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On the Zariski multiplicity conjecture for weighted homogeneous and Newton nondegenerate line singularities. (English) Zbl 1442.14013

O. Zariski posed in [Bull. Am. Math. Soc. 77, 481–491 (1971; Zbl 0236.14002)] the following question, which he supposed to have a quick answer by topologists: Given two hypersurface singularities with the same topological type, do they have the same multiplicity? This question could be answered only for a few special cases. For a survey see [C. Eyral, N. Z. J. Math. 36, 253–276 (2007; Zbl 1185.32018)]. New families of weighted homogeneous and Newton nondegenerate line singularities are presented that satisfy Zariski’s multiplicity conjecture.

MSC:

14B05 Singularities in algebraic geometry
14B07 Deformations of singularities
14J70 Hypersurfaces and algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
32S15 Equisingularity (topological and analytic)
32S25 Complex surface and hypersurface singularities
32S05 Local complex singularities
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References:

[1] Abderrahmane, O. M., On deformation with constant Milnor number and Newton polyhedron, Math. Z.284(1-2) (2016) 167-174. · Zbl 1369.14007
[2] L. Birbrair, A. Fernandes, J. E. Sampaio and M. Verbitsky, Multiplicity of singularities is not a bi-Lipschitz invariant, arXiv:1801.06849v1 [math.AG] 2018. · Zbl 1442.32038
[3] Brzostowski, S. and Oleksik, G., On combinatorial criteria for nondegenerate singularities, Kodai Math. J.39(2) (2016) 455-468. · Zbl 1353.32029
[4] Eyral, C., Zariski’s multiplicity question — A survey, New Zealand J. Math.36 (2007) 253-276. · Zbl 1185.32018
[5] Eyral, C., Topics in Equisingularity Theory, , Vol. 3 (Polish Academy of Sciences Institute of Mathematics, 2016). · Zbl 1354.14002
[6] Eyral, C., Topologically equisingular deformations of homogeneous hypersurfaces with line singularities are equimultiple, Internat. J. Math.28(5) (2017) 1750029, 8 pp. · Zbl 1373.14006
[7] Eyral, C. and Ruas, M. A. S., Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities, Nagoya Math. J.218 (2015) 29-50. · Zbl 1408.32029
[8] Fernandes, A. and Sampaio, J. E., Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomorphisms, J. Topol.9(3) (2016) 927-933. · Zbl 1353.14005
[9] de Bobadilla, J. Fernández, Topological equisingularity of hypersurfaces with \(1\)-dimensional critical set, Adv. Math.248 (2013) 1199-1253. · Zbl 1284.32018
[10] de Bobadilla, J. Fernández, Fernandes, A. and Sampaio, J. E., Multiplicity and degree as bi-Lipschitz invariants for complex sets, J. Topol.11(4) (2018) 957-965. · Zbl 1409.32024
[11] Fulton, W., Intersection Theory, , Vol. 2 (Springer-Verlag, Berlin, 1984). · Zbl 0541.14005
[12] A. M. Gabrièlov and A. G. Kušnirenko, Description of deformations with constant Milnor number for homogeneous functions, Funkcional. Anal. i Priložen.9(4) (1975) 67-68 (Russian). English translation: Functional Anal. Appl.9(4) (1975) 329-331. · Zbl 0325.32007
[13] Gaffney, T. and Gassler, R., Segre numbers and hypersurface singularities, J. Algebr. Geom.8(4) (1999) 695-736. · Zbl 0971.13021
[14] Greuel, G.-M., Der Gauss-Manin Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Math. Ann.214 (1975) 235-266. · Zbl 0285.14002
[15] Greuel, G.-M., Constant Milnor number implies constant multiplicity for quasihomogeneous singularities, Manuscr. Math.56(2) (1986) 159-166. · Zbl 0594.32021
[16] Hironaka, H., Normal cones in analytic Whitney stratifications, Inst. Hautes Études Sci. Publ. Math.36 (1969) 127-138. · Zbl 0219.57022
[17] Kouchnirenko, A. G., Polyèdres de Newton et nombres de Milnor, Invent. Math.32(1) (1976) 1-32. · Zbl 0328.32007
[18] D. T. Lê, Computation of the Milnor number of an isolated singularity of a complete intersection, Funkcional. Anal. i Priložen.8(2) (1974) 45-49 (Russian). English translation: Funct. Anal. Appl.8 (1974) 127-131. · Zbl 0351.32007
[19] Lê, D. T. and Ramanujam, C. P., The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. Math.98(1) (1976) 67-78. · Zbl 0351.32009
[20] Massey, D. B., A reduction theorem for the Zariski multiplicity conjecture, Proc. Amer. Math. Soc.106(2) (1989) 379-383. · Zbl 0674.32005
[21] Massey, D. B., The Lê-Ramanujam problem for hypersurfaces with one-dimensional singular sets, Math. Ann.282(1) (1988) 33-49. · Zbl 0657.32005
[22] Massey, D. B., The Lê varieties, I, Invent. Math.99(2) (1990) 357-376. · Zbl 0712.32020
[23] Massey, D. B., The Lê varieties, II, Invent. Math.104(1) (1991) 113-148. · Zbl 0727.32015
[24] Massey, D. B., Lê Cycles and Hypersurface Singularities, , Vol. 1615 (Springer-Verlag, Berlin, 1995). · Zbl 0835.32002
[25] Mather, J., Notes on topological stability, Bull. Amer. Math. Soc. (N.S.)49(4) (2012) 475-506. · Zbl 1260.57049
[26] Morgado, M. F. Z. and Saia, M. J., On the Milnor fiber and Lê numbers of semi-weighted homogeneous arrangements, Bull. Braz. Math. Soc. (N.S.)43(4) (2012) 615-636. · Zbl 1255.32004
[27] O’Shea, D., Topologically trivial deformations of isolated quasihomogeneous hypersurface singularities are equimultiple, Proc. Amer. Math. Soc.101(2) (1987) 260-262. · Zbl 0628.32029
[28] Saia, M. J. and Tomazella, J. N., Deformations with constant Milnor number and multiplicity of complex hypersurfaces, Glasg. Math. J.46(1) (2004) 121-130. · Zbl 1051.32018
[29] Sampaio, J. E., Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones, Selecta Math. (N.S.)22(2) (2016) 553-559. · Zbl 1338.32008
[30] Smale, S., On the structure of manifolds, Amer. J. Math.84 (1962) 387-399. · Zbl 0109.41103
[31] Thom, R., Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc.75 (1969) 240-284. · Zbl 0197.20502
[32] Timourian, J. G., The invariance of Milnor’s number implies topological triviality, Amer. J. Math.99(2) (1977) 437-446. · Zbl 0373.32003
[33] Vannier, J.-P., Sur les fibrations de Milnor de familles d’hypersurfaces à lieu singulier de dimension un, Math. Ann.287(3) (1990) 539-552. · Zbl 0687.32014
[34] A. N. Varchenko, A lower bound for the codimension of the \(\mu = \text{const}\) stratum in terms of the mixed Hodge structure, Vestnik Moskov. Univ. Ser. I Mat. Mekh.120(6) (1982) 28-31 (Russian). English translation: Moscow Univ. Math. Bull.37(6) (1982) 30-33. · Zbl 0517.32004
[35] Zariski, O., Some open questions in the theory of singularities, Bull. Amer. Math. Soc.77 (1971) 481-491. · Zbl 0236.14002
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