Specialization to the tangent cone and Whitney equisingularity

Arturo Giles Flores

Bulletin de la Société Mathématique de France (2013)

  • Volume: 141, Issue: 2, page 299-342
  • ISSN: 0037-9484

Abstract

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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.

How to cite

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Giles Flores, Arturo. "Specialization to the tangent cone and Whitney equisingularity." Bulletin de la Société Mathématique de France 141.2 (2013): 299-342. <http://eudml.org/doc/272587>.

@article{GilesFlores2013,
abstract = {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 $\{\mathfrak \{X\}\}^0$ of the specialization to the tangent cone $\varphi : \{\mathfrak \{X\}\}\rightarrow \mathbb \{C\}$ 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 $\mathbb \{C\}^3$ which establishes the Whitney equisingularity of $X$ and its tangent cone under these conditions.},
author = {Giles Flores, Arturo},
journal = {Bulletin de la Société Mathématique de France},
keywords = {equisingularity; Whitney conditions; specialization to the tangent cone},
language = {eng},
number = {2},
pages = {299-342},
publisher = {Société mathématique de France},
title = {Specialization to the tangent cone and Whitney equisingularity},
url = {http://eudml.org/doc/272587},
volume = {141},
year = {2013},
}

TY - JOUR
AU - Giles Flores, Arturo
TI - Specialization to the tangent cone and Whitney equisingularity
JO - Bulletin de la Société Mathématique de France
PY - 2013
PB - Société mathématique de France
VL - 141
IS - 2
SP - 299
EP - 342
AB - 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 ${\mathfrak {X}}^0$ of the specialization to the tangent cone $\varphi : {\mathfrak {X}}\rightarrow \mathbb {C}$ 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 $\mathbb {C}^3$ which establishes the Whitney equisingularity of $X$ and its tangent cone under these conditions.
LA - eng
KW - equisingularity; Whitney conditions; specialization to the tangent cone
UR - http://eudml.org/doc/272587
ER -

References

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  14. [14] L. D. ung Tráng – « Limites d’espaces tangents sur les surfaces », Nova Acta Leopoldina (N.F.) 52 (1981), p. 119–137. MR642701
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  16. [16] L. D. Tráng & B. Teissier – « Limites d’espaces tangents en géométrie analytique », Comment. Math. Helv.63 (1988), p. 540–578. Zbl0658.32010MR966949
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