Reduction of Singularities of Three-Dimensional Line Foliations
aa r X i v : . [ m a t h . AG ] S e p REDUCTION OF SINGULARITIES OF THREE-DIMENSIONALLINE FOLIATIONS
F. CANO; C. ROCHE; M. SPIVAKOVSKY
Dedicated to Heisuke Hironaka on the occasion of his 80th birthday.
Contents
0. Introduction 2
Part 1. Zero dimensional arquimedean valuations
41. Parameterized regular local models 41.1. Coordinate changes and blow-ups 51.2. Puiseux packages of blow-ups 51.3. Statements in terms of parameterized regular local models 72. The combinatorial case ( r = n ) 72.1. Newton polyhedron 82.2. The effect of a blow-up 82.3. End of the proof of Proposition 4. 83. The Newton-Puiseux Polygon 93.1. Newton-Puiseux Polygon of a foliation 93.2. The initial parts 103.3. The expression of the derivatives after a Puiseux package 114. Rational co-rank one 124.1. The effect of a Puiseux package 124.2. Getting a formal hypersurface of transversal maximal contact 154.3. The case of dimension two 155. Etale Puiseux packages 165.1. Review on etale neighborhoods 165.2. Etale Puiseux packages 166. Rational rank one 176.1. The independent coefficient 186.2. Invariants from the Newton Puiseux Polygon 196.3. Prepared situations in two variables 206.4. Effect of etale w -Puiseux packages 206.5. Preparations in three variables. 236.6. Critical initial part and critical polynomial 246.7. Stability of the main height. Dominant critical segment 266.8. Stability of the main height. Recessive critical segment 276.9. The condition of Tchirnhaus 286.10. Tchirnhausen preparation. Recessive case 286.11. Tchirnhausen preparation. Dominant case 31 Date : 2010 april, the 30th.
7. Maximal contact 317.1. Maximal contact with rational rank two. 327.2. Maximal contact with rational rank one 34
Part 2. Higher rank and higher dimensional valuations
Part 3. Globalization
Introduction
We give a birational reduction of singularities for one dimensional foliations inambient spaces of dimension three. To do this, we first prove the existence of aLocal Uniformization in the sense of Zariski [19]. The reduction of singularities isthen obtained by a gluing procedure for Local Uniformization similar to Zariski’sone in [20].Let K be the field of rational functions of a projective algebraic variety M ofdimension n over an algebraically closed field k of characteristic zero. We prove thefollowing theorem Theorem 1 (Local Uniformization) . Assume that n = 3 . Consider a k -valuation ν of K and a foliation by lines L ⊂
Der k K . There is a composition of a finite sequenceof blow-ups with non singular centers M → M such that L is log-elementary atthe center Y ⊂ M of ν . A foliation by lines (or simply a foliation ) is any 1-dimensional K -vector sub-space L ⊂
Der k K . Recall that space of k -derivations Der k K is a n -dimensional K -vector space. The notion of “log-elementary” comes from results in [4]. Letus explain it. Take a regular point P in a projective model M . We know thatDer k O M,P ⊂ Der k K is a free O M,P -module of rank n generated by the partialderivatives ∂/∂x i , i = 1 , , . . . , n , for any regular system of parameters x , x , . . . , x n of the local ring O M,P . Moreover L M,P = L ∩
Der k O M,P is a free rank one sub-module of Der k O M,P that we call the local foliation inducedby L at M, P . We say that L is non-singular at P if L M,P
6⊂ M
M,P
Der k O M,P ,where M M,P ⊂ O
M,P is the maximal ideal. We say that L is log-elementary at P if there is a regular system of parameters z , z , . . . , z n , an integer 0 ≤ e ≤ n and ξ ∈ L M,P of the form ξ = e X i =1 a i z i ∂∂z i + n X i = e +1 a i ∂∂z i , ( a i ∈ O M,P , i = 1 , , . . . , n )with a j / ∈ M M,P for at least one index j . If Y ⊂ M is an irreducible subvariety,we say that L is non-singular at Y , respectively, log-elementary at Y , if it is so ata generic point of Y . Note in particular that M must be non-singular at a genericpoint of Y . EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 3
Theorem 1 may be globalized as a consequence of a patching procedure developedby O. Piltant [12], which is an axiomatic adaptation of the one given by Zariski inthe case of varieties [19]. We obtain the following birational result of reduction ofsingularities of foliations in an ambient space of dimension three
Theorem 2.
Assume that n = 3 and let L ⊂
Der k K be a foliation. Consider abirational model M of K . There is a birational morphism M → M such that L is log-elementary at all the points of M . The reduction of singularities of foliations in an ambient space of dimension twois proved in the classical Seidenberg’s paper [14]. In dimension three or higherone would like to be able to obtain elementary singularities , that is singularitieswith a non-nilpotent linear part. This is not possible in a birational way as anexample of F. Sanz and F. Sancho shows (see for instance the introduction of [11]).There is no general result in dimension n ≥
4, except for the case of absolutelyisolated singularities [3]. In dimension three Panazzolo [11] gives a global but non-birational result over the real numbers, getting elementary singularities after doingramifications and blow-ups. There is also a preprint of Panazzolo and McQuillan,where they announce and adaptation to the results in [11] to the language of stacks.In [5] there is a local result, along a trajectory of a real vector field, obtained alsoby the use of ramifications and blow-ups. Finally, in [4] there is a strategy to solveby means of blow-ups a “formal version” of the local uniformization problem, whereformal non-algebraic centers of blow-up are allowed.Let us give an outline of the proof of Theorem 1. We organize the proof bytaking account of the ranks and dimension of the valuation and of the existence of“maximal contact” with a formal series.In Part I, we consider the case of a real valuation ν : K \ { } → R with residualfield κ ν = k . In the classical situations of Zariski’s Local Uniformization [19] thisone is considered to be the most difficult case. Note that since κ ν = k the centerof ν at any projective model is a closed point. Our first result is Theorem 3.
Assume that n = 3 and ν is a real k -valuation of K with residualfield κ ν = k . There is a finite composition of blow-ups with non-singular centers M → M such that M is non-singular at the center P of ν at M and and one ofthe following properties holds (1) L is log-elementary at P . (2) There is ˆ f ∈ b O M,P having transversal maximal contact with ν . A formal series ˆ f ∈ b O M,P has transversal maximal contact with ν if it is theKrull-limit of a sequence f i ∈ O M,P with strictly increasing values and moreoverwe have the following property of transversality: there is a part of a regular systemof parameters x , x , . . . , x r of O M,P such that the values ν ( x ) , ν ( x ) , . . . , ν ( x r )are Z -independent, where r is the rational rank of ν , and x , x , . . . , x r , ˆ f is a partof a regular system of parameters of the complete local ring b O M,P .In order to prove Theorem 3, we work over the rational rank r of ν and we studythe three following cases in an ordered way:(1) r = n . Here we get L elementary for any ambient dimension n . Thisis a combinatorial case with few differences with respect to the classicalsituations of varieties. F. CANO; C. ROCHE; M. SPIVAKOVSKY (2) r = n −
1. The statement of Theorem 3 is valid for any n . We use NewtonPolygon technics to give the proof. If n = 2 the result is slightly stronger:we get either maximal contact of a non-singular foliation. This will beuseful in the next case.(3) r = 1 , n = 3. This is the hardest situation. We have important difficultiesdue to the fact that ν is not a discrete valuation.We end Part I by giving a proof of Theorem 4.
Assume that n = 3 . Let ν be a real k -valuation of K with residualfield κ ν = k and suppose that ˆ f ∈ b O M,P has transversal maximal contact with ν .There is a finite composition of blow-ups with non-singular centers M → M suchthat L is log-elementary at the center P of ν . Part II is devoted to the remaining cases. We obtain many of the results by aninductive use of the technics in Part I. In Part III we prove the validity of Piltant’spatching axioms and hence we obtain the proof of Theorem 2.
Part Zero dimensional arquimedean valuations
In all this part ν : K \ { } → Γ denotes a valuation such that Γ ⊂ ( R , +)and κ ν = k . In other words, the (arquimedean) rank of ν is one and it is a zero-dimensional k -valuation of K . We denote by r the rational rank of ν , that is, themaximum number of Z -linearly independent elements in the value group of ν . Weknow that 1 ≤ r ≤ n by Abhyankar’s inequality. In particular, for the case n = 3we have the possibilities r = 3 , r = 2 and r = 1.1. Parameterized regular local models A parameterized regular local model A = ( O , z = ( x , y )) for K, ν is a pair with O = O M,P , where M is a projective model of K , the point P ∈ M is the center of ν in M and the sequence( z , z , . . . , z n ) = z = ( x , y ) = ( x , x , . . . , x r , y r +1 , y r +2 , . . . , y n )is a regular system of parameters of O such that ν ( x ) , ν ( x ) , . . . , ν ( x r ) are Z -linearly independent values. We call x = ( x , x , . . . , x r ) the independent variables and y = ( y r +1 , y r +2 , . . . , y n ) the dependent variables . The existence of parameter-ized regular local models is a consequence of Hironaka’s reduction of singularities[10]. More precisely, we have Proposition 1.
Given a projective model M of K , there is a composition of afinite sequence of blow-ups with non-singular centers M → M such that the center P of ν at M provides a local ring O = O M,P for a parameterized regular local model A = ( O , z = ( x , y )) .Proof. By Hironaka’s reduction of the singularities (see [10]) of M , we get a non-singular projective model M ′ of K jointly with a birational morphism M ′ → M that is the composition of a finite sequence of blow-ups with non-singular centers.Consider the local ring O M ′ ,P ′ of M ′ at the center P ′ of ν and chose elements f , f , . . . , f r ∈ O M,P such that ν ( f ) , ν ( f ) , . . . , ν ( f r ) are Z -linearly independent.Another application of Hironaka’s theorem gives a birational morphism M → M ′ ,that is also a composition of a finite sequence of blow-ups with non-singular centers,such that f = Q ri =1 f i , is a monomial (times a unit) in a suitable regular system EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 5 of parameters at any point of M and hence each of the f i , i = 1 , , . . . , r is also amonomial (times a unit) in that regular system of parameters. In particular, if P is the center of ν at M there is a regular system of parameters z = ( z , z , . . . , z n )of O M,P such that f i = U i z m i , U i ∈ O M,P \ M
M,P , for i = 1 , , . . . , n, where m i = ( m i, , m i, , . . . , m i,n ) ∈ Z n ≥ and z m i = z m i, z m i, · · · z m i,n n . In termsof values, we have ν ( f i ) = P nj =1 m ij ν ( z j ). This implies that there are r variablesamong the z j whose values are Z -linearly independent. (cid:3) Coordinate changes and blow-ups.
Take a parameterized regular localmodel A = ( O , z ). We will do “atomic” transformations of A of two types: coordi-nate changes in the dependent variables and coordinate blow-ups with codimensiontwo centers . Our “basic” transformations, called Puiseux packages will be certainsequences of coordinate changes and blow-ups.Let us describe the two types of transformations. Each one produces a parame-terized local model A ′ = ( O ′ , z ′ ). Coordinate changes in the dependent variables.
Consider j with r + 1 ≤ j ≤ n .A j -coordinate change is such that z ′ i = z i for i = j and y ′ j is one of the followinga) y ′ j = y j − c x a , ν ( y ′ j ) ≥ ν ( y j ) , c ∈ k, a ∈ Z r ≥ .b) y ′ j = y j + y s , for another s = j with r + 1 ≤ s ≤ n .If r = n we do not do coordinate changes. Coordinate blow-ups with codimension two centers.
Take a pair i, j of distinctindices with 1 ≤ i ≤ r and 1 ≤ j ≤ n . We say that A ′ = ( O , z ′ ) is obtained from A by an ( i, j )- blow-up if the following holds. First z ′ s = z s for any s / ∈ { i, j } . In ordertwo determine z ′ i , z ′ j we have three cases(1) ν ( x i ) < ν ( z j ). We put x ′ i = x i and z ′ j = z j /x i .(2) ν ( x i ) > ν ( z j ). We put x ′ i = x i /z j and z ′ j = z j .(3) ν ( x i ) = ν ( z j ). Note that in this case we necessarily have that j ≥ r + 1and hence z j = y j . Since κ ν = k , there is c ∈ k with ν ( y j /x i − c ) >
0. Weput x ′ i = x i and y ′ j = y j /x i − c .The first two cases above are called combinatorial and the third one correspondsto a blow-up with translation . If x i , x j are independent variables, we have always acombinatorial case, since ν ( x i ) = ν ( x j ).The local ring O ′ is the (algebraic) localization of O [ z ′ ] at the ideal ( z ′ ).In the case that j ≥ r + 1 the above blow-up will also be referred as a j - blow-up . Remark 1.
Let M be a projective model for K such that O = O M,P , where P isthe center of ν at M . There is a closed irreducible algebraic subvariety Y ⊂ M ofcodimension two defined by the equations x i = z j = 0 that is non singular at P .Let π : M ′ → M be the blow-up of M with center Y and let P ′ be the center of ν at M ′ . Then O ′ = O M ′ ,P ′ .1.2. Puiseux packages of blow-ups.
Let A = ( O , z = ( x , y )) be a parameterizedregular local model. Consider a dependent variable y j . Then ν ( y j ) can be expresseduniquely as a Q -linear combination of ν ( x ) , ν ( x ) , . . . , ν ( x r ). More precisely, thereare unique integer numbers d > p , p , . . . , p r such that dν ( y j ) = p ν ( x ) + p ν ( x ) + · · · + p r ν ( x r ) F. CANO; C. ROCHE; M. SPIVAKOVSKY and gcd ( d ; p , p , . . . , p r ) = 1. In particular, the rational functionΦ = y dj / x p , x p = x p x p · · · x p r r , has value equal to zero. We call this function the j -contact rational function and d is the j -ramification index for A . Note that there is a unique scalar c ∈ k suchthat ν (Φ − c ) >
0, since κ ν = k .A coordinate ( i, s )-blow-up is said to be j - admissible if either 1 ≤ s ≤ r with p i = 0 = p s or p i = 0 and s = j . Remark 2.
Assume that A ′ has been obtained from A by a j -admissible coordinate( i, s )-blow-up. There are two possibilities:A) The blow-up is combinatorial. In this case Φ is also the j -contact rationalfunction for A ′ .B) The blow-up has a translation. Then Φ = y j /x i and s = j . Moreover, wehave y ′ j = Φ − c . Definition 1. A j - Puiseux package starting at A is a finite sequence A = A → A → · · · → A N = A ′ where A t − → A t is a combinatorial j -admissible blow-up for t = 1 , , . . . , N − and A N − → A N is a j -admissible blow-up with translation. In this situation, wesay that A ′ has been obtained from A by a j -Puiseux package. Note that y ′ j = Φ − c , in view of the above Remark. Proposition 2.
Given A and j , with r < j ≤ n , there is at least one j -Puiseuxpackage starting at A .Proof. There are many known algorithms for doing this (see [10, 16, 15, 18, 2]). Weinclude a proof for the sake of completeness. Let us writeΦ = y dj x q x r , where q i = − p i if p i < q i = 0, otherwise and, in the same way, we put r i = p i if p i > r i = 0 otherwise. There are two possibilities: q = 0 or q = 0. Notethat we always have that r = 0, since ν ( z s ) > s . Assume first that q = 0.Let us choose indices 1 ≤ i, s ≤ r such that p i p s <
0. We do the ( i, s )-blow-up.The sum | p i | + | p s | decreases. We continue and one of the independent variables x i or x s disappears. In this way we get that q = 0. Now, we consider an index i with p i = 0 and we do the ( i, j )-blow-up. This blow-up is combinatorial except in thecase that Φ = y j /x i . If we are not in this case, then d + p i decreases and finally thevariable x i disappears. We obtain that Φ = y j /x i . The only possible j -admissiblecoordinate blow-up is the ( i, j )-blow-up. Moreover, ν ( y j ) = ν ( x i ) and hence it is acoordinate blow-up with translation. (cid:3) Remark 3.
We are interested in the following features of Puiseux packages. Letus start with A = ( O , z = ( x , y )) and assume that A ′ = ( O ′ , z ′ = ( x ′ , y ′ )) has beenobtained from A by a j -Puiseux package. Let Φ = y dj / x p be the j -contact functionand suppose that ν (Φ − c ) >
0. For s / ∈ { i ; p i = 0 } ∪ { j } we have that z s = z ′ s .Moreover y ′ j = Φ − c and there are monomial expressions z s = r Y i =1 x ′ ib si ! Φ b sj ; s ∈ { i ; p i = 0 } ∪ { j } . EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 7
This is proved by induction on the number of j -admissible coordinate blow-ups ofthe j -Puiseux package.1.3. Statements in terms of parameterized regular local models.
Considera foliation by lines
L ⊂
Der k K and a parameterized regular local model A = ( O , z ).The local foliation induced by L at A is defined by L A = L ∩
Der k O . Obviously L A = L M,P for any projective model M for K such that O = O M,P . Inthe next sections we shall prove the following proposition
Proposition 3.
Assume that n = 3 . Let ν be a real k -valuation of K with κ ν = k and take a foliation L ⊂
Der k K . Consider a parameterized regular local model A = ( O , z ) for K, ν . There is a finite sequence of coordinate changes and blow-ups such that the parameterized regular local model A ′ = ( O ′ , z ′ ) obtained from A satisfies one of the following properties: (1) The foliation L A ′ is log-elementary. (2) There is ˆ f ∈ b O ′ having transversal maximal contact with ν . This result implies Theorem 3. Indeed, we already know that there is a birationalmorphism M → M , composition of blow-ups with nonsingular centers, such that M is non-singular and the local ring O M,P of M at the center P of ν supportsa parameterized regular local model A . The sequence of blow-ups that gives A ′ may be substituted, by Hironaka’s reduction of singularities, by another sequenceof blow-ups with non-singular centers, since the original blow-ups are non-singular(in fact they are non-singular and two dimensional) at the corresponding centers ofthe valuation at each projective model.Next sections are devoted to proving Proposition 3.2. The combinatorial case ( r = n )The following Proposition 4 implies Proposition 3 for the case of maximal ra-tional rank. Let us note that in Proposition 4 there is no assumption about the(arquimedean) rank of the valuation nor on the fact that κ ν = k . Indeed if n = r we know that κ ν is an algebraic extension of k and thus κ ν = k since we assumethe base field k to be algebraically closed. Proposition 4.
Let ν be a k -valuation of K with maximal rational rank r = n .Take a foliation L ⊂
Der k K and a parameterized regular local model A for K, ν .There is a parameterized regular local model A ′ obtained from A by a finite sequenceof coordinate blow-ups such that L A ′ is elementary. Let us recall that L A is elementary if there is a vector field ξ ∈ L A having anon-nilpotent linear part. If ξ ∈ Der k O is singular, that is ξ ( O ) ⊂ M , the linearpart Lξ is intrinsically defined as the k = O / M -linear map Lξ : M / M → M / M given by f + M ξf + M . Note that “elementary” implies “log-elementary”.Note also that a vector field ξ ∈ Der k O of the form(1) ξ = n X i =1 f i x i ∂∂x i , b i ∈ O , has a non-nilpotent linear part if and only if one of the f i is a unit in O . F. CANO; C. ROCHE; M. SPIVAKOVSKY
Newton polyhedron.
Note that z = x , since all the variables have Q -linearlyindependent values. Any element f ∈ O can be expanded in a formal series f = X f a x a ; f a ∈ k. The support of f is defined by Supp( f ; x ) = { a ; f a = 0 } ⊂ Z n ≥ . For a vector field ξ ∈ Der k O written as in formula (1), the support isSupp( ξ ; x ) = ∪ ni =1 Supp( f i ; x ) . The
Newton polyhedron N ( ξ ; x ) is the convex hull in R n of the set Supp( ξ ; x )+ R n ≥ .The local foliation L A contains a vector field ξ of the form (1) such that thecoefficients f i ∈ O have no common factor in O , that we call an x -generator of L A . To see this, take any η ∈ L A , then ( Q ni =1 x i ) η is of the form (1) and now it isenough to divide by the gcd of the f i .We define the Newton polyhedron N ( L ; x ) by N ( L ; x ) = N ( ξ ; x ), where ξ is an x -generator of L A . Remark 4.
The Newton polyhedron N ( L ; x ) has vertices in Z n ≥ . Since the co-efficients f i have no common factor (and “a fortiori” they are free of a monomialcommon factor) the only v ∈ R n ≥ such that N ( L ; x ) ⊂ v + R n ≥ is v = 0. Thus, if N ( L ; x ) has only one vertex v , then v = 0 and the vector field ξ has a non-nilpotent linear part. This implies that L A is elementary.2.2. The effect of a blow-up.
Let A ′ = ( O , z ′ ) be obtained from A by an ( i, s )- blow-up . Recall that there are no dependent variables and hence it is a combinatorialblow-up. If ν ( x i ) < ν ( x s ), we have x ′ s = x s /x i and x ′ s = x s , for s = j . Considerthe affine function σ iis : R n → R n defined by σ iis ( a ) t = (cid:26) a i + a j , if t = sa s , if t = s Take v ∈ R n ≥ such that σ iij ( N ( L ; x )) is inscribed in the orthant v + R n ≥ . Thenthe Newton polyhedron N ( L ; x ′ ) is obtained as N ( L ; x ′ ) = (cid:0) σ iis ( N ( L ; x )) − v (cid:1) + R n ≥ . In fact, the behavior of the Newton polyhedron is the same one as the behavior ofthe Newton polyhedron of the ideal generated by the coefficients f i . In the case ν ( x i ) > ν ( x s ), we do the same argument with the corresponding affine map σ sis .2.3. End of the proof of Proposition 4.
We can use the same idea as in theproof of Proposition 2. Let N be the number of vertices of N ( L ; x ). After doingan ( i, j )-blow-up, we obtain that N ′ ≤ N . If N = 1, we are done. Assume that N ≥
2. Take two distinct vertices a and b of N ( L ; x ) and let v be the element in Z n ≥ such that the set { a , b } is inscribed in v + R n ≥ . In other terms, the monomial x v is the gcd of x a and x b . Put ˜ a = a − v and ˜ b = b − v . Note that for any index t we have ˜ a t ˜ b t = 0 and also ˜ a = 0 = ˜ b . Choose indices i, s with ˜ a i ˜ b s = 0. Do the( i, s )-blow-up. Assuming that N ′ = N , the set of indices { t ; ˜ a t = 0 or ˜ b t = 0 } is contained in the corresponding one after blow-up. If the two sets coincide, theamount ˜ a i + ˜ b s decreases strictly. This ends the proof. EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 9
Remark 5.
The same kind of combinatorial game, but using centers of any codi-mension, with a “permissibility” additional condition, is called the Weak Hironaka’sGame [15]. 3.
The Newton-Puiseux Polygon
Let us assume in this section that r = n − κ ν = k and take a parameterizedregular local model A = ( O , z = ( x , y )). Note that since r = n −
1, there is onlyone dependent variable y .Consider an element f ∈ y − O , that we write f = P ∞ s = − h s ( x ) y s , where h s ( x )is a formal series h s ( x ) ∈ k [[ x ]] ∩ O . The Newton-Puiseux support of f is the setNPSup( f ; x , y ) = { ( ν ( h s ) , s ); h s = 0 } ⊂ Γ × Z ≥− . We denote by α ( f ; x , y ) the minimum abscissa of the Newton Puiseux support,that is α ( f ; x , y ) = min { ( ν ( h s )) } . The main height ~ ( f ; x , y ) is the minimum ofthe s such that ν ( h s ) = α ( f ; x , y ). Let δ ( f ; x , y ) be the minimum of the values ν ( h s ) + sν ( y ). The critical segment C ( f ; x , y ) is the set of the s such that ν ( h s ) + sν ( y ) = δ ( f ; x , y ) . The main height χ ( f ; x , y ) is the highest s in the critical segment. Let us note that χ ( f ; x , y ) ≤ ~ ( f ; x , y ).Consider a finite list f = ( f , f , . . . , f t ) of elements f j ∈ y − O . The Newton-Puiseux support NPSup( f ; x , y ) is the set of ( u, s ), where u is the minimum of the u j such that ( u j , s ) ∈ NPSup( f j ; x , y ), for j = 1 , , . . . , t . We obtain in this waya definition for α ( f ; x , y ), ~ ( f ; x , y ) , δ ( f ; x , y ) and χ ( f ; x , y ) since these invariantsdepend only on the Newton-Puiseux support.3.1. Newton-Puiseux Polygon of a foliation.
Consider the free O -moduleDer k O [log x ] whose elements are the vector fields of the form(2) ξ = n − X i =1 f i ( x , y ) x i ∂∂x i + g ( x , y ) ∂∂y where g ∈ O , f i ∈ O , i = 1 , , . . . , n −
1. Such vector fields will be called x -logarithmic vector fields , or simply x -vector fields . Let us denote f n = g/y and f = ( f , f , . . . , f n ). We define NPSup( ξ ; x , y ) = NPSup( f ; x , y ) and α ( ξ ; x , y ) = α ( f ; x , y ); ~ ( ξ ; x , y ) = ~ ( f ; x , y ) δ ( ξ ; x , y ) = δ ( f ; x , y ); χ ( ξ ; x , y ) = χ ( f ; x , y ) . Given a foliation
L ⊂
Der k K , we consider the local x -logarithmic foliation L A [log x ] at A defined by(3) L A [log x ] = L ∩
Der k O [log x ] . We define the main height ~ ( L ; A ), respectively the critical height χ ( L ; A ), to bethe minimum of the ~ ( ξ ; x , y ), respectively χ ( ξ ; x , y ), where ξ ∈ L A [log x ]. Notethat ~ ( L ; A ) ≥ χ ( L ; A ) ≥ − . These ones are the main invariants we shall use to control the singularity of L afterperforming a Puiseux package. The initial parts.
Consider an element h = P m λ m x m ∈ O ∩ k [[ x ]]. Sincethe values ν ( x i ), i = 1 , , . . . , n − Q -linearly independent, there is exactly oneexponent m such that ν ( λ m x m ) = ν ( h ). Moreover, if ˜ h = h − λ m x m then ν (˜ h ) > ν ( h ). Take an element γ ∈ Γ with γ ≤ ν ( h ). We define the γ - initial form In γ ( h ) by In γ ( h ) = 0 if γ < ν ( h ) and In ν ( h ) ( h ) = λ m x m if γ = ν ( h ). Givena list h = ( h , h , . . . , h n ) of elements h j = h j ( x ) ∈ k [[ x ]] ∩ O , and γ ∈ Γ with γ ≤ min { ν ( h j ( x )); j = 1 , , . . . , n } we putIn γ ( h ; x ) = (In γ ( h ; x ) , In γ ( h ; x ) , . . . , In γ ( h n ; x )) . If we have a vector field of the form η = n − X j =1 h j ( x ) x j ∂∂x j + h n ( x ) y ∂∂y and γ ≤ min { ν ( h j ( x )); j = 1 , , . . . , n } we putIn γ ( η ; x ) = n − X j =1 In γ ( h j ; x ) x j ∂∂x j + In γ ( h n ; x ) y ∂∂y . Take an x -vector field ξ ∈ Der O [log x ] that we write as in equation (2). Put f j = P ∞ s = − h js ( x ) y s , j = 1 , , . . . , n . We have ξ = P ns = − y s η s , where(4) η s = n − X j =1 h js ( x ) x j ∂∂x j + h ns ( x ) y ∂∂y ; s = − , , , . . . . Put δ = δ ( ξ ; x , y ) = min j,s { ν ( y s h js ( x )) } . We define the initial form In( ξ ; x , y ) asIn( ξ ; x , y ) = ∞ X s = − y s In δ − sν ( y ) ( η s ; x ) . Let us note that if ˜ ξ = ξ − In( ξ ; x , y ), then δ ( ˜ ξ ; x , y ) > δ ( ξ ; x , y ). Note also thatif χ = χ ( ξ ; x , y ) is the critical height, then In δ − sν ( y ) ( η s ; x ) = 0 for s > χ andIn δ − χν ( y ) ( η χ ; x ) = 0. In particular In( ξ ; x , y ) is a finite sumIn( ξ ; x , y ) = χ X s = − y s In δ − sν ( y ) ( η s ; x ) . Now we are going to give a particular expression of In( ξ ; x , y ) in terms of the contactrational function Φ = y d / x p .Let us take an index s such that In δ − sν ( y ) ( η s ; x ) = 0 and in particular s ≤ χ .Write In δ − sν ( y ) ( η s ; x ) = x q ( s ) Λ s , where Λ s is the linear vector fieldΛ s = n − X j =1 λ js x j ∂∂x j + λ ns y ∂∂y and q ( s ) ∈ Z n − ≥ . Put r ( s ) = q ( s ) − q ( χ ). We have ν ( x r ( s ) ) = ( χ − s ) ν ( y ) = χ − sd ν ( x p ) , this implies that (( χ − s ) /d ) p = r ( s ) and thus (( χ − s ) /d ) p ∈ Z n − . Since thecoefficients p , p , . . . , p n − have no common factor, we have that ( χ − s ) /d ∈ Z . EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 11
Put t = ( χ − s ) /d ∈ Z ≥ ; note that t ≤ ̺ , where ̺ ∈ Z ≥ is the biggest integerbounded above by ( χ + 1) /d .We may write In( ξ ; x , y ) as follows:(5) In( ξ ; x , y ) = x q ( χ ) y χ χ X s = − ( χ − s ) /d Λ s = x q ( χ ) y χ ̺ X t =0 t Λ χ − dt . In order to simplify the notation, let us rename ∆ t = Λ χ − dt . Then(6) x − q ( χ ) y − χ Φ ̺ In( ξ ; x , y ) = ̺ X t =0 Φ ̺ − t ∆ t . We recall that ∆ = 0.3.3. The expression of the derivatives after a Puiseux package.
Assumethat A ′ = ( O ′ , z ′ = ( x ′ , y ′ )) has been obtained from A = ( O , z = ( x , y )) by aPuiseux package. Let Φ = y d / x p be the contact rational function. By remark 3 wehave that y ′ = Φ − c and there is a matrix B = ( b si ) with determinant 1 or − z s = n − Y i =1 x ′ ib si ! Φ b sn ; s = 1 , , . . . , n. Moreover if p s = 0 we know that x s = x ′ s , that is b si = 0 if i = s and b ss = 1. Thisimplies that x ′ i ∂∂x ′ i = n − X s =1 b si x s ∂∂x s + b ni y ∂∂y ; i = 1 , , . . . , n − , (7) Φ ∂∂y ′ = n − X s =1 b sn x s ∂∂x s + b nn y ∂∂y (8)Let B − = (˜ b is ) be the inverse matrix of B = ( b si ). We obtain x s ∂∂x j = n − X i =1 ˜ b is x ′ i ∂∂x ′ i + ˜ b ns Φ ∂∂y ′ ; s = 1 , , . . . , n − , (9) y ∂∂y = n − X i =1 ˜ b in x ′ i ∂∂x ′ i + ˜ b nn Φ ∂∂y ′ (10)Note that the ˜ b si are integer (may be negative) numbers. Moreover, we have(11) ˜ b nn = 1Φ y ∂∂y (Φ) = d = 0 . Finally, a given linear vector field ∆ = P ni =1 µ i z i ∂/∂z i , we have(12) ∆ = ( n − X i =1 ˜ µ i x ′ j ∂∂x ′ j + ˜ µ n y ′ ∂∂y ′ ) + c ˜ µ n ∂∂y ′ . where (˜ µ , ˜ µ , . . . , ˜ µ n ) = ( µ , µ , . . . , µ n ) B − . Rational co-rank one
In this section we also assume that r = n − κ ν = k . We take a parameterizedregular local model A = ( O , z = ( x , y )) and a foliation L ⊂
Der k K . We will provethe following result Proposition 5.
