniformization. Let ν be a valuation of the function field of X , ν dominates a unique point p , on any complete variety Y , which birationally dominates X . The problem of local uniformization is to show that, given a valuation ν of the function field of X , there exists a complete variety Y , which birationally dominates X such that the center of ν on Y , is a regular local ring. Zariski proved local uniformization (in characteristic zero) in 1944. His proof gives a very detailed analysis of rank 1 valuations, and produces a resolution which re ects invariants of the valuation. We extend Zariski's methods to higher rank to give a proof of local uniformization which reflects important properties of the valuation. We simultaneously resolve the centers of all the composite valuations, and resolve certain formal ideals associated to the valuation.