Zariski's multiplicity question and aligned singularities

Eyral, Christophe

doi:10.1016/j.crma.2005.12.008

Cited by 8 publications

(8 citation statements)

References 16 publications

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“…(For an updated survey paper, see for instance [5].) Given a hypersurface singularity (V, 0) germ, we denote by C V its projectivized tangent cone and by BV be the blowup of V at p. The second problem by Zariski is to find out if the following assertion is true: B-problem.…”

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confidence: 99%

Milnor Number of Weighted-Le-Yomdin Singularities

Bartolo

Bobadilla

Luengo

et al. 2010

International Mathematics Research Notices

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Correspondence to be sent to: javier@mat.csic.es At the beginning of the seventies, O. Zariski proposed several problems related with the (embedded) topology of a germ of a n-dimensional hypersurface singularity defined by the zero locus of a germ of a complex analytic function. The second one was roughly stated as "if two analytic hypersurface germs are topologically equivalent then their tangent cones must be homeomorphic and the homeomorphism must respect the topological equisingularity type at any point." In this paper, we give counterexamples for n = 3 and 4 (even in a family). Our proof is mainly based on the study of the topology of weighted-Lê-Yomdin surface singularities which are a generalization of the well-known Lê-Yomdin singularities. We obtain a formula for the Milnor number of a weighted-Lê-Yomdin surface singularity and derive an equisingularity criterion for them.In [19], Zariski proposed to study a series of problems (from A to H) related with the (embedded) topology of a germ of a hypersurface singularity (V, 0) ⊂ (C n , 0) defined by the zero locus of a germ of a complex analytic function f : (C n , 0) → (C, 0). He defined

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confidence: 99%

Milnor Number of Weighted-Le-Yomdin Singularities

Bartolo

Bobadilla

Luengo

et al. 2010

International Mathematics Research Notices

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“…(2) T w (ln r2 r1 + α 2 i) = τ 2 + ϑ 1 (0)i, where τ 2 is minimal with ln r 2 r 1 − 2 ln L ≤ τ 2 ≤ ln r 2 r 1 + 2 ln L. 6 Here ϑ 1 is the function given by Proposition 11…”

Section: Proof Of Proposition 22 and Lema 23mentioning

confidence: 99%

Bilipschitz Invariants for Germs of Holomorphic Foliations

Rosas

2015

Int Math Res Notices

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“…where ξ and are holomorphic functions with ξ(0) = 0; (C3) h 0 is a semiquasihomogeneous polynomial with respect to z ; (C4) n = 3 (i.e., {h t } is a family of plane curve singularities). In [3], the first author proved the following equimultiplicity result for nonisolated singularities. n k t (z).…”

Section: Deformations Of the Form F Tmentioning

confidence: 99%

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

Eyral

Ruas

2015

Nagoya Mathematical Journal

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We show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers are “almost” constant. In the special case of families with isolated singularities—a case for which the constancy of the Lê numbers is equivalent to the constancy of the Milnor number—the result was proved by Greuel, Plénat, and Trotman.As an application, we prove equimultiplicity for new families of nonisolated hypersurface singularities with constant topological type, answering partially the Zariski multiplicity conjecture.

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Zariski's multiplicity question and aligned singularities

Cited by 8 publications

References 16 publications

Milnor Number of Weighted-Le-Yomdin Singularities

Milnor Number of Weighted-Le-Yomdin Singularities

Bilipschitz Invariants for Germs of Holomorphic Foliations

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

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