There is a a parameterized regular local model A ′ obtained from A by a finite sequence of coordinate changes in the dependent variable and coordinateblow-ups with codimension two centers, such that one of the following propertiesholds: (1) There is ˆ f ∈ b O ′ having transversal maximal contact with ν . (2) The local foliation L A ′ is non-singular if n = 2 and elementary if n ≥ . Now, Proposition 5 is a consequence of the following five lemmas.
Lemma 1.
Assume that A ′ has been obtained from A by a coordinate change inthe dependent variable. Then ~ ( L ; A ′ ) = ~ ( L ; A ) .Proof. Left to the reader. (cid:3)
Lemma 2.
Assume that A ′ has been obtained from A by a Puiseux package. Then ~ ( L ; A ′ ) ≤ χ ( L ; A ) . Moreover, we have ~ ( L ; A ′ ) < χ ( L ; A ) if χ ( L ; A ) ≥ and d ( A ) ≥ , where d ( A ) is the the ramification index of A . Lemma 3.
Assume that ~ ( L ; A ) ∈ {− , } . We have the following properties (1) If ~ ( L ; A ) = − , after performing a finite sequence of coordinate blow-upsin the independent variables, we obtain A ′ such that L A ′ is non-singular. (2) If ~ ( L ; A ) = 0 , after performing a finite sequence of coordinate blow-ups inthe independent variables, we obtain A ′ such that L A ′ is elementary. Lemma 4. If n = 2 , after performing a finite sequence of Puiseux packages weobtain A ′ such that either L A ′ is non-singular or there is ˆ f ∈ b O ′ having transversalmaximal contact with ν . Lemma 5.
Assume that ~ ( L A ) ≥ and that the following property holds:“After any finite sequence of coordinate blow-ups in the independentvariables, Puiseux packages and coordinate changes in the depen-dent variable we have that d ( A ) = 1 and ~ ( L A ′ ) = ~ ( L A ) ”.Then there is ˆ f ∈ b O having transversal maximal contact with ν . In order to show that Lemmas 1, 2, 3, 4 and 5 imply Proposition 5, let us onlyrecall that χ ( L ; A ) < ~ ( L ; A ). So, unless we have a transversal maximal contact,we arrive to the situation of Lemma 3 by a repeated application of Lemma 2 andwe are done.Let us prove the above lemmas.4.1. The effect of a Puiseux package.
Let us consider A ′ = ( O ′ , z ′ = ( x ′ , y ′ ))obtained from A by a Puiseux package. Take an x -vector field ξ ∈ L A [log x ] suchthat χ ( ξ ; x , y ) = χ ( L ; A ) and let us write ξ = P ns = − y s η s as in equations (2)and (4). In order to simplify the notation, put χ = χ ( ξ ; x , y ) and δ = δ ( ξ ; x , y ).Moreover, we denote d = d ( A ) the ramification index associated to A . Let us write˜ ξ = ξ − In( ξ ; x , y ). We recall that δ ( ˜ ξ ; x , y ) > δ . EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 13
Next we express In( ξ ; x , y ) and ˜ ξ in terms of the coordinates z ′ = ( x ′ , y ′ ). Lemma 6. α ( ˜ ξ ; x ′ , y ′ ) > δ .Proof. Left to the reader. (cid:3)
Let us consider now In( ξ ; x , y ) and let us express it in the coordinates z ′ . Let usrecall equation 6, where x − q ( χ ) y − χ Φ ̺ In( ξ ; x , y ) = P ̺t =0 Φ ̺ − t ∆ t and∆ t = Λ χ − dt = n X i =1 λ i,χ − dt z i ∂/∂z i = n X i =1 µ it z i ∂/∂z i , with ∆ = 0. Let us put ζ = P ̺t =0 Φ ̺ − t ∆ t . We can write ζ = P s ≥ β ′ y ′ s ϑ s , where ϑ β ′ = 0 and all the ϑ s are z ′ -linear vector fields ϑ s = P nj =1 α js z ′ j ∂/∂z ′ j . Lemma 7.
We have β ′ ≤ χ . If χ ≥ and d ≥ , then β ′ < χ .Proof. Looking at the equation 12, we see that ζ = P ̺t =0 ( y ′ + c ) ̺ − t ∆ t and∆ t = n X j =1 ˜ µ jt z ′ j ∂∂z ′ j + c ˜ µ nt ∂∂y ′ ; (˜ µ t , ˜ µ t , . . . , ˜ µ nt ) = ( µ t , µ t , . . . , µ nt ) B − . Let ς = max { t ; ∆ t = 0 } ≤ ̺ . Then ζ = Φ ̺ − ς P ςt =0 Φ ς − t ∆ t . Recalling thatΦ = y ′ + c , and dividing the above expression by Φ ̺ − ς , we obtain β ′ ≤ ς . Rememberthat ̺ is the greatest integer bounded above by ( χ +1) /d . Then, if d ≥ χ ≥ d ≥ χ = 1 we obtain β ′ ≤ ς ≤ ̺ < χ . If χ = 0 and d ≥ ̺ = 0and then β ′ ≤
0. It remains to study the cases with d = 1, the case d = 2, χ = 1and the case χ = − The case χ = − . In this case ̺ = ς = 0. In particular ζ = ∆ . Moreover∆ = Λ − = µ n y∂/∂y . Recalling that ˜ b nn = d in view of equation 11, we have ζ = µ n y∂/∂y = µ n n X j =1 ˜ b jn z ′ j ∂∂z ′ j + µ n dc ∂∂y ′ . This implies that α n, − = µ n dc = 0 and thus β ′ = − Cases with d = 1 , χ ≥ . We reason by contradiction, assuming that β ′ ≥ χ + 1.This implies that ς = ̺ = χ + 1. In particular, we have ∆ χ +1 = 0 and ∆ χ +1 = Λ − .Note that Λ − = µy∂/∂y , where µ = µ n,χ +1 = λ n, − . Now, our contradictionhypothesis β ′ ≥ χ + 1 implies that ζ ( y ′ ) is divisible by y ′ χ +2 . We have ζ ( y ′ ) = χ +1 X t =0 Φ χ +1 − t ∆ t ( y ′ ) == Φ (cid:0) Φ χ +1 ˜ µ n + Φ χ ˜ µ n + Φ χ − ˜ µ n + · · · + Φ˜ µ nχ + ˜ µ n,χ +1 (cid:1) . Recall that Φ = y ′ + c , then we necessarily have that ζ ( y ′ ) = ˜ µ n y ′ χ +2 , since thebiggest possible power of y ′ in the above expression is y ′ χ +2 and its coefficient is˜ µ n . Moreover we also have that Φ = y ′ + c divides ζ ′ ( y ′ ). The only possibility isthat ζ ( y ′ ) = 0 and hence all the coefficients ˜ µ nt are zero, for t = 1 , , . . . , χ + 1.This is a contradiction, since ˜ µ n,χ +1 = ˜ b nn µ = dµ = 0. Case d = 2 , χ = 1. Let us reason by contradiction, assuming that β ′ ≥ χ . Then ς = ̺ = χ = 1. We have ζ = Φ∆ + ∆ and y ′ must divide ζ ( y ′ ). That is ζ ( y ′ ) = Φ (Φ˜ µ n + ˜ µ n ) = y ′ ˜ µ n . We deduce as above that ζ ( y ′ ) = 0 and thus ˜ µ n = ˜ µ n = 0. Note that 0 = ∆ ,since ς = 1. Moreover, in our case ∆ t = Λ − t and thus ∆ = Λ − = µy∂/∂y = 0.Now we have ˜ µ n = 2 µ and we obtain that ˜ µ n = 0 and ˜ µ n = 0 simultaneously,contradiction. (cid:3) Lemma 8. ~ ( L ; A ′ ) ≤ β ′ .Proof. It is enough to show that β ′ = ~ ( ξ ; x ′ , y ′ ). We have β ′ = ~ ( ζ ; x ′ ; y ′ ) and α ( ζ ; x ′ ; y ′ ) = 0. Recall that In( ξ ; x , y ) = x q ( χ ) y χ Φ − ̺ ζ , where ν ( x q ( χ ) y χ Φ − ̺ ) = ν ( x q ( χ ) y χ ) = δ. Moreover, in view of Remark 3, we have that x q ( χ ) y χ Φ − ̺ = x ′ q ′ Φ r ′ , and ν ( x ′ q ′ ) = δ. Noting that In( ξ ; x , y ) = x ′ q ′ Φ r ′ ζ , we deduce that α (In( ξ ; x , y ); x ′ , y ′ ) = δ and ~ (In( ξ ; x , y ); x ′ , y ′ ) = ~ ( ζ ; x ′ , y ′ ) = β ′ . Moreover, by Lemma 6, we have δ = α (In( ξ ; x , y ); x ′ , y ′ ) < α ( ˜ ξ ; x ′ , y ′ ) . Recalling that ξ = In( ξ ; x , y ) + ˜ ξ , we have that α ( ξ ; x ′ , y ′ ) = δ and ~ ( ξ ; x ′ , y ′ ) = ~ (In( ξ ; x , y ); x ′ , y ′ ) = β ′ . This ends the proof. (cid:3)
Remark 6.
Lemma 1 follows from Lemma 7, in view of Lemma 8.Before giving a proof of Lemma 3, we explain the effect of the blow-ups in theindependent variables in the following result.
Lemma 9.
Given A and L , after performing finitely many coordinate blow-ups inthe independent variables with centers of codimension two, we can obtain A ′ suchthat α ( L ; A ′ ) = 0 . Moreover ~ ( L ; A ′ ) ≤ ~ ( L ; A ) .Proof. Write ξ = P ∞ s = − y s η s with η s = P nj =1 h js ( x ) z j ∂/∂z j . Let us do a blow-upin the independent variables and let x ′ , y be the obtained variables. Then the samedecomposition as above acts in this new set of variables, that is ξ = P ∞ s = − y s η s where we can write η s = n X j =1 h ′ js z ′ j ∂∂z ′ j ; h ′ js ∈ k [[ x ′ ]] . Moreover the ideal I ′ s ⊂ k [[ x ′ ]] generated by { h ′ js } nj =1 is I ′ s = I s k [[ x ]], where I s isthe ideal of k [[ x ]] generated by { h js } nj =1 . This already implies that α ( η s ; x ) = α ( η s ; x ′ ); s = − , , , . . . . In particular we have that ~ ( ξ ; x , y ) = ~ ( ξ ; x ′ , y ).Moreover, the ideal I ′ = P ∞ s = − I ′ s ⊂ k [[ x ′ ]] is also given by I ′ = Ik [[ x ′ ]],where I = P ∞ s = − I s ⊂ k [[ x ]]. Thus, we can apply classical results of reduction ofsingularities under combinatorial blow-ups, that can be proved as in Proposition 4(see also [7]) to assure that after a finite number of blow-ups in the independentvariables with centers of codimension two, the ideal I is generated by a singlemonomial, say x ′ q ′ . We obtain an x ′ -vector field ξ ′ = x ′− q ′ ξ ∈ L A ′ [log x ′ ] suchthat α ( ξ ′ ; x ′ , y ) = 0. (cid:3) EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 15
Remark 7.
In the above lemma we have ~ ( L ; A ′ ) = ~ ( L ; A ′ ). Anyway, we do notneed to use this fact.We obtain an immediate proof of Lemma 3. By Lemma 9, we may suppose thatthere is ξ ∈ L A [log x ] such that ~ ( ξ ; x , y ) = ~ ( L ; A ) and α ( ξ ; x , y ) = 0. Now, it isevident that(1) If ~ ( ξ ; x , y ) = −
1, then ξ is non-singular.(2) If ~ ( ξ ; x , y ) = 0, then ξ is elementary (or non-singular).4.2. Getting a formal hypersurface of transversal maximal contact.
Letus give a proof of Lemma 5. In view of Lemma 9, after performing finitely manyblow-ups in the independent variables, we can assume that there is ξ ∈ L A [log x ]such that ~ ( ξ ; x , y ) = ~ ( L ; A ) and α ( ξ ; x , y ) = 0. Moreover, we also have that χ ( ξ ; x , y ) = ~ ( ξ ; x , y ) , since otherwise, an application of Lemma 2 allows us to decrease ~ ( L ; A ). Moreover,in view of our hypothesis, we have d ( A ) = 1 and ~ ( L ; A ) ≥ Lemma 10.
Let
Φ = y/ x p be the contact rational function. We have p ∈ Z n − ≥ .Proof. Let us keep the notations of subsection 3.2. Recall thatIn( ξ ; x , y ) = χ X s = − y s x q ( s ) Λ s . Since α ( ξ ; x , y ) = 0 and χ ( ξ ; x , y ) = ~ ( ξ ; x , y ), we have q ( χ ) = 0. Thus, for any s such that Λ s = 0 we have ν ( x q ( s ) ) = ( χ − s ) ν ( y )and hence ( χ − s ) p = q ( s ). Noting that q ( s ) ∈ Z n − ≥ , it is enough to show thatthere is at least an index s < χ such that Λ s = 0. Assume the contrary. ThenIn( ξ ) = y χ Λ χ , where χ = ~ ( ξ ; x ; y ) ≥
1. Let us do a Puiseux package, taking the notations of theproof of Lemma 7, we obtain ς = 0 and hence χ ′ ≤ β ′ ≤ ς = 0. Contradiction. (cid:3) In this situation, we have ν ( y − c x p ) > ν ( y ). Let us do the coordinate change y ′ = y − c x p . The situation repeats. In this way we can produce a sequence ofelements y ( j ) ∈ M \ M , such that y (0) = y and y ( j ) = y ( j − − c j x p ( j ) ; ν ( y ( j ) ) > ν ( y ( j − ) , j = 1 , , . . . . Taking ˆ f = lim j y ( j ) , we obtain the desired formal hypersurface.4.3. The case of dimension two.
The statement of Lemma 4 is a consequenceof Seidenberg’s reduction of singularities in dimension two [14]. Let us see this.Assuming that we do not get non-singular points, after finitely Puiseux packages,we obtain a “simple singularity” in the sense of Seidenberg. It is given by an x -vector field of the form ξ = ( λ + a ( x, y )) x ∂∂x + ( αx + µy + ˜ b ( x, y )) ∂∂x ; a (0 ,
0) = 0 , ˜ b ( x, y ) ∈ M where, ( λ, µ ) = (0 ,
0) and if λ = 0 then µ/λ / ∈ Q > . Such singularity has exactlytwo formal invariant curves: x = 0 and ˆ f = 0, where ˆ f = y − ˆ φ ( x ). They arenon-singular and transversal one to the other. After doing one more blow-up, the exceptional divisor is invariant and we obtain exactly two simple singularities, oneof them corresponds to the strict transform of x = 0, it is a corner , and the other oneis in the strict transform of ˆ f = 0. This shows that blowing-up a corner producesonly corners as singularities, thus, since the valuation has rational rank one and wehave nontrivial Puiseux packages, we necessarily do blow-ups outside the corners.Hence we follow the infinitely near points of ˆ f = 0. “A fortiori”, we obtain thatˆ f is non-algebraic (otherwise the value of ˆ f would be infinite) and has maximalcontact with ν . 5. Etale Puiseux packages
Review on etale neighborhoods.
Let us recall the definition of a local etalemorphism as one can see in [1]. Let us fix the local ring O = O M,P of a projectivemodel M of K at the center P in M of the k -valuation ν of K and assume that P is a regular point of M . Here we assume that ν is a real valuation with κ ν = k .Consider a morphism O → e O of local rings. We say that O → e O is local-etale or that e O is a local-etale extension of O if we have the following properties:(1) The local rings O and e O have the same residual field.(2) e O is the localization at a prime ideal of an etale O -algebra.An etale O -algebra is an O -algebra of the type B = O [ t , t , . . . , t n ] / ( f , f , . . . , f n ),where the Jacobian matrix of the f i is invertible in B . This is equivalent to saythat B is a finitely generated A -flat algebra and Ω A B = 0. Note that e O is also aregular local ring and its fraction field e K is a finitely generated algebraic extensionof K . Recall also that e O ⊂ O h ⊂ b O , where O h is the henselian closure of O .We say that the pair ( e O , ˜ ν ) is a local etale extension of ( O , ν ) if e O is a local-etaleextension of O and ˜ ν is a k -valuation of e K centered at e O such that ˜ ν | K = ν . Notethat ˜ ν is a real k -valuation and κ ˜ ν = k .In the following proposition, we summarize the properties that allow us to work“up to local-etale extensions”. Proposition 6.
Consider a foliation
L ⊂
Der k K and a real k -valuation ν of K such that κ ν = k . Let ( e O , ˜ ν ) is a be a local etale extension of ( O , ν ) and denote e L = e K L ⊂
Der k e K the induced foliation on e K . Assume that we respectively have: (1) The foliation e L is log-elementary at e O . (2) There is a formal ˆ f ∈ b O with transversal maximal contact relatively to e O .Then, up to perform a finite sequence of local blow-ups of O we respectively have: (1) The foliation L is log-elementary at O . (2) There is a formal ˆ f ∈ b O with transversal maximal contact relatively to O .Proof. Let ˜ x , ˜ x , . . . , ˜ x n be a regular system of parameters of e O . Consider ˜ h = Q ni =1 ˜ x i . The ideal ˜ h e O gives a principal ideal h O = O ∩ ˜ h e O . We can do the localuniformization of h by using centers that respect the fact that e L is log elementary(relatively to ˜ x ) (see [4] to the definition of permissible centers and the neededproperties). Finally we get that h is a monomial and we are done. (cid:3) Etale Puiseux packages.
We introduce here an etale version of Puiseuxpackages for the case r = 1. It has the same effect over a foliation as the Puiseuxpackages introduced in Section 1, but it will allow us to do an accurate control of EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 17 the foliation. Indeed, the study of the case n = 3, r = 1 will be done under the useof etale Puiseux packages.We assume that ν is a valuation with rational rank r = 1 and κ ν = k .Let A = ( O , z = ( x, y )) be a regular parameterized model. Consider a dependentvariable y j . Let Φ = y dj /x p be the contact rational function and c ∈ k such that ν (Φ − c ) >
0. Recall that d is the y j -ramification index of A . Remark 8.
In the case that d = 1, all the blow-ups in a Puiseux package are “inthe first chart” in the sense that we always have ν ( y j ) ≥ ν ( x ).Let us consider the ring O ♮ = O [ T ] / ( T d − x ) = O [ t ], where T is an indeterminateand let e K be the fraction field of O ♮ . We know [17] that there are k -valuations ˜ ν of e K such that R ˜ ν ∩ K = R ν . Note that all the ˜ ν have the same group of values.Let us choose one of them, say ˜ ν . The ring O ♮ is a regular local ring that supportsa parameterized regular local model A ♮ = ( O ♮ , z ♮ = ( t, y ))relative to ˜ K and ˜ ν . We have k ⊂ O ⊂ O ♮ and M ♮ ∩ O = M . Moreover, k = κ ˜ ν and t d = x . Let us note that ˜ ν ( y/t p ) = 0. In particular d ( e A ) = 1. Let ˜ c ∈ k besuch that ˜ ν ( y/t p − ˜ c ) >
0, we see that ˜ c d = c . Definition 2.
We say that ( e A , ˜ ν ) has been obtained from ( A , ν ) by an etale j -Puiseux package if and only if e A has been obtained from A ♮ by a j -Puiseux package. Proposition 7.
Assume that e A = ( e O , ˜ z = ( t, ˜ y )) has been obtained from A by anetale j -Puiseux package. There is A ′ = ( O ′ , z ′ = ( x ′ , y ′ )) obtained from A by a j -Puiseux package such that ( e O , ˜ ν ) is a local-etale extension of ( O ′ , ν ) .Proof. Consider the j -Puiseux package A ♮ e A . Put Φ = y dj /x p and ˜Φ = y j /t p ,the respective contact rational functions for A and A ♮ . Note that ˜Φ d = Φ. Let c, ˜ c ∈ k be such that ˜ ν ( ˜Φ − ˜ c ) > c d = c . We have˜ y j = ˜Φ − ˜ c ; y ′ j = Φ − c. Moreover ˜ y j is a simple root of a polynomial over O ′ as the following relation shows y ′ j = (˜ y j + ˜ c ) d − c. Now ˜ t is of the form ˜ t = x ′ P (˜ y j ), where P (0) = 0. This is enough to obtain theconclusion. (cid:3) Remark 9. If e A = ( e O , ˜ z = ( t, ˜ y )) has been obtained from A = ( O , z = ( x, y )) byan etale Puiseux package, then O ⊂ e O and t d = x . Definition 3.
We say that ( A , ν ) ( e A , ˜ ν ) is an etale standard transformation ifand only if ( e A , ˜ ν ) has been obtained from ( A , ν ) by an etale Puiseux Package or bya coordinate change in the dependent variables. Rational rank one
We end here the proof of Theorem 3. To do this we consider the followingproposition
Proposition 8.
Let
L ⊂
Der k K be a foliation over K , where n = 3 . Take a k -valuation ν of K of rational rank one and such that κ ν = k . Assume that A is aparameterized regular local model for K and ν . Then, there is a finite sequence ofetale standard transformations ( A , ν ) = ( A , ν ) ( A , ν )
7→ · · · 7→ ( A N , ν N ) = ( A ′ , ν ′ ) such that either the transformed foliation L ′ is log-elementary in A ′ or there is ˆ f ∈ O ′ having transversal maximal contact. Proposition 8 gives the end of the proof of Proposition 3 and hence it completesthe proof of Theorem 3. Indeed, by propositions 4 and 5 we obtain Proposition 3for rational rank r = 2 ,
3. For the case of rational rank r = 1 and n = 3, we obtainProposition 3 from Proposition 8 in view of propositions 6 and 7.This section is devoted to the proof of Proposition 8. In all this section weassume implicitly that we do not get a formal transversal maximal contact.Recall that we in this section we have n = 3, the rational rank of ν is equal toone and κ ν = k . We start with a parameterized regular local model A = ( O ; z =( x, w, y )) and a foliation L ⊂
Der k K .6.1. The independent coefficient.
Let ξ be an O -generator of L A [log x ]. Let usput H = ξ ( x ) /x ∈ O . Consider an etale standard transformation ( A , ν ) ( A , ν ′ )where A ′ = ( O ′ , z ′ = ( t, w ′ , y ′ )) . Recall that t d = x , for d ≥
1. We know that L ′A ′ [log t ] is generated by a germ ofvector field of the form ξ ′ = t q ξ where q ∈ Z . Moreover, we have that ξ ( t ) /t = λ ′ ξ ( x ) /x, where λ ′ = 1 /d ∈ Q > . This implies that(13) H ′ = ξ ′ ( t ) /t = λ ′ t q H ∈ O ′ . In particular, the coefficient H is transformed essentially “as a function” under theetale standard transformations. This allows us to obtain the following result Proposition 9.
After finitely many etale standard transformations we can chosean O -generator ξ of L A [log x ] such that ξ ( x ) /x = λx m , where λ ∈ Q > .Proof. We apply to H the usual local uniformization for functions. We obtain that H = x m U , where U is a unit. Now we divide ξ by U . (cid:3) Moreover, the above form of H is persistent under etale standard transforma-tions. This justifies the next definition. Definition 4.
We say that L is x -prepared relatively to A if there is an O -generator ξ of L A [log x ] such that ξ ( x ) /x = λx q , for = λ ∈ Q . Such generators ξ will becalled x -privileged generators . In view of Proposition 9 we can obtain that L is x -prepared after a finite numberof etale-standard transformations and this property is persistent under new etale-standard transformations. EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 19
Invariants from the Newton Puiseux Polygon.
Take f ∈ w − k [[ x, w ]]that we write f = P ∞ t = − w t f t ( x ). We put(14) λ ( f ; x, w ) = min t { ν ( f t ( x )) + tν ( w ) } ; α ( f ; x, w ) = min t { ν ( f t ( x )) } . Consider a vector field η of the form(15) η = a ( x, w ) x ∂∂x + b ( x, w ) ∂∂w + c ( x, w ) y ∂∂y . We denote λ ( η ; x, w ) = min { λ ( a ; x, w ) , λ ( b/w ; x, w ) , λ ( c ; x, w ) } (16) α ( η ; x, w ) = min { α ( a ; x, w ) , α ( b/w ; x, w ) , α ( c ; x, w ) } (17)Let us note that α ( b/w ; x, w ) = α ( b ; x, w ). We also write(18) Λ( η ; x, w ) = λ ( η ; x, w ) − α ( η ; x, w ) . Note that Λ( η ; x, w ) ≥ − ν ( w ). Remark 10.
We can draw a Newton-Puiseux polygon N for f , or for η , by consid-ering the support { ( ν ( f t ( x ) , t )) } ⊂ Γ × Z ≥− ⊂ R × Z ≥− . Then α is the abscissa ofthe highest vertex and λ corresponds to the smallest value a + ν ( w b ), where ( a, b )is in the support. In particular, we have that Λ = − ν ( w ) if and only if N has thesingle vertex ( α, − ξ = P ∞ s = − y s η s ∈ Der k O [log x ], where(19) η s = a s ( x, w ) x ∂∂x + b s ( x, w ) ∂∂w + c s ( x, w ) y ∂∂y . We denote α ( ξ ; A ) = α ( ξ ; x, w, y ) = min ∞ s = − { α ( η s ; x, w ) } . Let us note that α ( ξ ; A ) = 0 when ξ is a generator of L A [log x ], since x is not a common factorof the coefficients. The main height ~ ( ξ ; A ) is the minimum of the s such that α ( η s ; x, w ) = α ( ξ ; A ). When ξ is a generator of L , we put ~ ( L , A ) = ~ ( ξ, A ).Denote δ ( ξ ; A ) = min ∞ s = − { α ( η s ; x, w ) + sν ( y ) } . We say that s belongs to thecritical segment C ( ξ ; A ) if α ( η s ; x, w ) + sν ( y ) = δ ( ξ ; A ). The critical height χ ( ξ ; A )is the greatest s ∈ C ( ξ ; A ). Note that χ ( ξ ; A ) ≤ ~ ( ξ ; A ). Remark 11.
We can draw a Newton-Puiseux polygon N ( ξ ; A ) ⊂ R × Z ≥− bytaking as support the set { ( α ( η s ; x, w ) , s ); s = − , , , . . . } . Then ( α ( ξ ; x, w, y ) , ~ ( ξ ; x, w, y )) is the main vertex of the Newton-Puiseux polygon.We also have that δ ( ξ ; x, w, y ) is the smallest value a + ν ( y b ), where ( a, b ) is inthe support. Nevertheless, this Newton-Puiseux polygon needs to be prepared byperforming preliminary transformations in the variables x, w in order to be a usefultool in the control of the transformations in the variables x, y .The invariants in three variables make sense also for f ( x, w, y ) = P s y s f s ( x, w ).Thus, we write(20) α ( f ; x, w, ) = min s { ν ( f s ; x, w ) } ; δ ( f, x, w, y ) = min s { ν ( f s ; x, w ) + sν ( y ) } . Prepared situations in two variables.
Take a vector field η as in equation(15). We say that η is ( x, w ) -prepared if there is q ∈ Z ≥ such that( a, b, c ) = x q (˜ a ( x, w ) , ˜ b ( x, w ) , ˜ c ( x, w )); (˜ b (0 , , ˜ c (0 , = (0 , . We say that η is dominant if ˜ b s (0 , = 0 and recessive if ˜ b (0 ,
0) = 0 , ˜ c (0 , = 0. Remark 12.
The condition Λ( η ; x, w ) = − ν ( w ) is equivalent to say that η isprepared-dominant. If η is prepared-recessive, then 0 ≥ Λ( η ; x, w ) > ν ( w ). Definition 5.
Take η as in equation (15). We say that η is strongly ( x, w )-prepared if there is a decomposition (21) η = x ρ U ( x, w ) θ + x τ V ( x, w ) y ∂∂y ; θ = xh ( x, w ) x ∂∂x + ∂∂w satisfying the following properties (1) ρ, τ ∈ Z ∪ { + ∞} , with ρ = τ . Here ρ = + ∞ , respectively or τ = + ∞ ,indicates that U ( x, w ) , respectively V ( x, w ) , is identically zero. (2) We can write U = λ + xf ( x, w ) and V = µ + xg ( x, w ) , where λ, µ ∈ k .Moreover, if ρ = + ∞ then λ = 0 and if τ = + ∞ then µ = 0 . Let us note that “strongly prepared” implies “prepared”. The dominant casecorresponds to r < t and the recessive case to r > t .6.4.
Effect of etale w -Puiseux packages. Let us perform an etale w -Puiseuxpackage and let ( t, w ′ , y ) be the obtained coordinates. Recall that t d = x and ν ( w/t p ) = 0, where p, d are without common factor. Moreover, we have w ′ = w/t p − c , with c = 0, and hence(22) x ∂∂x = 1 d (cid:26) t ∂ ′ ∂t − p ( w ′ + c ) ∂ ′ ∂w ′ (cid:27) ; ∂∂w = 1 t p ∂ ′ ∂w ′ ; ∂∂y = ∂ ′ ∂y . Consider η as in equation (15) and write η = a ′ ( t, w ′ ) t ∂ ′ ∂t + b ′ ( t, w ′ ) ∂ ′ ∂w ′ + c ′ ( t, w ′ ) y ∂ ′ ∂y in the coordinates t, w ′ , y . Then we have(23) a ′ = η ( t ) /t = (1 /d ) ab ′ = η ( w ′ ) = ( w ′ + c ) { b/w − ( p/d ) a } = t − p { b − ( p/d ) t p ( w ′ + c ) a } c ′ = η ( y ) /y = c. From these considerations, we obtain the following results:
Lemma 11.
Consider f = P ∞ ℓ = − w ℓ f ℓ ( x ) ∈ w − k [[ x, w ]] . We have α ( f ; t, w ′ ) = λ ( f ; x, w ) . As a consequence, we also have that α ( η ; t, w ′ ) = λ ( η ; x, w ) .Proof. Take a monomial w a x b . Note that w a x b = t ap + bd ( w ′ + c ) a where ( w ′ + c ) a is a unit, hence λ ( w a x b ; x, w ) = ν ( w a x b ) == ν ( t ap + bd ( w ′ + c ) a ) = ν ( t ap + bd ) = α ( w a x b ; t, w ′ ) . Note that both λ ( − ; x, w ) and α ( − ; t, w ′ ) have the usual valuative properties. Thisgives in particular that α ( f ; t, w ′ ) ≥ λ ( f ; x, w ), as a consequence of the above EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 21 property for monomials. Put λ = λ ( f ; x, w ) and let us decompose f = L( f ) + f ∗ ,where λ ( f ∗ ; x, w ) > λ and L( f ) is of the formL( f ) = X ν ( w a x b )= λ µ ab w a x b . Now it is enough to prove that α (L( f ); x ′ , w ′ ) = λ . We know that if ν ( w a x b ) = λ then m = ad + bp is independent of ( a, b ), since ν ( t m ) = ν ( x a w b ). We haveL( f ) = t m X µ ab ( w ′ + c ) a = 0 . Then α (L; t, w ′ ) = λ . The last statement comes from the above arguments and theequations 23. (cid:3) Corollary 1.
Consider η as in equation (15). We have α ( η ; t, w ′ ) ≥ α ( η ; x, w ) − ν ( w ) , and the equality holds exactly when η is ( x, w ) -prepared and dominant.Proof. We know that α ( η ; t, w ′ ) = λ ( η ; x, w ) by Lemma 11. Now, it is a directconsequence of the definitions that λ ( η ; x, w ) ≥ α ( η ; x, w ) − ν ( w ), and the equalityholds exactly when η is ( x, w )-prepared and dominant, in view of Remark 12. (cid:3) Lemma 12.
Consider η as in equation (15). We have (1) If Λ( η ; x, w ) < , then Λ( η ; t, w ′ ) = − ν ( w ) and hence η is ( t, w ′ ) -preparedand dominant. (2) If η is ( x, w ) -prepared and dominant, then η is also ( t, w ′ ) -prepared anddominant. (3) If Λ( η ; x, w ) = 0 , then Λ( η ; t, w ′ ) ≤ . (4) If η is ( x, w ) -prepared and recessive, then η is ( t, w ′ ) -prepared.Proof. Write η = x m η ′ , where ν ( x m ) = α ( η ; x, w ). If we substitute η by η ′ wecan assume without loss of generality that α ( η ; x, w ) = 0 and thus Λ( η ; x, w ) = λ ( η ; x, w ). Let us put b = b/w = P ∞ ℓ = − b ℓ ( x ) w ℓ . Consider first the case that λ ( η ; x, w ) <
0. This implies that(24) η = b − ( x ) ∂∂w + η ∗ = x r U ( x ) ∂∂w + η ∗ , U (0) = 0 , where 0 ≤ ν ( b − ( x )) = ν ( x r ) = ν ( t rd ) < ν ( w ) = ν ( t p ) and η ∗ has the form η ∗ = ∞ X ℓ =0 w ℓ (cid:26) a ℓ ( x ) x ∂∂x + b ℓ ( x ) w ∂∂w + c ℓ ( x ) y ∂∂y (cid:27) By equations (22) we see that α ( η ∗ ; t, w ′ ) ≥ x r U ( x ) ∂∂w = t rd − p U ( t d ) ∂∂w ′ . Note that rd − p <
0. We obtainΛ( η ; t, w ′ ) = − ν ( w ′ )and thus η is ( t, w ′ )-prepared and dominant. This proves statement 1. Now, state-ment 2 is a direct consequence of statement 1.Assume that Λ( η ; x, w ) = 0. Write η as in Equation (24), where ν ( x r ) ≥ ν ( w )(we accept the case r = + ∞ to denote that w divides η ( w )). If ν ( x r ) = ν ( w ),by the same argument as above we obtain that Λ( η ; t, w ′ ) = − ν ( w ′ ), hence η is ( t, w ′ )-prepared and dominant. If ν ( x r ) > ν ( w ), we have λ ( η ∗ ; x, w ) = 0. Thus, wecan write η = (cid:18) µ x ∂∂x + µ w ∂∂w + µ y ∂∂y (cid:19) + η ∗∗ , where ( µ , µ , µ ) = (0 , ,
0) and λ ( η ∗∗ ; x, w ) >
0. By equations (23) we have η − η ∗∗ = µ d x ′ ∂ ′ ∂x ′ + dµ − pµ d ( w ′ + c ) ∂ ′ ∂w ′ + µ y ∂∂y , where α ( η ∗∗ ; t, w ′ ) = λ ( η ∗∗ ; t, w ′ ) >
0. We obtain0 = α ( η − η ∗∗ ; t, w ′ ) = α ( η ; t, w ′ )(25) 0 ≥ λ ( η − η ∗∗ ; t, w ′ ) = λ ( η ; t, w ′ ) . (26)This ends the proof of statement 3. Note that if dµ − pµ = 0 then η is ( t, w ′ )-prepared and dominant. If dµ − pµ = 0 and µ = 0, we have that η is ( t, w ′ )-prepared and recessive. Now, if η where ( x, y )-prepared and recessive, then µ = 0.This proves statement 4. (cid:3) Proposition 10.
Consider η as in equation (15). After performing finitely manyetale w -Puiseux packages, either we get transversal formal maximal contact or weobtain one of the following properties: a) The vector field η is strongly ( x, w ) -prepared dominant and this propertypersists under new etale w -Puiseux packages. b) The vector field η is strongly ( x, w ) -prepared recessive and this propertypersists under new etale w -Puiseux packages.Proof. By the two dimensional desingularization for vector fields [14] and since wedo not get maximal contact, we can obtain η written down as η = f ( x, w ) θ + g ( x, w ) y ∂∂y ; θ = xh ( x, w ) x ∂∂x + ∂∂w . Under new etale w -Puiseux packages, this form persist. Let us see it. First, weknow that θ = 1 d (cid:26) t d h ′ ( t, w ′ ) t ∂ ′ ∂t + (cid:18) dt d − p ( w ′ + c ) t d h ′ ( t, w ′ ) (cid:19) ∂ ′ ∂w ′ (cid:27) where h ′ ( t, w ′ ) = h ( t d , t p ( w ′ + c )). This allows us to write θ = t − d W ( t, w ′ ) θ ′ , where W ( t, w ′ ) = 1 − t d ( p/d )( w ′ + c ) h ′ ( t, w ′ )is a unit and θ ′ has the same form as θ . Note that W ( t, w ′ ) − t .Now, we write η = f ′ ( t, w ′ ) t − d W ( t, w ′ ) θ ′ + g ′ ( t, w ′ ) y ∂∂y , where f ′ ( t, w ′ ) = f ( t d , t p ( w ′ + c )) and g ′ ( t, w ′ ) = g ( t d , t p ( w ′ + c )). By the standarddesingularization of functions, we can perform new etale w -Puiseux packages toobtain that f = x ρ U ( x, w ); g = x τ V ( x, w ) , where U, V and ρ, τ satisfy to the properties in Definition 5 (note that it is possiblethat ρ or τ are negative; to recover a non-meromorphic vector field we can multi-ply by a suitable power of x ). By performing new etale w -Puiseux packages, thedifference ν ( x τ ) − ν ( x ρ ) increases the positive amount ν ( w ). If this difference is EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 23 positive, we are in case a), if it is always negative, we obtain case b). If it is zero,in the next step it is positive. (cid:3)
Remark 13.
The above proof also shows that if η is strongly ( x, w )-prepared anddominant, it is so with respect to ( t, w ′ ). Nevertheless, it is not always true that if η is strongly ( x, w )-prepared and recessive the same holds with respect to ( t, w ′ ).We start with ρ > τ , by it can happen that ρ ′ ≤ τ ′ and in this case, after a newetale w -Puiseux package we would obtain a dominant situation. Remark 14.
Assume that we are in one of the situations a) or b) described inProposition 10. Let us perform an etale w -Puiseux package. Then we have α ( η ; t, w ′ ) = (cid:26) α ( η ; x, w ) − ν ( w ) dominant case a) α ( η ; x, w ) recessive case b)To see this, the only difficulty is the recessive case. Note that since the recessivesituation is stable under any finite sequence of etale w -Puiseux packages, we have ν ( x ρ − τ ) > ν ( w ), and this is enough to assure the above formula.6.5. Preparations in three variables.
Consider a vector field ξ ∈ Der k O [log x ],that we write ξ = P ∞ s = − y s η s where the η s are like in equation (19). Definition 6.
Let h = ~ ( ξ ; A ) be the main height. We say that ξ is main-vertexprepared with respect to A when η h is ( x, w ) -prepared and dominant. If in addition η h is strongly ( x, w ) -prepared, we say that ξ is strongly main-vertex prepared . Wesay that L is well prepared with respect to A if there is ξ ∈ L A [log x ] that is x -prepared and strongly main-vertex prepared. Remark 15. If ξ is main-vertex prepared, we have ~ ( ξ ; A ) ≥
0. Moreover, if α ( ξ ; A ) = 0 and ξ is main-vertex prepared with ~ ( ξ ; A ) = 0, then ξ is a non-singular vector field. Proposition 11.
Assume that ξ is x -prepared and strongly main-vertex preparedwith respect to A . Let us perform an etale w -Puiseux package to obtain A ′ =( O ′ , ( t, w ′ , y )) . Then ξ is t -prepared, strongly main-vertex prepared with respect to A ′ and the main height does not vary, that is ~ ( ξ ; A ′ ) = ~ ( ξ ; A ) .Proof. The fact that ξ is t -prepared has been proved in subsection 6.1. The decom-position ξ = P ∞ s = − y s η s is the same one with respect to x, w, y and with respect to t, w ′ , y . Let us put h = ~ ( L ; A ). By hypothesis η h is strongly ( x, w )-prepared anddominant and hence it is also strongly ( t, w ′ )-prepared and dominant, in view ofRemark 13. Now we have only to show that h is also the main height ~ ( ξ ; t, w ′ , y )relatively to A ′ . By Corollary 1 we have α ( η h ; t, w ′ ) = α ( η h ; x, w ) − ν ( w ) , since η h is ( x, w )-prepared and dominant. For any other index s we have α ( η s ; t, w ′ ) ≥ α ( η s ; x, w ) − ν ( w ) , and this is enough to see that η h gives the main height for ξ with respect to A ′ . (cid:3) Proposition 12.
Assume that ξ = P ∞ s = − y s η s is x -prepared and strongly main-vertex prepared with respect to A . Let us put h = ~ ( ξ ; x, w, y ) . By performingfinitely many etale w -Puiseux packages we have the following properties: (1) For any s < h , the vector field η s is strongly ( x, w ) -prepared and this isstable, with the same character dominant or recessive, under any new finitesequence of etale w -Puiseux packages. (2) The critical segment C ( ξ ; A ) does not vary under any new finite sequenceof etale w -Puiseux packages, and all the levels s ∈ C ( ξ ; A ) have the samecharacter dominant or recessive.Proof. The first statement is a corollary of Proposition 10. Let us prove the secondstatement, assuming that statement 1 holds. Let us perform an etale w -Puiseuxpackage. In view of Remark 14 we have α ( η s ; t, w ′ ) = (cid:26) α ( η s ; x, w ) − ν ( w ) dominant case α ( η s ; x, w ) recessive caseThus, the critical segment thus not vary if η s is dominant for all s ∈ C ( ξ ; A ). Ifthere is an s ∈ C ( ξ ; A ) such that η s is dominant, then all the recessive s in thecritical segment disappear under a new w -Puiseux package and we are in the firstcase. Finally, if under any finite sequence of etale w -Puiseux packages there isno dominant η s that appears in the critical segment, the elements in the criticalsegment are also stable. (cid:3) Definition 7.
We say that ξ = P ∞ s = − y s η s is completely prepared with respect to A if it is x -prepared, strongly main-vertex prepared and the properties 1 and 2 ofProposition 12 hold. We have two possible situations: a) Dominant critical segment . The η s corresponding to s in the critical seg-ment are strongly ( x, w ) -prepared and dominant. b) Recessive critical segment . The η s corresponding to s in the critical segmentare strongly ( x, w ) -prepared and recessive. Remark 16.
Assume that ξ = P ∞ s = − y s η s is completely prepared with respect to A . Let χ = χ ( ξ ; A ) be the critical height and h = ~ ( ξ ; A ) the main height. Wehave χ ≤ h . Moreover, since η h is strongly ( x, w )-prepared and dominant, in thecase of a recessive critical segment we have χ ≤ h − Critical initial part and critical polynomial.
Let us consider a vectorfield ξ = P ∞ s = − y s η s ∈ Der k ( O )[log x ] and write it as ξ = ∞ X s = − X j y s x j (cid:26) a sj ( w ) x ∂∂x + b sj ( w ) ∂∂w + c sj ( w ) y ∂∂y (cid:27) . We know that δ ( ξ ; x, w, y ) = min (cid:8) ν ( x j y s ); ( a sj ( w ) , b sj ( w ) , c sj ( w )) = (0 , , (cid:9) . Put δ = δ ( ξ ; x, w, y ). We define the critical initial part of ξ by(27) Crit( ξ ; x, w, y ) = X ν ( x j y s )= δ y s x j (cid:26) a sj ( w ) x ∂∂x + b sj ( w ) ∂∂w + c sj ( w ) y ∂∂y (cid:27) . Obviously, if we put ξ ∗ = ξ − Crit( ξ ; x, w, y ) we have δ ( ξ ∗ ; x, w, y ) > δ ( ξ ; x, w, y ). Definition 8.
Take δ ∈ Γ . A monic polynomial P ( x, y ) ∈ k [ x, y ] given by P ( x, y ) = y m + λ m − x n y m − + λ m − x n y m − + · · · + λ x n m is called ν -homogeneous or degree δ if and only if ν ( x n j y m − j ) = ν ( y m ) = δ for any j such that λ j = 0 . It is called a Tchirnhausen polynomial if λ m − = 0 . EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 25
Remark 17.
Let us perform an etale y -Puiseux package, to obtain coordinates t, w, y ′ such that t d = x and y ′ = y/t p − c . Consider a monic ν -homogeneouspolynomial P = P ( x, y ) or degree δ . Then P = P ( t d , t p ( y ′ + c )) = t q ′ P (1 , y ′ + c ) , where ν ( t q ′ ) = δ and P (1 , y ′ + c ) is a monic polynomial of degree m in the variable y ′ . Write P (1 , y ′ + c ) = y ′ h ′ Q ( y ′ ), where Q (0) = 0. We have h ′ ≤ m . Moreover, theonly possibility to have m = h ′ is that P ( x, y ) = ( y − cx n ) m . This cannot occurwhen P ( x, y ) is a Tchirnhausen polynomial. Hence if P ( x, y ) is a Tchirnhausenpolynomial we have h ′ < m . This argument is crucial in most of the procedures ofreduction of singularities in characteristic zero. Lemma 13.
Assume that ξ = P ∞ s = − y s η s is completely prepared relatively to A .Then, the critical initial part ξ = Crit ( ξ ; x, w, y ) satisfies that (1) ξ ( x ) = ξ ( y ) = 0 , in the case of dominant critical segment. (2) ξ ( x ) = ξ ( w ) = 0 , in the case of recessive critical segment.More precisely, the critical initial part ξ takes one of the following forms ξ = λx q χ X s =0 λ s y s x q s ∂∂w ; dominant critical segment case (28) ξ = λx q χ X s = − λ s y s x q s y ∂∂y ; recessive critical segment case (29) where λ = 0 , λ χ = 1 and ν ( y s x q s ) = δ − ν ( x q ) for each s with λ s = 0 .Proof. Put h = ~ ( ξ ; x, w, y ). Recall that for any s ≤ h the vector field η s = X j x j (cid:26) a sj ( w ) x ∂∂x + b sj ( w ) ∂∂w + c sj ( w ) y ∂∂y (cid:27) = X j x j η sj is ( x, w )-strongly prepared. Put α s = α ( η s ; x, w ) and let us take r s such that ν ( x r s ) = α s . Write η s = x r s ˜ η s . In view of definition 5, we have that˜ η s = (cid:26) µ s ∂/∂w + xη s (dominant case) µ s y∂/∂y + xη s (recessive case)We end by putting λ s = µ s /λ if s is in the critical segment and λ s = 0 otherwise. (cid:3) Definition 9.
In the situation of Lemma 13, we define the critical polynomial P ξ ( x, y ) of ξ with respect to x, w, y to be P ξ ( x, y ) = (cid:26) ξ ( w ) /λx q = P χs =0 λ s y s x q s (dominant critical segment) ξ ( y ) /λx q = P χs = − λ s y s +1 x q s (recessive critical segment) (It is a ν -homogeneous monic polynomial of ν -degree χν ( y ), respectively ( χ +1) ν ( y ),in the case of a dominant, respectively recessive critical segment.) Remark 18.
The critical initial part is obtained from the critical polynomial bythe formulaCrit( ξ ; x, w, y ) = (cid:26) λx q P ξ ( x, y ) ∂/∂w (dominant critical segment) λx q P ξ ( x, y ) ∂/∂y (recessive critical segment) Stability of the main height. Dominant critical segment.
In this sub-section we start to study the effect of an etale y -Puiseux package on the mainheight. Let us consider ξ ∈ Der k O [log x ] and denote h = ~ ( ξ ; x, w, y ); χ = χ ( ξ ; x, w, y ); δ = δ ( ξ ; x, w, y ); ξ = Crit( ξ ; x, w, y ) . Let us perform an etale y -Puiseux package, to obtain t, w, y ′ such that t d = x and y ′ = y/t p − c . We recall that(30) x ∂∂x = 1 d (cid:26) t ∂ ′ ∂t − p ( y ′ + c ) ∂ ′ ∂y ′ (cid:27) ; ∂∂w = ∂ ′ ∂w ; y ∂∂y = ( y ′ + c ) ∂ ′ ∂y ′ . Lemma 14. α ( ξ ; t, w, y ′ ) ≥ δ = δ ( ξ ; x, w, y ) .Proof. In view of the valuative behavior of the invariant α ( − ; x, w ) and because ofthe “monomial” definition of δ ( − ; x, w ), it is enough to verify the case that ξ is ofone of the following monomial types ξ = y s x m w n x ∂∂x ; ξ = y s x m w n ∂∂w ; ξ = y s x m w n y ∂∂y , where ν ( y s x m ) ≥ δ . Note that y s x m w n = x ′ sp + dm ( y ′ + c ) w n , where ν ( x ′ sp + dm ) = ν ( y s x m ) ≥ δ . Now, in view of the equations 30 we have that ξ = x ′ sp + dm ξ ∗ , where α ( ξ ∗ ; t, w, y ′ ) ≥ (cid:3) Proposition 13.
Assume that ξ is completely prepared with a dominant criticalsegment and h ≥ . Let us perform an etale y -Puiseux package. After performingfinitely many subsequent etale w -Puiseux packages, we obtain A ′ such that ξ iscompletely prepared with respect to A ′ and h ′ = h ( ξ ; A ′ ) ≤ χ . Moreover, if thecritical polynomial P ξ ( x, y ) is a Tchirnhausen polynomial, we have h ′ < χ ≤ h .Proof. Denote ξ = ξ + ξ ∗ . We know that δ ( ξ ∗ ; x, w, y ) > δ and hence, by Lemma14 we have α ( ξ ∗ ; t, w, y ) > δ . On the other hand ξ = λx q P ξ ( x, y ) ∂∂w . Afterperforming the etale y -Puiseux package, we obtain ξ = λx ′ q ′ P ξ (1 , y ′ + c ) ∂ ′ ∂w where ν ( t q ′ ) = δ . If ξ ∗′ = λ − t − q ′ ξ ∗ , we have α ( ξ ∗′ ; t, w, y ′ ) >
0. Write(31) ξ ′ = λ − t − q ′ ξ = P ξ (1 , y ′ + c ) ∂ ′ ∂w + ξ ∗′ , Then α ( ξ ′ ; t, w, y ′ ) = 0. Let h ′ ≤ χ be such that P ξ (1 , y ′ + c ) = y ′ h ′ χ X s = h ′ λ ′ s y ′ s − h ′ ; λ ′ h ′ = 0 . It is obvious that h ′ ≤ χ ≤ h and, in view of Remark 17, we have that h ′ < χ ≤ h in the case that P ξ ( x, y ) is a Tchirnhausen polynomial. Moreover, we see that h ′ = ~ ( ξ ′ ; A ′ ) = ~ ( ξ ; A ′ ). Write ξ = P ∞ s = − y ′ s η ′ s , as usual, with η ′ s = a ′ s ( t, w ) x ′ ∂∂ ′ t + b ′ s ( t, w ) ∂ ′ ∂w + c ′ s ( t, w ) y ′ ∂ ′ ∂y ′ . Then η ′ h ′ is ( t, w )-prepared and dominant in view of Equation 31. By performingnew etale w -Puiseux packages to obtain a completely prepared ξ , the main height h ′ is not modified and we are done. (cid:3) EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 27
Stability of the main height. Recessive critical segment.
Take herethe situation and notations of the previous Subsection 6.7.Let us assume that ξ is completely prepared with respect to A with a recessivecritical segment and h ≥
1. Recall that χ ≤ h − ξ = ξ + ξ ∗ where ξ = λx q P ξ ( x, y ) ∂∂y = λx q P ξ ( x, y ) y y ∂∂y , where P ξ ( x, y ) = y χ +1 + P χ − s = − λ s y s +1 x q s is the critical polynomial. After per-forming an etale y -Puiseux package, we have ξ = λt q ′ P ξ (1 , y ′ + c ) ∂ ′ ∂y ′ where ν ( t q ′ ) = δ . Write ξ ∗′ = λ − t − q ′ ξ ∗ , as in the proof of Proposition 13. Wehave α ( ξ ∗′ ; t, w, y ′ ) > ξ ′ = 1 λt q ′ ξ = P ξ (1 , y ′ + c ) ∂∂y ′ + ξ ∗′ = ξ ′ + ξ ∗′ . Let − ≤ h ′ ≤ χ be such that P ξ (1 , y ′ + c ) = y ′ h ′ +1 χ X s = h ′ λ ′ s y ′ s − h ′ = y ′ h ′ +1 Q ( y ′ ); Q (0) = 0 . It is obvious that h ′ ≤ χ ≤ h −
1. Moreover, if P ξ ( x, y ) is a Tchirnhausen polynomialwe have h ′ < χ ≤ h −
1, in view of Remark 17.
Remark 19.
We have that h ′ = ~ ( ξ ′ ; t, w, y ′ ), but the main vertex is not dominant.For this reason, we will do a coordinate change in the dependent variables of thetype w ′′ = w + y ′ .Let us do a coordinate change w ′′ = w + y ′ to obtain A ′′ = ( O ′ , z ′′ = ( t, w ′′ , y ′ )).We have(33) ξ ′ = P ξ (1 , y ′ + c ) ∂ ′ ∂y ′ = y ′ h ′ +1 Q ( y ′ ) (cid:26) ∂ ′′ ∂w ′′ + ∂ ′′ ∂y ′ (cid:27) . Let us write ξ ′ = P ∞ s = − y ′ s η ′′ s , where η ′′ s = a ′′ s ( x ′ , w ′′ ) x ′ ∂ ′′ ∂x ′ + b ′′ s ( x ′ , w ′′ ) ∂ ′′ ∂w ′′ + c ′′ s ( x ′ , w ′′ ) y ′ ∂ ′′ ∂y ′ . Recalling that ξ ′ = ξ ′ + ξ ∗′ and α ( ξ ∗′ ; x ′ , w ′′ , y ′ ) >
0, we see from Equations 33that α ( η ′′ h ′ +1 ; x ′ , w ′′ , y ′ ) = 0 and η ′′ h ′ +1 is dominant and prepared with respect to( t, w ′′ , y ′ ). In particular ~ ( ξ ′ ; t, w ′′ , y ′ ) ≤ h ′ + 1.As a consequence, by performing new etale w ′′ -Puiseux packages, we obtain ˜ A such that ξ ′ is completely prepared and h ( ξ ′ ; ˜ A ) ≤ h ′ + 1. Now, recalling that h ′ ≤ χ ≤ h − h ′ < χ ≤ h −
1, we have proved the following statement:
Proposition 14.
Let ξ be completely prepared with a recessive critical segmentand assume h = ~ ( ξ ; A ) ≥ . Let us perform an etale y -Puiseux package. Afterperforming a coordinate change in the dependent variables and finitely many sub-sequent etale w -Puiseux packages, we obtain ˜ A such that ξ is completely preparedwith respect to ˜ A and ˜ h = ˜ ~ ( ξ ; ˜ A ) ≤ χ + 1 ≤ h . Moreover, if the critical polynomial P ξ ( x, y ) is a Tchirnhausen polynomial, we have ˜ h < χ + 1 ≤ h . The condition of Tchirnhaus.
Let ξ ∈ Der k O [log x ] be completely preparedwith respect to A . Put h = h ( ξ ; A ), χ = χ ( ξ ; A ) and assume h ≥
1. We have thefollowing possible cases:A) The critical polynomial is not Tchirnhausen and the critical segment isdominant with χ = h .B) The critical polynomial is not Tchirnhausen and the critical segment isrecessive with χ = h − χ < h − χ < h .The last case C corresponds to a winning situation in the sense of the followingproposition Proposition 15.
Assume we are in case C above. Let us perform an etale y -Puiseux package. By performing a subsequent coordinate change in the dependentvariables (if it is necessary) and finitely many etale w -Puiseux packages, we obtain ˜ A such that ξ is completely prepared with respect to ˜ A and ˜ h = h ( ˜ L ; ˜ A ) < h .Proof. Direct consequence of Propositions 13 and 14. (cid:3)
Next subsections are devoted to the study of situations B and A.6.10.
Tchirnhausen preparation. Recessive case.
In this subsection we in-troduce a recessive Tchirnhausen preparation algorithm in order to deal with thecase B of the preceding subsection. This algorithm is based on the following twodefinitions.
Definition 10.
Let ς > be a positive element of the value group Γ ⊂ R . Consider A = ( O , ( x, w, y )) . We say that ( x, w ) is recessive for ς if and only if we have ς > P Ni =0 ν ( w i ) for any finite sequence of etale w -Puiseux Packages A = A
7→ A
7→ · · · A N , where ( x i , w i , y ) is the coordinate system in A i , i = 0 , , . . . , N . An example of this situation is obtained if we are in the case B of Subsection6.9. More generally, let ξ be completely prepared with recessive critical segmentand put h = ~ ( ξ ; x, w, y ), χ = χ ( ξ ; x, w, y ). Let ( α, h ) be the main vertex and ( β, χ )the “critical” vertex. Then ( x, w ) is recessive for ς = ( h − χ ) ν ( y ) − β + α . Remark 20.
Assume that ξ and A are in the situation of case B of Subsection6.9. Then, there is an integer number p ∈ Z > such that ν ( y ) = ν ( x p ). Indeed, thisis always true when we have a ν -homogeneous Tchirnhausen polynomial, that wewrite P ( x, y ) = y m + λ x n y m − + · · · + λ m x n m since λ = 0 implies that ν ( y ) = ν ( x n ). Definition 11.
Consider a vector field ξ ∈ Der k ( O )[log x ] and A with coordinates ( x, w, y ) . Write ξ = x ˜ q ξ ′ , where α ( ξ ′ ; x, w, y ) = 0 . Consider two elements ǫ , γ ofthe value group Γ and take h ∈ Z ≥ . Let us do the decomposition ξ ′ = P ∞ s = − y s η ′ s associated to ( x, w, y ) . We say that ( ξ ; A ; ǫ, γ , h, p ) is a recessive preparation stepof order p ∈ Z ≥ if the following properties hold EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 29 (1) h = ~ ( ξ ; x, w, y ) and γ ≤ ν ( y ) . (2) There is q ∈ Z ≥ such that ǫ = ν ( x q ) and ( x, w ) is recessive for γ − ǫ . (3) γ ≤ ν ( x p )(4) There are units U ( x, w ) , V ( x, w ) such that the levels η ′ h , η ′ h − , and η ′ h − take the forms (34) η ′ h = U ( x, w ) { ∂/∂w + xc h ( x, w ) y∂/∂y } ,η ′ h − = x q V ( x, w ) { xa h − ( x, w ) x∂/∂x + xb h − ( x, w ) ∂/∂w + y∂/∂y } ,η ′ h − = a h − ( x, w ) x∂/∂x + b h − ( x, w ) ∂/∂w + x q + p c h − ( x, w ) y∂/∂y, We say that ( ξ ; A ; ǫ, γ , h, p ) is a final recessive step if in addition we have that ν ( y ) < ν ( x p ) . Remark 21.
Assume that ξ and A are in the situation of case B of Subsection6.9. Take p ∈ Z > such that ν ( y ) = ν ( x p ) and γ = ν ( y ). Let ( α, h ) be the mainvertex and ( β, h −
1) the critical vertex and put ǫ = β − α . Then ( ξ ; A ; ǫ, γ , h, p )is a (non-final) recessive preparation step or order p . Proposition 16.
Assume that ( ξ, A ; ǫ, γ , h, p ) is a recessive preparation step oforder p . There is a coordinate change y ∗ = y − x p g ( x, w ) such that ( ξ, A ∗ ; ǫ, γ , h, p ∗ ) is a recessive preparation step of order p ∗ > p .Proof. Take g ( x, w ) in the Hensel closure of O and let us write y ∗ = y − x p g ( x, w ).Note that γ ≤ ν ( y ∗ ) since γ ≤ ν ( y ) and γ ≤ ν ( x p g ( x, y )). The property that( x, w ) is recessive for γ − ǫ does not depend on y ∗ . We have x ∂∂x = x ∂ ∗ ∂x + x p (cid:18) pg ( x, w ) + x ∂g ( x, w ) ∂x (cid:19) ∂ ∗ ∂y ∗ (35) ∂∂w = ∂ ∗ ∂w + x p ∂g ( x, w ) ∂w ∂ ∗ ∂y ∗ (36) y ∂∂y = y ∗ ∂ ∗ ∂y ∗ + x p ∂ ∗ ∂y ∗ (37)Let us decompose ξ ′ = P ∞ s = − y ∗ s η ′∗ s as usual with respect to ( x, w, y ∗ ). Notingthat q < p since ǫ = ν ( x q ) < γ ≤ ν ( x p ), it is a straightforward computation fromequations 35, 36 and 35 that ~ ( ξ ′ ; x, w, y ∗ ) = h and η ′∗ h , η ′∗ h − and η ′∗ h − take theforms η ′∗ h = U ∗ ( x, w ) { ∂/∂w + xc ∗ h ( x, w ) y∂/∂y } ,η ′∗ h − = x q V ∗ ( x, w ) { xa ∗ h − ( x, w ) x∂/∂x + xb ∗ h − ( x, w ) ∂/∂w + y∂/∂y } ,η ′∗ h − = a ∗ h − ( x, w ) x∂/∂x + b ∗ h − ( x, w ) ∂/∂w + x q + p c ∗ h − ( x, w ) y∂/∂y where U ∗ (0 , = 0 and V ∗ (0 , = 0. In order to end our proof it is enough to showthat g ( x, w ) may be chosen in such a way that x divides c ∗ h − .Let us put(38) F = ξ ′ ( y ) = P ∞ s =0 y s F s ( x, w ) = P ∞ s =0 y ∗ s F ∗ s ( x, w ) H = ξ ′ ( x p g ( x, w )) = P ∞ s =0 y s H s ( x, w ) = P ∞ s =0 y ∗ s H ∗ s ( x, w ) G = ξ ′ ( y ∗ ) = F − H = P ∞ s =0 y ∗ s G ∗ s ( x, w )We have to prove that G ∗ h − ( x, w ) is divisible by x q + p +1 after a suitable choice of g ( x, w ). Let us decompose(39) F = ˜ F + F ; ˜ F = y h − F h − + y h F h H = ˜ H + H ; ˜ H = y h − H h − + y h H h We have that F ∗ h − and H ∗ h − are divisible by x q + p +1 , since they are is divisible by x p and 2 p ≥ p + q + 1. Note also that˜ H = J + K ; J = y h − η ′ h − ( x p g ( x, w )) , K = y h η ′ h ( x p g ( x, w )) . Moreover, J is divisible by x q + p +1 in view of form of η ′ h − in Definition 11. Wealso have that x p divides K ∗ h − and 2 p > p + q + 1.Thus, we have only to prove that after a suitable choice of g ( x, w ) we can obtainthat ˜ F ∗ h − is divisible by x q + p +1 . Recall that ˜ F = y h − ( yF h + F h − ) where(40) yF h = η ′ h − ( y ) = yV ( x, w ) x q F h − = η ′ h − ( y ) /y = x q + p c h − ( x, w ) . Now, write c h − ( x, w ) = f ( w ) + xf ( x, w ). If we put g ( x, w ) = − f ( w ) /V ( x, w )we are done. (cid:3) Let us show how to obtain a final recessive step. We start with ξ completelyprepared with respect to A in the case B of Subsection 6.9. Thus we have arecessive preparation step ( ξ, A ; ǫ, γ , h, p ) of order p , where γ = ν ( y ) = ν ( x p ).Since ν ( x p ) = ν ( y ), it is not a final recessive step. We do a coordinate change y = y − x p g ( x, w ) as in Proposition 16 to obtain a new recessive preparation step( ξ, A ; ǫ, γ , h, p ) with p > p . We repeat to obtain y j +1 = y j − x p j g j ( x, w ) , where ( x, w, y j ) are the coordinates for a recessive preparation step ( ξ ; A j ; ǫ, γ , h, p j )of order p j and p j +1 > p j . There are two possibilities:(1) We have ν ( y j ) ≥ ν ( x p j ) for all j . In this case we obtain a transversal formalmaximal contact element ˆ f ∈ ˆ O as the limit of the y j .(2) There is an index j such that ν ( y j ) < ν ( x p j ). In this case we obtain afinal recessive step ( ξ ; A j ; ǫ, γ , h, p j ). Proposition 17.
