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Revista Matemática Complutense

, Volume 29, Issue 2, pp 439–454 | Cite as

Geometry and equisingularity of finitely determined map germs from \({\mathbb {C}}^n\) to \({\mathbb {C}}^3\), \(n >2\)

  • A. J. Miranda
  • V. H. Jorge Pérez
  • E. C. RizziolliEmail author
  • M. J. Saia
Article
  • 124 Downloads

Abstract

In this article we describe the geometry and the Whitney equisingularity of finitely determined map germs \(f{:} ({\mathbb {C}}^n,0) \rightarrow ( {\mathbb {C}}^3,0)\) with \(n \ge 3\). In the study of the geometry, we first investigate the critical locus \(\Sigma (f)\) of the germ, which is in the source. Then the discriminant \(\Delta (f)\), the image of the critical locus by the germ f, is studied. Last, but not least we investigate the set X(f), which is the inverse image by f of the discriminant. If the critical locus is not empty, the set X(f) is an hypersurface in the source that has nonisolated singularity at the origin. Concerning the Whitney equisingularity of families, we use some of the properties of the strata to prove that the Whitney equisingularity of an unfolding F is equivalent to the constancy of the Lê numbers of the hypersurfaces \(\Delta (f)\) and X (f). From this study we describe some relationship among the invariants needed to describe the Whitney equisingularity of families in these dimensions, we reduce the number of invariants needed to a total of \(2n+2\), which improves substantially the number required by Gaffney’s theorem.

Keywords

Geometry of map germs Whitney equisingularity Numerical invariants Lâ numbers 

Mathematics Subject Classification

Primary 32S15 14B05 

References

  1. 1.
    Gaffney, T.: Polar multiplicities and equisingularity of map germs. Topology 32(1), 185–223 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gaffney, T., Vohra, R.: A numerical characterization of equisingularity for map germs from \(n\)-space, \((n \ge 3)\), to the plane. J. Dyn. Syst. Geom. Theor. 2, 43–55 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Jorge Pérez, V.H., Rizziolli, E.C., Saia, M.J.: Whitney equisingularity, Euler obstruction and invariants of map germs from \({\mathbb{C}}^n\). In: Real and Complex Singularities, Trends in Mathematics, pp. 263–287. Birkhäuser, Basel (2007)Google Scholar
  4. 4.
    Holzer, S., Labs, O.: surfex 0.90, University of Mainz and University of Saarbrücken (2008). http://www.surfex.AlgebraicSurface.net
  5. 5.
    Wall, C.T.C.: Lectures on \(C^{\infty }\)-stability and Classification. In: Lecture Notes in Mathematics, vol. 192, pp. 178–206. Springer, Berlin (1971)Google Scholar
  6. 6.
    Wall, C.T.C.: Finite determinacy of smooth map germs. Bull. Lond. Math. Soc. 13, 481–539 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Morin, B.: Calcul Jacobian. Ann. Sci. École Norm. Sup. 8, 1–98 (1975)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Jorge Pérez, V.H., Levcovitz, D., Saia, M.J.: Invariants, equisingularity and Euler obstruction of map germs. Journal fur die Reine und Angewandte Mathematik, Crelle’s Journal 587, 145–161 (2005)Google Scholar
  9. 9.
    Mond, D., Pellikaan, R.: Fitting ideals and multiple points of analytic mappings. In: Algebraic Geometry and Complex Analysis (Pátzcuaro, 1987), pp. 107–161. Lecture Notes in Mathematics, vol. 1414. Springer, Berlin (1989)Google Scholar
  10. 10.
    Jorge Pérez, V.H., Miranda, A.J., Saia, M.J.: Counting singularities via fitting ideals. Int. J. Math. 23(6), 1250062-1–1250062-18 (2012)Google Scholar
  11. 11.
    Hernandes, M.E., Miranda, A.J., Peñafort-Sanchis, G.: An algorithm to compute a presentation of pushforward modules (2014) (Pre-print). https://sites.google.com/site/aldicio/publicacoes/presentation-matrix-algorithm
  12. 12.
    Buchweitz, R.-O., Greuel, G.-M.: The Milnor number and deformations of complex curve singularities. Invent. Math. 58(3), 241–281 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Greuel, G.-M., Pfister, G., Schönnemann, H.: Singular: A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern, free software under the GNU General Public Licence 1990–2007. http://www.singular.uni-kl.de
  14. 14.
    Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie. Publications Mathématiques de l’IHÉS 24 (1965). doi: 10.1007/bf02684322
  15. 15.
    Matsumura, H.: Commutative Algebra, 2nd edn. The Benjamim/Cummings Publishing Company, Inc., Reading (1980)Google Scholar
  16. 16.
    Massey, D.: Lê cycles and hypersurface singularities. In: Lecture Notes in Mathematics, vol. 1615. Springer, Berlin (1995)Google Scholar
  17. 17.
    Gaffney, T., Massey, D.: Trends in equisingularity theory. In: Bruce, J.W., Mond. D. (eds.) Singularity Theory. London Mathematical Society Lecture Note Series 263, pp. 207–248. Cambridge University Press, Cambridge (1999)Google Scholar
  18. 18.
    Gaffney, T., Gassler, R.: Segre numbers and hypersurfaces singularities. J. Algebraic Geom. 08, 695–736 (1999)MathSciNetzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2016

Authors and Affiliations

  • A. J. Miranda
    • 1
  • V. H. Jorge Pérez
    • 2
  • E. C. Rizziolli
    • 3
    Email author
  • M. J. Saia
    • 2
  1. 1.Faculdade de MatemáticaUniversidade Federal de UberlândiaUberlândiaBrazil
  2. 2.Departamento de Matemática, Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São PauloSão CarlosBrazil
  3. 3.Departamento de Matemática, Instituto de Geociências e Ciências ExatasUniversidade Estadual Paulista Júlio Mesquita FilhoRio ClaroBrazil

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