Elimination of resonances in codimension one foliations

Duque, Miguel Fernández

doi:10.5565/publmat_59115_05

Cited by 9 publications

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“…We can find proofs of this result in another contexts in [10] and [11]. Anyway, we provide a complete proof next.…”

Section: Statementsmentioning

confidence: 84%

Combinatorial aspects of classical resolution of singularities

Molina-Samper

2018

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We describe combinatorial aspects of classical resolution of singularities that are free of characteristic and can be applied to singular foliations and vector fields as well as to functions and varieties. In particular, we give a combinatorial version of Hironaka's maximal contact theory in terms of characteristic polyhedra systems and we show the global existence of maximal contact in this context.Moreover, if there is an equivalence φ between two support fabrics F 1 and F 2 , then for every J ∈ H 1 , the blow-ups π J (F 1 ) and π φ(J) (F 2 ) are also equivalent.There is a surjective map π # J : H ′ → H between the strata sets of π J (F ) and F respectively, given by π # J (H ′K ∞ ) = {K}, for each K ∈ {J} ∩ H and by π #Proposition 1. The map π # J : H ′ → H is continuous.Proof. Remark that for every J ′ ∈ H ′ , we have π # J P(J ′ ) ⊂ P π # J (J ′ ) . Corollary 1. Given an open setwhere M is a complex analytic variety and E is a strong normal crossings divisor of M . That is, E is the union of a finite family {E i } i∈I of irreducible smooth hypersurfaces E i , where we fix an order in the index set I and the next properties hold:Let us consider a polyhedra system D = (F ; {∆ J } J∈H , d) and a stratum T ∈ H. We introduce now a new polyhedra system D T , using Hironaka's projection of D from T , that plays an important role in Section 7.3.Let F T = (I T , H T ) be the support fabric obtained by projection of F from T . Given J * ∈ H T , let us take the stratum J = J * ∪ T ∈ H and let us consider the subset4 Definition 2. The total transform Λ 0 J (D) of a polyhedra system D = (F ; {∆ J } J∈H , d) centered in a stratum J ∈ H is the polyhedra system Λ 0The characteristic transform Λ J (D) is a polyhedra system again, because the center of the blow-up is singular and then δ(∆ 0 {∞} ) ≥ 1. Let N = (F ; {N J } J∈H , 1) be a Newton polyhedra system and take an integer number d ≥ 1. Consider the polyhedra system D = N /d and a singular stratum J ∈ Sing(D). The d-moderatedRemark 4. If there is an equivalence φ between two polyhedra systems D 1 and D 2 , then for all J ∈ Sing(D 1 ), the characteristic transforms Λ J (D 1 ) and Λ φ(J) (D 2 ) are also equivalent. Reduction of Singularities.In this section we prove Theorem 2 for Hironaka quasi-ordinary polyhedra systems.Given a Hironaka quasi-ordinary system D = (F ; {∆ J } J∈H , d) and a singular stratum J,Lemma 1. The set A of decreasing sequences of natural numbers is well-ordered for the lexicographical order.Proof. Given a decreasing sequence ϕ 1 ≥ ϕ 2 ≥ · · · ≥ ϕ k ≥ · · · in A, we take k 0 = 0 andLet us consider the decreasing sequence ϕ : N → N given by ϕ(n) = ϕ kn (n). There is a positive number l ∈ N such that ϕ(n) = ϕ(l), for all n ≥ l. As a consequence we have ϕ j = ϕ k l , for all j ≥ k l .Proposition 3. Every Hironaka quasi-ordinary polyhedra system has reduction of singularities.Proof. Choose a bijective map s D : {1, 2, . . . , #H} → H such thatTake the decreasing sequence ϕ : N → N given by ϕ(j) = dδ(∆ sD (j) ) if j ≤ #H and ϕ(j) = 0 if j > #H. We choose ...

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“…We can find proofs of this result in another contexts in [10] and [11]. Anyway, we provide a complete proof next.…”

Section: Statementsmentioning

confidence: 84%

Combinatorial aspects of classical resolution of singularities

Molina-Samper

2018

RACSAM

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“…More precisely, we show how to obtain pre-simple points as they were defined in the Introduction. In view of the results in [19], this is enough, since we can pass from pre-simple to simple points.…”

Section: Definition 97 Consider a Pair (A F) Where F ∈mentioning

confidence: 99%

“…The definition of a simple formal integrable differential 1-form ω is compatible with the above one given for functions, in the sense that that if f is not a unit and it is simple, then d f will be simple. The precise definition, the existence of normal formal forms and other properties may be found in [5][6][7]19], where "simple points" are "pre-simple points" with a diophantine additional condition, that is automatically satisfied in the case of functions.…”

Section: Introductionmentioning

confidence: 99%

“…By a paper of Fernández-Duque [19], it is possible to get only simple singularities when we start with pre-simple ones in any dimension; in other words, we can globally obtain the diophantine condition that makes the difference between pre-simples and simples. This means that it is only necessary to obtain pre-simple points in Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

“…By Proposition 5.2, we obtain ν A ( α1 ) > 0 by a suitable -nested transformation.Recall that the -nested transformations are a degenerate type of normalized coordinate changes, see Remark 32. Lemma 9 19. Assume that α 1 = α * 1 + α1 , where dα * 1 = 0 and ν A ( α1 ) > 0.…”

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Truncated Local Uniformization of Formal Integrable Differential Forms

Cano

Fernández-Duque

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Qual. Theory Dyn. Syst.

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We prove the existence of Local Uniformization for rational codimension one foliations along rational rank one valuations, in any ambient dimension. This result is consequence of the Truncated Local Uniformization of integrable formal differential 1-forms, that we also state and prove in the paper. Thanks to the truncated approach, we perform a classical inductive procedure, based both in the control of the Newton Polygon and in the possibility of avoiding accumulations of values, given by the existence of suitable Tschirnhausen transformations.

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Resolution of Singularities: An Introduction

Spivakovsky

2020

Handbook of Geometry and Topology of Singularities I

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The problem of resolution of singularities and its solution in various contexts can be traced back to I. Newton and B. Riemann. This paper is an attempt to give a survey of the subject starting with Newton till the modern times, as well as to discuss some of the main open problems that remain to be solved. The main topics covered are the early days of resolution (fields of characteristic zero and dimension up to three), Zariski's approach via valuations, Hironaka's celebrated result in characteristic zero and all dimensions and its subsequent strenthenings and simplifications, existing resutls in positive characteristic (mostly up to dimension three), de Jong's approach via semi-stable reduction, Nash and higher Nash blowing up, as well as reduction of singuarities of vector fields and foliations. In many places, we have tried to summarize the main ideas of proofs of various results without getting too much into technical details.

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Elimination of resonances in codimension one foliations

Cited by 9 publications

References 10 publications

Combinatorial aspects of classical resolution of singularities

Combinatorial aspects of classical resolution of singularities

Truncated Local Uniformization of Formal Integrable Differential Forms

Resolution of Singularities: An Introduction

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