Skip to main content Accessibility help
×
Home
Hostname: page-component-544b6db54f-2p87r Total loading time: 0.413 Render date: 2021-10-22T11:21:36.309Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

Published online by Cambridge University Press:  11 January 2016

Christophe Eyral
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, 00-656 Warsaw, Poland, eyralchr@yahoo.com
Maria Aparecida Soares Ruas
Affiliation:
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 13566-590 São Carlos - SP, Brazil, maasruas@icmc.usp.br
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Abderrahmane, O. M., On the deformation with constant Milnor number and Newton polyhedron, preprint, http://www.rimath.saitama-u.ac.jp/lab.jp/Fukui/ould/dahm.pdf (accessed 21 November 2004).Google Scholar
[2] de Bobadilla, J. Fernández and Gaffney, T., The Lê numbers of the square of a function and their applications, J. Lond. Math. Soc. (2) 77 (2008), 545557. MR 2418291. DOI 10.1112/jlms/jdm101.CrossRefGoogle Scholar
[3] Eyral, C., Zariski's multiplicity question and aligned singularities, C. R. Math. Acad. Sci. Paris 342 (2006), 183186. MR 2198190. DOI 10.1016/j.crma.2005.12.008.CrossRefGoogle Scholar
[4] Eyral, C., Zariski's multiplicity question—A survey, New Zealand J. Math. 36 (2007), 253276. MR 2476643.Google Scholar
[5] Fulton, W., Intersection Theory, Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1984. MR 0732620. DOI 10.1007/978-3-662-02421-8.Google Scholar
[6] Greuel, G.-M., Constant Milnor number implies constant multiplicity for quasihomogeneous singularities, Manuscripta Math. 56 (1986), 159166. MR 0850367. DOI 10.1007/BF01172153.CrossRefGoogle Scholar
[7] Greuel, G.-M. and Pfister, G., Advances and improvements in the theory of standard bases and syzygies, Arch. Math. (Basel) 66 (1996), 163176. MR 1367159. DOI 10.1007/BF01273348.CrossRefGoogle Scholar
[8] Hamm, H. A. and Tráng, Lê Dũng, Un théorème de Zariski du type de Lefschetz, Ann. Sci. Éc. Norm. Supér. (4) 6 (1973), 317355. MR 0401755.CrossRefGoogle Scholar
[9] Kouchnirenko, A. G., Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 131. MR 0419433.Google Scholar
[10] Tráng, Lê Dũng, “Topologie des singularités des hypersurfaces complexes” in Singularités à Cargèse (Cargèse, 1972), Astérisque 7/8, Soc. Math. France, Paris, 1973, 171182. MR 0361147.Google Scholar
[11] Tráng, Lê Dũng and Ramanujam, C. P., The invariance of Milnor number implies the invariance of the topological type, Amer. J. Math. 98 (1976), 6778. MR 0399088.CrossRefGoogle Scholar
[12] Tráng, Lê Dũng and Saito, K., La constance du nombre de Milnor donne des bonnes stratifications, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), 793795. MR 0350063. Google Scholar
[13] Massey, D. B., Lê cycles and hypersurface singularities, Lecture Notes in Math. 1615 , Springer, Berlin, 1995. MR 1441075.Google Scholar
[14] Massey, D. B., Numerical Control over Complex Analytic Singularities, Mem. Amer. Math. Soc. 163 (2003), no. 778. MR 1962934. DOI 10.1090/memo/0778.Google Scholar
[15] O'Shea, D., Topologically trivial deformations of isolated quasihomogeneous hypersurface singularities are equimultiple, Proc. Amer. Math. Soc. 101 (1987), 260262. MR 0902538. DOI 10.2307/2045992.CrossRefGoogle Scholar
[16] Plénat, C. and Trotman, D., On the multiplicities of families of complex hypersurface-germs with constant Milnor number, Internat. J. Math. 24 (2013), article no. 1350021. MR 3048008. DOI 10.1142/S0129167X13500213.CrossRefGoogle Scholar
[17] Saia, M. J. and Tomazella, J. N., Deformations with constant Milnor number and multiplicity of complex hypersurfaces, Glasg. Math. J. 46 (2004), 121130. MR 2034839. DOI 10.1017/S0017089503001599.CrossRefGoogle Scholar
[18] Teissier, B., “Cycles évanescents, sections planes et conditions de Whitney” in Singularités à Cargése (Cargése, 1972), Astérisque 7/8, Soc. Math. France, Paris, 1973, 285362. MR 0374482.Google Scholar
[19] Trotman, D., Partial results on the topological invariance of the multiplicity of a complex hypersurface, lecture, Université Paris 7, France, 1977.Google Scholar
[20] Zariski, O., Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481491. MR 0277533.CrossRefGoogle Scholar
[21] Zariski, O., On the topology of algebroid singularities, Amer. J. Math. 54 (1932), 453465. MR 1507926. DOI 10.2307/2370887.CrossRefGoogle Scholar
You have Access
3
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *