×

zbMATH — the first resource for mathematics

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities. (English) Zbl 1408.32029
Summary: 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.

MSC:
32S15 Equisingularity (topological and analytic)
32S25 Complex surface and hypersurface singularities
32S05 Local complex singularities
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] O. M. Abderrahmane, On the deformation with constant Milnor number and Newton polyhedron , preprint, (accessed 21 November 2004). · www.rimath.saitama-u.ac.jp
[2] J. Fernández de Bobadilla and T. Gaffney, The Lê numbers of the square of a function and their applications , J. Lond. Math. Soc. (2) 77 (2008), 545-557. · Zbl 1140.14002 · doi:10.1112/jlms/jdm101 · arxiv:math/0508151
[3] C. Eyral, Zariski’s multiplicity question and aligned singularities , C. R. Math. Acad. Sci. Paris 342 (2006), 183-186. · Zbl 1089.32019 · doi:10.1016/j.crma.2005.12.008
[4] C. Eyral, Zariski’s multiplicity question-A survey , New Zealand J. Math. 36 (2007), 253-276. · Zbl 1185.32018
[5] W. Fulton, Intersection Theory , Ergeb. Math. Grenzgeb. (3) 2 , Springer, Berlin, 1984. · Zbl 0541.14005
[6] G.-M. Greuel, Constant Milnor number implies constant multiplicity for quasihomogeneous singularities , Manuscripta Math. 56 (1986), 159-166. · Zbl 0594.32021 · doi:10.1007/BF01172153 · eudml:155163
[7] G.-M. Greuel and G. Pfister, Advances and improvements in the theory of standard bases and syzygies , Arch. Math. (Basel) 66 (1996), 163-176. · Zbl 0854.13015 · doi:10.1007/BF01273348
[8] H. A. Hamm and Lê Dũng Tráng, Un théorème de Zariski du type de Lefschetz , Ann. Sci. Éc. Norm. Supér. (4) 6 (1973), 317-355.
[9] A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor , Invent. Math. 32 (1976), 1-31. · Zbl 0328.32007 · doi:10.1007/BF01389769
[10] Lê Dũng Tráng, “Topologie des singularités des hypersurfaces complexes” in Singularités à Cargèse (Cargèse, 1972) , Astérisque 7/8 , Soc. Math. France, Paris, 1973, 171-182.
[11] Lê Dũng Tráng and C. P. Ramanujam, The invariance of Milnor number implies the invariance of the topological type , Amer. J. Math. 98 (1976), 67-78. · Zbl 0351.32009 · doi:10.2307/2373614
[12] Lê Dũng Tráng and K. Saito, La constance du nombre de Milnor donne des bonnes stratifications , C. R. Acad. Sci. Paris Sér. A-B 277 (1973), 793-795. · Zbl 0283.32007
[13] D. B. Massey, Lê cycles and hypersurface singularities , Lecture Notes in Math. 1615 , Springer, Berlin, 1995. · Zbl 0835.32002 · doi:10.1007/BFb0094409
[14] D. Massey, Numerical Control over Complex Analytic Singularities , Mem. Amer. Math. Soc. 163 (2003), no. 778. · doi:10.1090/memo/0778
[15] D. O’Shea, Topologically trivial deformations of isolated quasihomogeneous hypersurface singularities are equimultiple , Proc. Amer. Math. Soc. 101 (1987), 260-262. · Zbl 0628.32029 · doi:10.2307/2045992
[16] C. Plénat and D. Trotman, On the multiplicities of families of complex hypersurface-germs with constant Milnor number , Internat. J. Math. 24 (2013), article no. 1350021. · Zbl 1267.14007 · doi:10.1142/S0129167X13500213 · arxiv:1202.5177
[17] M. J. Saia and J. N. Tomazella, Deformations with constant Milnor number and multiplicity of complex hypersurfaces , Glasg. Math. J. 46 (2004), 121-130. · Zbl 1051.32018 · doi:10.1017/S0017089503001599
[18] B. Teissier, “Cycles évanescents, sections planes et conditions de Whitney” in Singularités à Cargèse (Cargèse, 1972) , Astérisque 7/8 , Soc. Math. France, Paris, 1973, 285-362.
[19] D. Trotman, Partial results on the topological invariance of the multiplicity of a complex hypersurface , lecture, Université Paris 7, France, 1977.
[20] O. Zariski, Some open questions in the theory of singularities , Bull. Amer. Math. Soc. 77 (1971), 481-491. · Zbl 0236.14002 · doi:10.1090/S0002-9904-1971-12729-5
[21] O. Zariski, On the topology of algebroid singularities , Amer. J. Math. 54 (1932), 453-465. · Zbl 0004.36902 · doi:10.2307/2370887
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.