Limites d’espaces tangents en géométrie analytique

Tráng, Dũng Lê; Teissier, Bernard

doi:10.1007/bf02566778

Cited by 42 publications

(3 citation statements)

References 19 publications

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“…Proof. -The first step is to prove that (X, 0) does not have exceptional cones, however by [16,Prop. 2.1.4.2,p.…”

Section: For Every J We Have Thatmentioning

confidence: 99%

“…The space (X, 0) → (C, 0) has been used to study Whitney conditions in [10], and to study the structure of the set of limits of tangent spaces in [16] and [15]. In [16], the authors prove the existence of a finite family {V α } of subcones of the reduced tangent cone |C X,0 | that determines the set of limits of tangent spaces to X at 0.…”

Section: Introductionmentioning

confidence: 99%

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Specialization to the tangent cone and Whitney equisingularity

Flores¹

2013

Bul. Soc. Math. France

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Let (X, 0) be a reduced, equidimensional germ of an analytic singularity with reduced tangent cone (C X,0 , 0). We prove that the absence of exceptional cones is a necessary and sufficient condition for the smooth part X 0 of the specialization to the tangent cone ϕ : X → C to satisfy Whitney's conditions along the parameter axis Y. This result is a first step in generalizing to higher dimensions Lê and Teissier's result for hypersurfaces of C 3 which establishes the Whitney equisingularity of X and its tangent cone under these conditions. Résumé (Spécialisation sur le cône tangent et équisingularité à la Whitney) Soit (X, 0) un germe de singularité analytique complexe, réduit et équidimensionel tel que son cône tangent (C X,0 , 0) est réduit. On montre que l'absence des cônes exceptionnels est une condition nécessaire et suffisante pour que la partie lisse X 0 de la spécialisation sur le cône tangent ϕ : X → C satisfasse les conditions de Whitney le long l'axe des paramètres Y. Ce résultat est un premier pas vers la généralisation aux dimensions supérieures du résultat de Lê et Teissier pour les hypersurfaces de C 3 qui établit la équisingularité à la Whitney de X et son cône tangent sous ces conditions.

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“…Proof. -The first step is to prove that (X, 0) does not have exceptional cones, however by [16,Prop. 2.1.4.2,p.…”

Section: For Every J We Have Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Specialization to the tangent cone and Whitney equisingularity

Flores¹

2013

Bul. Soc. Math. France

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“…The integral closure of ideals is related to Whitney equisingularity. For instance, Teissier [19] gave an algebraic description for Whitney's condition (b) using the integral closure of the sheaf of ideals, which started the modern equisingularity theory. Gaffney [4,5,6] generalized the theory of integral closure of ideals to modules, and made many applications in Whitney equisingularity.…”

Section: Introductionmentioning

confidence: 99%

The integral closure of a primary ideal is not always primary

Li¹,

Li²,

Yang³

et al. 2022

Preprint

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In 1936, Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic p = 2. In this paper, we give counterexamples to Krull's question in terms of polynomial rings with any characteristics. It is amazing that the Jacobian ideal of the polynomial f = x 6 + y 6 + x 4 zt + z 3 given by Briançon and Speder in 1975 also provides a counterexample to Krull's question. This example was used by Briançon and Speder to show that the pair (V1 \ V2, V2) satisfies Whitney's conditions but fails Zariski equisingularity conditions, where V1 is the hypersurface defined by f = 0 and V2 is the singular locus of V1. Most surprisingly, we show that the pair (V1 \ V2, V2) does not satisfy Whitney's conditions: there are three points (0, 0, 0, − 3 27/4ω) (ω 3 = 1) on V2 such that Whitney's conditions fail! Moreover, we also show that Whitney stratification of this hypersurface is different from the stratification of isosingular sets given by Hauenstein and Wampler, which is related to Thom-Boardman singularity.

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Open books decompositions of links of minimally elliptic singularities

Bhupal

2021

Mathematische Nachrichten

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We present an explicit Milnor open book decomposition supporting the canonical contact structure on the link of each minimally elliptic singularity whose fundamental cycle satisfies −3 ≤ ⋅ ≤ −1. For the Milnor open books whose pages have genus less than three, we give a factorization of the monodromy which does not involve any left-handed Dehn twists around interior curves. Necessary results regarding the roots of reducible, and irreducible elements in mapping class groups are proved, and some new relations in the mapping class groups are presented.

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Limites d’espaces tangents en géométrie analytique

Cited by 42 publications

References 19 publications

Specialization to the tangent cone and Whitney equisingularity

Specialization to the tangent cone and Whitney equisingularity

The integral closure of a primary ideal is not always primary

Open books decompositions of links of minimally elliptic singularities

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