Assume that we have a final recessive step ( ξ ; A ; ǫ, γ , h, p ) . Afterperforming finitely many w -Puiseux packages we obtain A ′ such that ξ is completelyprepared with respect to A ′ and we are in the winning situation C of Subsection 6.9.Proof. Note that if U ( x, w, y ) is a unit, then ( U ( x, w, y ) ξ ; A ; ǫ, γ , h, p ) is still a finalrecessive step.Let us perform an etale w -Puiseux package to obtain A whose coordinates are( t, w , y ), where t d = x and w = w/t ˜ p − c . In view of Equations 34 we see that ξ ismain-vertex prepared with respect to ( x, w, y ) and hence the main height h is notchanged under the etale w -Puiseux package. The form of Equations 34 persists,with the following observations:(1) The parameter ǫ is transformed into ǫ = ǫ + ν ( w ). Anyway, we still havethat ( t, w ) is recessive for γ − ǫ (see Definition 10).(2) The order p is transformed into p = pd .In particular ( ξ ; A ; ǫ , γ , h, p ) is a final recessive preparation step.Thus, we can multiply by a unit ξ an do successive etale w -Puiseux packages inorder to obtain that in addition ξ is completely prepared with respect to A . Let uslook at Equations 34. Let us put α h − = α ( η ′ h − ; x, w ) α h − = α ( η ′ h − ; x, w ) . EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 31
From the form of η ′ h − in Equations 34 we have that α h − = ν ( x q ) = ǫ . Since ǫ < γ ≤ ν ( y ) we see that χ < h . In particular, if we are in the case of a dominantcritical segment, we are in one of the winning situations C. Assume that χ = h − η ′ h − = a h − ( x, w ) x∂/∂x + b h − ( x, w ) ∂/∂w + x q + p c h − ( x, w ) y∂/∂y and we are assuming moreover that η ′ h − is prepared. If this level h − h − η ′ h − we deduce that α h − = ν ( x q + p ) = ǫ + ν ( x p ). But weknow that ν ( x p ) > ν ( y ) and thus the level h − (cid:3) Tchirnhausen preparation. Dominant case.
In this Subsection we as-sume we are in case A of Subsection 6.9. That is, we have ξ completely preparedwith respect to A , the critical segment is dominant with χ = h and the criticalpolynomial is not Tchirnhausen. We also assume that h ≥ h ≤ Proposition 18.
We can perform a coordinate change y ∗ = y − x p g ( x, w ) , with ν ( x p ) = ν ( y ) to obtain A ∗ is such a way that after performing finitely many etale w -Puiseux packages, we get A ′ such that ξ is completely prepared with respect to A ′ with h = ~ ( ξ ; A ′ ) and we are in one of the situations B or C of 6.9.Proof. Since the critical polynomial is not Tchirnhaus, we have that ν ( y ) = ν ( x p )for some p ∈ Z ≥ . Up to multiply ξ by a power of x , let us assume without loss ofgenerality that α ( ξ ; x, w, y ) = 0. Denote F = ξ ( w ) = P ∞ s =0 y s F s ( x, w ). We knowthat F h (0 , = 0. Moreover, in view of our hypothesis, we have F h − ( x, w ) = x p G h − ( x, w ), where G h − (0 , = 0. By an argument like in Proposition 16, wecan find a coordinate change of the form y ∗ = y − x p g ( x, w ) such that F ∗ ( x, w, y ∗ ) = F ( x, w, y ∗ + x p g ( x, w )) = ∞ X s =0 y ∗ s F ∗ s ( x, w )satisfies that F ∗ h − = 0. This condition eliminates the level h − w -Puiseux packages. (cid:3) Maximal contact
In this section we prove Theorem 4. Recall that we consider the case when n = 3and ν is a valuation of arquimedean rank one with κ ν = k . We have a projectivemodel M of K , where P is the center of ν at M . We assume that P is a regularpoint of M and there is ˆ f ∈ b O M ,P that has transversal maximal contact with ν . The rational rank r can be supposed to be r = 1 or r = 2, since if r = 3 thedefinition of transversal maximal contact makes no-sense (see the Introduction).The computations in this section are essentially contained in the paper [5], butwe include them for the sake of completeness. Maximal contact with rational rank two.
Take a regular system of pa-rameters ( x , x , y ) of O M ,P such that ν ( x ) , ν ( x ) are Z -linearly independentand ˆ f = y + X i,j λ ij x i x j . Since ν is arquimedian, we may write ˆ f as the Krull limit ˆ f = lim µ →∞ f µ , where f µ = y + X iν ( x )+ jν ( x ) ≤ µ λ ij x i x j ∈ O M ,P . Note that ν ( f µ ) > µ and, more precisely we have ν ( f µ ) = min { ν ( x i x j ); λ ij = 0 , ν ( x i x j ) > µ } . In this paragraph we denote Y = { P } , Y = { x = ˆ f = 0 } and Y = { x = ˆ f = 0 } .The next Lemma 15 may be proved by standard computations in terms of blow-ups and valuations and we leave the verification to the reader: Lemma 15.
Let π : M ′ → M be the blow-up of M with one of the centers Y , Y or Y and assume that if we use Y , respectively Y , as a center, then ˆ f (0 , x , y ) ∈ O M ,P , respectively ˆ f ( x , , y ) ∈ O M ,P . Let P ′ ∈ M ′ be the centerof ν at M ′ . Then P ′ belongs to the strict transform of ˆ f = 0 . More precisely , wehave the following cases: T-01:
The center is Y and µ = ν ( x ) < ν ( x ) . In this case P is in thestrict transform of the formal curve x = ˆ f = 0 and there is a regularsystem of parameters ( x ′ , x ′ , y ∗ ) at O M ′ ,P ′ such that x ′ = x , x ′ = x /x , y ∗ = ˆ f µ /x . Moreover ˆ f ′ = ˆ f /x = y ∗ + X iν ( x )+ jν ( x ) >µ λ ij x ′ i + j − x ′ j ∈ O M ′ ,P ′ has transversal maximal contact with ν . T-02:
The center is Y and µ = ν ( x ) < ν ( x ) . Similar to T-01. T-1:
The center is Y , where µ = ν ( x ) . In this case P is the only pointover P in the strict transform of ˆ f = 0 and there is a regular systemof parameters ( x ′ , x ′ , y ∗ ) at O M ′ ,P ′ such that x ′ = x , x ′ = x , y ∗ =( f µ + ˆ h µ (0 , x )) /x , where ˆ h µ ( x , x ) = ˆ f − f µ . Moreover ˆ f ′ = ˆ f /x = y ∗ + X ν ( x i x j ) >µ ; i ≥ λ ij x ′ i − x ′ j ∈ O M ′ ,P ′ has transversal maximal contact with ν . T-2:
The center is Y , where µ = ν ( x ) . Similar to T-1. Take a generator ξ of L M ,P [log x x ]. Define the formal vector field ˆ ξ to beˆ ξ = ξ if ˆ f divides ξ ( ˆ f ) (this corresponds to saying that ˆ f = 0 defines a formalinvariant hypersurface) and ˆ ξ = ˆ f ξ if ˆ f does not divide ξ ( ˆ f ). Let us writeˆ ξ = ˆ a x ∂∂x + ˆ a x ∂∂x + ˆ b ˆ f ∂∂ ˆ f . EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 33
Note that ˆ a , ˆ a , ˆ b have no common factors. The adapted (or logarithmic) order of L at P with respect to x x ˆ f isLogOrd( L , O M ,P ; x x ˆ f ) = ord c M M ,P (ˆ a , ˆ a , ˆ b ) ∈ Z ≥ , where ord c M M ,P ( − ) means the c M M ,P -adic order (see also [4]).Put ζ = LogOrd( L , O M ,P ; x x ˆ f ). We say that Y is permissible for L adaptedto x x if the two following properties hold:(1) ˆ f (0 , x , y ) ∈ O M ,P . (Hence Y is a subvariety of M )(2) ord ( x , ˆ f ) (ˆ a , ˆ a , ˆ b ) = ζ .We give a symmetric definition for Y being permissible. By definition Y is alwayspermissible. Remark 22. If ζ ≥
2, the condition 2 above implies condition 1, since in this case,the curve Y must be contained in the locus ξ ( x ) /x = ξ ( x ) /x = ξ ( y ) = 0 . If ζ = 1 and ˆ ξ = ξ , the same argument holds. Lemma 16.
Let π : M ′ → M be the blow-up of M with a permissible center Y , Y or Y . Let P ′ ∈ M ′ be the center of ν at M ′ . Then LogOrd( L , O M ,P ; x x ˆ f ) ≥ LogOrd( L , O M ′ ,P ′ ; x ′ x ′ ˆ f ′ ) . Proof.
We may assume that either the center of the blow-up is Y or it is Y and ν ( x ) < ν ( x ) (the other cases follow from these by interchanging the roles of x , x ). Then we have ˆ f ′ = ˆ f /x and ˆ ξ ′ = x − ζ ˆ ξ whereˆ a ′ = x − ζ ˆ a ; ˆ b ′ = x − ζ (ˆ b − ˆ a ′ ) , and ˆ a ′ = x − ζ (ˆ a − ˆ a ) if Y , ˆ a ′ = x − ζ ˆ a if Y . The rest of the proof is given by thestandard results on the blow-up of equimultiple centers. (cid:3) We proceed by induction on ζ . First, consider the case ζ ≥ g = P s ˆ f s g ijs x i x j ∈ b O M ,P = k [[ x , x , ˆ f ]] and an integer η ∈ Z > .The Hironaka’s characteristic polygon ∆(ˆ g ; x , x , ˆ f ; η ) is the positive convex hullin R ≥ of the points of the form ( i/ ( η − s ) , j/ ( η − s )), where g ijs = 0 and s < η .Given a list { ˆ g l } we define ∆( { ˆ g l } ; x , x , ˆ f ; η ) to be the convex hull of the unionof the ∆(ˆ g l ; x , x , ˆ f ; η ). Now, we define∆( L ; x , x , ˆ f ; η ) = ∆( { ˆ a , ˆ a , ˆ b } ; x , x , ˆ f ; η ) . Let us list the properties of ∆ η = ∆( L ; x , x , ˆ f ; η ), similar to those used by Hiron-aka in his Bowdoin College Memoir [8]:(1) ∆ η = ∅ . Otherwise ˆ a , ˆ a , ˆ b would be divisible by ˆ f .(2) ∆ η ⊂ { ( u, v ); u + v ≥ } iff ζ ≥ η .(3) ∆ ζ ⊂ { ( u, v ); u ≥ } iff condition 2 of permissibility holds for Y .(4) ∆ ζ ⊂ { ( u, v ); v ≥ } iff condition 2 of permissibility holds for Y . The characteristic polygon behaves under blow-up as in the classical case of vari-eties, as we show in the next Lemma 17. To see this, let us introduce the linearmappings σ , σ , σ , σ defined as follows σ ( u, v ) = ( u + v − , v ) , σ ( u, v ) = ( u − , v ) ,σ ( u, v ) = ( u, u + v − , σ ( u, v ) = ( u, v − . Lemma 17.
Keep notations as in Lemma 15. Let π : M ′ → M be the blow-up of M with a permissible center Y , Y or Y . Let P ′ ∈ M ′ be the center of ν at M ′ .Put ∆ = ∆( L ; ( x , x , ˆ f ); ζ ) . Then the characteristic polygon ∆( L ; ( x ′ , x ′ , ˆ f ′ ); ζ ) isthe positive convex hull of σ (∆) , σ (∆) , σ (∆) , σ (∆) if we are respectively in the cases T-01 , T-02 , T-1 and
T-2 of Lemma 15.Proof.
Let I ⊂ b O M ,P be the ideal generated by ˆ a , ˆ a , ˆ f . Then the ideal I ′ ⊂ b O ′ M ,P generated by ˆ a ′ , ˆ a ′ , ˆ f ′ is I ′ = x − ζ I , respectively I ′ = y − ζ I if we are inthe cases (01) , (1), respectively (02) , (2). Now we apply the classical remarks ofHironaka in his Bowdoin College seminar [8]. (cid:3) Now, we choose the following strategy to blow up. We select the blow-up center Y until the characteristic polygon has only one vertex, this occurs after finitelymany steps. Then, since we are in the case ζ ≥
2, at least one of the centers Y , Y is permissible, since it is equimultiple. Blow-up this curve. After finitely manyoperations the characteristic polygon intersects { ( u, v ); u + v < } and hence thelogarithmic order drops. We arrive in this way to the case ζ ≤ ζ ≤
1. If ζ = 0 and ˆ ξ = ξ , we get an elementary singularityand if ˆ ξ = ˆ f ξ the foliation is in fact non-singular. Assume that ζ = 1. By Remark22, the case ˆ ξ = ξ can be handled as before. So we consider only the case ˆ ξ = ˆ f ξ .Blowing-up the origin (that is, we take the center Y each time), we get as abovethat the characteristic polygon has exactly one vertex of integer coordinates, say( α, β ) ∈ Z ≥ , where α + β ≥
1. Assume that α + β ≥
2; since ζ = 1, we have eitherord x ,x , ˆ f ( ξ ( x ) , ξ ( x )) = 0 or ˆ b (0 , , ˆ f ) = ˆ f U ( ˆ f ), with U (0) = 0. In both cases ξ is non-nilpotent and we obtain an elementary singularity. It remains to study thecase α + β = 1. We have two possibilities ( α, β ) = (1 ,
0) and ( α, β ) = (0 , α, β ) = (1 , ν ( x ) > ν ( x ), we are done by blowing-up the origin, since we get ζ = 0; if ν ( x ) < ν ( x ) the situation repeats itself, but this cannot occur infinitely manytimes, since we are dealing with an arquimedian valuation ν .This ends the proof of Theorem 4 in the case of rational rank r = 2.7.2. Maximal contact with rational rank one.
