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On polar invariants of hypersurface singularities. (English) Zbl 1009.32017

The author proves a generalization of the Teissier theorem [B. Teissier, Invent. Math. 40, 267-292 (1977; Zbl 0446.32002)] about expressing the topological determinacy order by the polar invariants. In this generalization the main assumption is that the projective tangent cone of the hypersurface has at most isolated singularities. Some consequences of the theorem are studied in the last section of the paper.

MSC:

32S25 Complex surface and hypersurface singularities
32S05 Local complex singularities
14B07 Deformations of singularities

Citations:

Zbl 0446.32002

References:

[1] Campillo, A.). - Algebroid Curves in Positive Characteristic, , vol. 813, Springer-Verlag, Berlin-Heidelberg-New York, 1980. · Zbl 0451.14010
[2] Casas-Alvero, E.). - ’Infinitely near imposed singularities and singularities of polar curves’, Math. Ann.287, 3 (1990), 429-454. · Zbl 0675.14009
[3] Casas-Alvero, E.). - ’Base points of polar curves’, Ann. Inst. Fourier (Grenoble)41, 1 (1991), 1-10. · Zbl 0707.14024
[4] Chang, S.H.) AND Lu, Y.C.). - ’On C0-sufficiency of complex jets’, Can. J. Math.25 (1973), 874-880. · Zbl 0258.58004
[5] Delgado De La Mata, F.). - ’A factorization theorem for the polar of a curve with two branches’, Compositio Math.92, 3 (1994), 327-375. · Zbl 0816.14012
[6] García Barroso, E.). - ’Sur les courbes polaires d’une courbe plane réduite’, Proc. London Math. Soc. (3)81 (2000), no. 1, 1-28. · Zbl 1041.14008
[7] Greuel, G.M.). - ’Constant Milnor number implies constant multiplicity for quasihomogeneous singularities’, Manuscripta Math.56 (1986), no. 2, 159-166. · Zbl 0594.32021
[8] Henry, J.P.) AND Merle, M.). - ’Sections planes, limites d’espaces tangents et transversalité de variétés polaires’, C.R. Acad. Sc.Paris t. 291 Serie A, (1980), 291-294. · Zbl 0472.32005
[9] Hermann, M.), Ikeda, S.) AND Orbanz, U.). - Equimultiplicity and Blowing up, Springer-Verlag, Berlin-Heidelberg-New York, 1988. · Zbl 0649.13011
[10] Kaup, L.) AND Kaup, B.). - Holomorphic Functions of Variables, Gruyter Studies in Math.3, De Gruyter, Berlin, 1983 · Zbl 0528.32001
[11] Lê, D.T.). - ’Une application d’un théorème d’A’Campo à l’équisingularité’, Indag. Math., 35, 5 (1973), 403-409. · Zbl 0271.14001
[12] Lê, D.T.), Michel, F.) AND (Weber, C.). - ’Sur le comportement des polaires associées aux germes de courbes planes’, Compositio Math., 72 (1989), 87-113. · Zbl 0705.32021
[13] Lê, D.T.) AND Teissier, B.). - ’Sur la géométrie des surfaces complexes I. tangentes exceptionnelles’, Amer. J. Math.101 (1979), 420-452. · Zbl 0427.32012
[14] Looijenga, E.J.N.). - Isolated singular points on complete intersections. , 77. Camb. Univ. Press, Cambridge-New York, 1984. · Zbl 0552.14002
[15] Luengo, I.). - ’The μ-stratum is not smooth’, Invent. Math.90 (1987), 139-152. · Zbl 0627.32018
[16] Luengo, I.) AND Melle Hernández, A.). - ’Une formule pour le nombre de Milnor’, C.R. Acad. Sc. Paris t. 321, Série I (1995), 1473-1478. · Zbl 0848.32029
[17] Melle Hernández, A.). - ’Milnor numbers for surface singularities’, Israel J. Math.115 (2000) 29-50. · Zbl 0956.32027
[18] O’Shea, D.). - ’Topologically trivial deformations of isolated quasihomogeneous hypersurface singularities are equimultiple’,Proc. Amer. Math. Soc.101 (1987), no. 2, 260-262. · Zbl 0628.32029
[19] Merle, M.). - ’Invariants polaires des courbes planes’, Invent. Math.41 (1977), 103-111. · Zbl 0371.14003
[20] Teissier, B.). - ’Cycles Evanescents, Sections Planes et Conditions de Whitney’. Singularités a Cargase1972,Asterisque7-8, (1973), 363-391 · Zbl 0295.14003
[21] Teissier, B.). - ’Variétés polaires I: invariants polaires des singularités d’hypersurfaces’, Invent. Math.40 (1977), 267-292. · Zbl 0446.32002
[22] Teissier, B.). - ’Polyèdre de Newton jacobien et équisingularité’, Seminar on Singularities (Paris, 1976/1977), Publ. Math. Univ. Paris VII, 7, Univ. Paris VII, Paris, (1980) 193-221.
[23] Teissier, B.). - ’A bouquet of bouquets for a birthday’, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, (1993) 93-122. · Zbl 0802.32029
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