Reduction of the singularities of codimension one singular foliations in dimension three

Cano, Felipe

doi:10.4007/annals.2004.160.907

Cited by 71 publications

(127 citation statements)

References 18 publications

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“…If P is pre-simple for (F, D) then it is also pre-simple for (F, D nd ). This follows directly from equation (3). In fact, if…”

Section: Definitionmentioning

confidence: 68%

“…In case II of Definition 4, the trivializing vector fields may be obtained thanks to the integrability condition. Details can be found in Lemma 5 of [3].…”

Section: Propositionmentioning

confidence: 99%

“…Contents of this section can be essentially found in [3]. Let M be a germ around a non singular compact analytic subset of a complex analytic variety of dimension n. Given a point P ∈ M we denote by O M,P the ring of analytic functions at P and by M M,P its maximal ideal.…”

Section: Preliminariesmentioning

confidence: 99%

“…In [4] and [3] the authors have identified two kinds of points for codimension one singular foliations in higher dimensional ambient spaces. They are the pre-simple points and the simple ones.…”

Section: Introductionmentioning

confidence: 99%

“…

Abstract: The problem of reduction of singularities for germs of codimension one foliations in dimension three has been solved by Cano in [3]. The author divides the proof in two steps.

…”

mentioning

confidence: 99%

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Elimination of resonances in codimension one foliations

Duque

2015

Publicacions Matemàtiques

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The problem of reduction of singularities for germs of codimension one foliations in dimension three has been solved by Cano in [3]. The author divides the proof in two steps. The first one consists in getting pre-simple points and the second one is the passage from pre-simple to simple points. In arbitrary dimension of the ambient space the problem is open. In this paper we solve the second step of the problem.

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“…If P is pre-simple for (F, D) then it is also pre-simple for (F, D nd ). This follows directly from equation (3). In fact, if…”

Section: Definitionmentioning

confidence: 68%

“…In case II of Definition 4, the trivializing vector fields may be obtained thanks to the integrability condition. Details can be found in Lemma 5 of [3].…”

Section: Propositionmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

Abstract: The problem of reduction of singularities for germs of codimension one foliations in dimension three has been solved by Cano in [3]. The author divides the proof in two steps.

…”

mentioning

confidence: 99%

See 3 more Smart Citations

Elimination of resonances in codimension one foliations

Duque

2015

Publicacions Matemàtiques

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Toward classification of codimension 1 foliations on threefolds of general type

Aleksei

2023

Mathematische Nachrichten

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The aim of this paper is to classify codimension 1 foliations ℱ with canonical singularities and 𝜈(𝐾 ℱ ) < 3 on threefolds of general type. I prove a classification result for foliations satisfying these conditions and having nontrivial algebraic part. We also describe purely transcendental foliations ℱ with the canonical class 𝐾 ℱ being not big on manifolds of general type in any dimension, assuming that ℱ is nonsingular in codimension 2.

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Handbook of Geometry and Topology of Singularities I

2020

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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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Reduction of the singularities of codimension one singular foliations in dimension three

Cited by 71 publications

References 18 publications

Elimination of resonances in codimension one foliations

Elimination of resonances in codimension one foliations

Toward classification of codimension 1 foliations on threefolds of general type

Handbook of Geometry and Topology of Singularities I

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