Take a regular system of pa-rameters ( x, w, y ) of O M ,P such thatˆ f = y + X i,j λ ij x i w j . In this paragraph we denote Y = { P } and Y = { x = ˆ f = 0 } . The next Lemma18 may be proved by standard computations in terms of blow-ups and valuationsand we left the verification to the reader: EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 35
Lemma 18.
Let π : M ′ → M be the blow-up of M with one of the centers Y or Y and assume that if we use Y as a center, then ˆ f (0 , w, y ) ∈ O M ,P . Then, thecenter P ′ ∈ M ′ of ν at M ′ belongs to the strict transform of ˆ f = 0 . More precisely,we have the following cases: T : The center is Y and µ = ν ( x ) < ν ( w ) . In this case P is in the stricttransform of the formal curve w = ˆ f = 0 and there is a regular system ofparameters ( x ′ , w ′ , y ∗ ) of O M ′ ,P ′ such that x ′ = x , w ′ = w/x , ˆ f ′ = ˆ f /x hastransversal maximal contact with ν and has the form ˆ f ′ = y ∗ + X i,j λ ′ ij x ′ i w ′ j .T : The center is Y and µ = ν ( w ) < ν ( x ) . Similar to the previous case, bythe roles of x, w interchanged. T , c : The center is Y and µ = ν ( x ) = ν ( w ) . Take a parameter c ∈ k suchthat ν ( w − cx ) > ν ( x ) . We do the coordinate change w ∗ = w − cx and weproceed as in the case (01) . T : The center is Y . In this case P is in the strict transform of ˆ f = 0 and there is a regular system of parameters ( x ′ , w ′ , y ∗ ) of O M ′ ,P ′ such that x ′ = x , w ′ = w , ˆ f ′ = ˆ f /x has transversal maximal contact with ν and it iswritten as ˆ f ′ = y ∗ + X i,j λ ′ ij x ′ i w ′ j . We define an { x, w, y, ˆ f } - formal Puiseux package to be a sequence of blow-ups M ← M ← · · · ← M N = M ′ such that:(1) Each blow-up has center at the center P i ∈ M i of the valuation in theprojective model M i .(2) We get ( x i , w i , y i , ˆ f i ) at each P i , obtained as in Lemma 18, starting from( x , w , y , ˆ f ) = ( x, w, y, ˆ f ).(3) Each blow-up is given by T or T , except the last blow-up that is givenby ( T , c ), with c = 0.A { x, w, y, ˆ f } -formal Puiseux package exists and is unique. More precisely, if weput ν ( w d /x p ) = 0 and c is such that ν ( w d /x p − c ) >
0, the sequence of blow-ups isthe reduction of singularities of the formal curve w d − cx p = ˆ f = 0.Now, let us consider a generator ξ of L M ,P [log x ], that we write a follows: ξ = a ( x, w, ˆ f ) x ∂∂x + b ( x, w, ˆ f ) ∂∂w + ˆ h ( x, w, ˆ f ) ∂∂ ˆ f , where a = ξ ( x ) /x , b = ξ ( w ), ˆ h = ξ ( ˆ f ). Note that a, b and ˆ h have no commonfactors. There are two cases that we will consider separately(1) The formal hypersurface ˆ f = 0 is invariant by ξ . Then ˆ f divides ˆ h and wecan put ˆ h = ˆ g ˆ f .(2) The formal hypersurface ˆ f = 0 is not invariant by ξ . Then ˆ f does notdivide ˆ h and thus ˆ f a, ˆ f b, ˆ h have no common factors. Let us put ˆ ξ = ˆ ξ if ˆ f = 0 is invariant and ˆ ξ = ˆ f ξ if ˆ f is not invariant. In bothcases we denote ˆ a = ˆ ξ ( x ) /x ; ˆ b = ˆ ξ ( w ); ˆ g = ˆ ξ ( ˆ f ) / ˆ f . Then ˆ a , ˆ b , ˆ g have no common factors. Define the logarithmic order asLogOrd( L ; x ˆ f ) = ord c M (ˆ a , ˆ b , ˆ g ) . Lemma 19.
Let π : M ′ → M be given by the { x, w, y, ˆ f } -formal Puiseux packageand let ( x ′ , w ′ , y ′ , ˆ f ′ ) be the resulting list at the center P ′ of ν in M ′ . Then LogOrd( L ; x ′ ˆ f ′ ) ≤ LogOrd( L ; x ˆ f ) . Proof.
The result is true under each of the blow-ups of the sequence given by the { x, w, y, ˆ f } -formal Puiseux package. This is a standard verification which is also apart of the proof of the vertical stability of the adapted order given in [4]. (cid:3) Consider an expansion ˆ ξ = P s ≥ ˆ f s ˆ η s ( x, w ), whereˆ η s ( x, w ) = ˆ a s ( x, w ) x ∂∂x + ˆ b s ( x, w ) ∂∂w + ˆ g s ( x, w ) ˆ f ∂∂ ˆ f . We say that ˆ η s is formally strongly prepared if we can write(41) ˆ η s = x ρ ˆ U ( x, w ) θ + x τ ˆ V ( x, w ) ˆ f ∂∂ ˆ f ; θ = x ˆ h ( x, w ) x ∂∂x + ∂∂w satisfying the same properties as in Definition 5, that is(1) ρ, τ ∈ Z ∪ { + ∞} , with ρ = τ .(2) ˆ U = λ + x ( · · · ) and ˆ V = µ + x ( · · · ), where λ, µ ∈ k \{ } . (Except if ρ = + ∞ or τ = ∞ , that indicates that ˆ U , respectively ˆ V is identically zero)By the same proof as in 10, we have Proposition 19.
Assume that ˆ η s = 0 , then after finitely many formal Puiseuxpackages we obtain ˆ η s that is formally strongly prepared. Let us work by induction on ̺ = LogOrd( L ; x ˆ f ). If ̺ ≤ ̺ ≥
2. By Proposition 19, after finitely manyformal Puiseux packages, the vector field ˆ ξ can be written as ˆ ξ = P ≤ s ˆ f s ˆ η ′ s ( x, w ),where ˆ η s = x ρ s ˆ U s ( x, w ) θ s + x τ s ˆ V s ( x, w ) ˆ f ∂∂ ˆ f ; θ s = x ˆ h s ( x, w ) x ∂∂x + ∂∂w is formally strongly prepared for any s ≤ ̺ . Let us put m s = min { ρ s , τ s } . let usalso define δ = min (cid:26) m s ̺ − s ; s < ̺ (cid:27) . It is clear that 1 ≤ δ < ∞ , since the adapted order is ̺ .Consider the ideal ( x, ˆ f ). Since ̺ ≥
2, this ideal gives a curve in the singularlocus of L . Thus we can blow-up it. After blowing-up, we get that ̺ ′ ≤ ̺ and δ ′ = δ − ̺ ′ = ̺ . This ends the proof of Theorem 4. EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 37
Part Higher rank and higher dimensional valuations
In this part we complete the proof of Theorem 1 by considering valuations ofhigher arquimedean rank or of dimension bigger than zero. In fact these casescorrespond to situations simpler than in Part 1, since they are “essentially” ofambient dimension two. 8.
Higher rank valuations
In this section we assume that n = 3 and κ ν = k but ν has rank bigger than one,that is, the value group Γ is not arquimedean. If the rational rank r = 3, there isno difference with the computations in the case of an arquimedean valuation donein Section 2. The only remaining situation is r = 2. Let us consider this situation.We can work in terms of parameterized regular local models A = ( O , z = ( x , y ))as in the case of a real valuation of rational rank two (Sections 3-4). Let us considerthe following statement TRI: Trivial ramification index assumption:
After perform-ing any finite sequence of y -Puiseux packages, coordinate blow-ups in the independent variables and coordinate changes in thedependent variable, we obtain A = ( O , z = ( x , y )) such that theramification index is equal to one. That is ν ( y ) = ν ( x p x p ) for( p , p ) ∈ Z .Following the same arguments as in Sections 3-4 we obtain Proposition 20.
Assume that the
Trivial Ramification Index Assumption does nothold after performing any finite sequence of y -Puiseux packages, coordinate blow-ups in the independent variables and coordinate changes in the dependent variable.Then we can obtain a log-elementary L A after performing such a finite sequence oftransformations. Thus, we assume that TRI holds. We can work by induction on the main height h = ~ ( ξ ; x , y ) of a generator of L . By the same arguments in Sections 3-4, if thecritical polynomial is not Thchirnhaus, we can win. So, we find an element at thelevel h − p , p ) ∈ Z ≥ ,since this point of the support is associated to a monomial x p x p appearing in thecoefficients of ξ . Now, we can do the coordinate change y = y − c x p x p ; ν ( y ) > ν ( y ) . The situation repeats. We obtain a formal element ˆ f = y − P c i x p i x p i . Now wecan apply to ˆ f the same arguments as in Subsection 7.1.9. Higher dimensional valuations
In this section we assume that n = 3 and κ ν = k . We look for a projectivemodel M of K and a birational morphism M → M such that the center Y of ν at M has dimension ≥ Y is a regular point of M whichis log-elementary for L . Since k is algebraically closed, the assumption κ ν = k implies that dim ν ≥
1, where dim ν is the transcendence degree of κ ν /k . ApplyingHironaka’s reduction of singularities to M , we may assume that all the points in M are nonsingular. Also by classical results on reduction of singularities, we obtainthe following statement: Lemma 20.
There is a birational morphism M → M such that the center Y of ν at M has dimension equal to dim ν .Proof. See for instance Vaqui´e’s paper [17]. (cid:3)
Thus we may assume that M is non-singular and the center Y of ν at M hasdimension equal to dim ν . If dim ν = 2, then Y is a hypersurface and a genericpoint of Y is always nonsingular for L , since the singular locus of L has codimensionat least 2 in any nonsingular ambient space.Consider the case dim ν = dim Y = 1. We blow-up M with center Y to get M → M . The new center Y of ν at M is a curve that applies surjectively over Y . We repeat the procedure to get an infinite sequence M ← M ← M ← · · · where the center Y i of ν at M i is a curve that applies surjectively over Y i − . In thissituation we can apply the equireduction arguments in [5], (see also [13]) to obtainan elementary L at a generic point of Y i for i >>
0. These arguments are actuallyof two-dimensional nature and the invoked equireduction results are very similar tothe original Seidenberg’s result in [14].
Part Globalization
In this Part 3 we prove the global result stated in Theorem 2. To do this we willapply the axiomatic version of the Zariski’s Patching of Local Uniformizations [19]that has been developed by O. Piltant in [12].Let us state the axiomatic version of the patching of local uniformizations thatwe need to use. Fix a field of rational functions
K/k of transcendence degree threeover k . We take k an algebraically closed field of characteristic zero, even if Piltant’sresult is more general than that. Assume that we have an assignation M RegP( M ) ⊂ M that chooses a nonempty Zariski open subset RegP( M ) ⊂ M for each projectivemodel M of K . This map can be thought of by saying that RegP( M ) is the set ofpoints of M that satisfy the property “P” . Let us introduce now a list of axioms forglobalization. Axiom I . For each projective model M of K the set RegP( M ) isa nonempty Zariski open set contained in the set of regular pointsReg( M ) of M . Moreover the definition of RegP( M ) is local in thesense that given two projective models M and M ′ , two Zariski opensets U ⊂ M and U ′ ⊂ M ′ and an isomorphism φ : U → U ′ , then φ (RegP( M ) ∩ U ) = RegP( M ′ ) ∩ U ′ .The next axiom says that RegP( M ) has a good behavior under blow-up. Axiom II . Let Y ⊂ M be an irreducible algebraic subvariety of M such that Y ∩ RegP( M ) = ∅ . Let π : M ′ → M be the blow-upwith center Y . There is a nonempty Zariski open subset V Y of Y ∩ RegP( M ) defined by the property that π − ( V Y ) ⊂ RegP( M ′ ).In what follows we take V Y to be the largest possible between the subsets V ⊂ Y ∩ RegP( M ) such that π − ( V ) ⊂ RegP( M ′ ). This determines V Y uniquely. Asa consequence of Axiom II, if we blow-up a point P ∈ RegP( M ) then we have π − ( P ) ⊂ RegP( M ′ ). The open set V Y is called set of permissibility for Y . We say EDUCTION OF SINGULARITIES OF THREE-DIMENSIONAL LINE FOLIATIONS 39 that Y is permissible if V Y = Y . We need also the notion of strong permissibility .If Y ⊂ M is a point or a hypersurface that cuts RegP( M ), we define the open set ofstrong permissibility W Y as W Y = V Y . Assume that Y is an irreducible curve andlet P ∈ V Y . We say that Y is strongly permissible at P iff the following propertyholds Let M = M ← M ← M ← · · · ← M N be a finite sequence ofblow-ups centered at points P i ∈ M i , such that P = P and P i projects over P i − and is in the strict transform Y i of Y . Then Y N is permissible at P N (that is P N ∈ V Y N ).We denote by W Y ⊂ V Y the set of points where Y is strongly permissible and wesay that Y is strongly permissible iff W Y = Y . Axiom III . Let Y be a curve in M such that Y ∩ RegP( M ) = ∅ .There is a finite sequence M = M ← M ← M ← · · · ← M N = M ′ of blow-ups with center in closed points such that the strict trans-form Y ′ of Y is strongly permissible.In fact, the centers in Axiom III can be chosen in Y − W Y at each step; this alsoshows that W Y is a nonempty open set of Y .We also need another axiom (of principalization), that can be seen as a resulton conditionated desingularizationAxiom IV [Principalization] . Given a (normal) projective model M of K and an ideal sheaf I ⊂ O M , there is a projective birationalmorphism π : M → M such that(1) IO M is locally principal in π − (RegP( M )).(2) π − (RegP( M )) ⊂ RegP( M ).(3) The induced map π − (RegP( M ) ∩ U ) → RegP( M ) ∩ U is anisomorphism where U is the open set of the points p of M such that I p is principal.The last axiom states the existence of Local Uniformization. Axiom V [Local Uniformization] . Let ν be a k -valuation of K .There is a projective model M of K such that the center Y of ν in M cuts RegP( M ), that is Y ∩ RegP( M ) = ∅ .With this axioms, it is possible to reproduce Zariski’s arguments in [20] for thepatching of local uniformizations and we can state the following result Theorem 5 (Piltant) . Assume that the assignation M RegP ( M ) satisfies to theaxioms I,II,III, IV and V above. Consider a projective model M of K . Then thereis a birational projective morphism M → M such that RegP ( M ) = M . This statement is slightly more restrictive that the result proved by Piltant in[12]. It is the result we need to get our global statement for the case of foliations.Now, in order to prove Theorem 2, we just need to prove the following statement
Proposition 21.
Let us consider a foliation
L ⊂
Der k K . The assignation M RegLog L ( M ) = { P ∈ M ; P is log-elementary } satisfies to the axioms I,II,III,IV and V. The Local Uniformization Axiom is given by Theorem 1. The first axiom isevident from the local definition of log-elementary points, let us just point thatRegLog L ( M ) is non-empty since the non singular points of L in Reg( M ) are in thecomplement of a closed subset of codimension bigger or equal than two.The axioms II and III come from the general computations done in [4] concerningthe definition and properties of permissible centers in terms of the adapted multi-plicity. More precisely, in theorem 3.1.4. of [4] is proved the stability of the adaptedorder under blow-up (the log-elementary singularities are defined to have adaptedorder less or equal to one). The permissibilyzing and permissibility properties comefrom the results on stationary sequences in the section 3.3. of [4].Finally axiom IV of principalization has been explicitly proved for the caseRegP( M ) = RegLog L ( M ) by Piltant in [12], Proposition 4.2.Now, Theorem 2 is a consequence of Proposition 21 and Theorem 5. References [1] Artin, M.:
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