Skip to main content
SpringerLink
Log in

Equimultiple deformations of isolated singularities

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study equimultiple deformations of isolated hypersurface singularities, introduce a blow-up equivalence of singular points, which is intermediate between topological and analytic ones, and give numerical sufficient conditions for the blow-up versality of the equimultiple deformation of a singularity or multisingularity induced by the space of algebraic hypersurfaces of a given degree. For singular points, which become Newton nondegenerate after one blowing up, we prove that the space of algebraic hypersurfaces of a given degree induces all the equimultiple deformations (up to the blow-up equivalence) which are stable with respect to removing monomials lying above the Newton diagrams. This is a generalization of a theorem by B. Chevallier.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. I. Arnol’d, S. M. Gusein-zade and A. N. Varchenko,Singularities of Differentiable Maps, Vol. 1, Birkhäuser, Boston, 1985.

    MATH  Google Scholar 

  2. M. F. Atiyah,Convexity and commuting Hamiltonians, The Bulletin of the London Mathematical Society14 (1982), 1–15.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. F. Atiyah,Angular momentum, convex polyhedra and algebraic geometry, Proceedings of the Edinburgh Mathematical Society26 (1983), 121–138.

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Chevallier,Secteurs et déformations locales des courbes réelles, Mathematische Annalen307 (1997), 1–28.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Diaz and J. Harris,Ideals associated to deformations of singular plane curves, Transactions of the American Mathematical Society309 (1988), 433–467.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. A. du Plessis,Versality properties of projective hypersurfaces, Preprint, 1998.

  7. A. A. du Plessis and C. T. C. Wall,Versal deformations in spaces of polynomials of fixed weight, Compositio Mathematica114 (1998), 113–124.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. A. du Plessis and C. T. C. Wall,Singular hypersurfaces, versality and Gorenstein algebras, Journal of Algebraic Geometry9 (2000), 309–322.

    MATH  MathSciNet  Google Scholar 

  9. G.-M. Greuel and U. Karras,Families of varieties with prescribed singularities, Composito Mathematica69 (1989), 83–110.

    MATH  MathSciNet  Google Scholar 

  10. G.-M. Greuel and C. Lossen,Equianalytic and equisingular families of curves on surfaces, Manuscripta Mathematica91 (1996), 323–342.

    Article  MATH  MathSciNet  Google Scholar 

  11. G.-M. Greuel, C. Lossen and E. Shustin,New asymptotics in the geometry of equisingular families of curves, International Mathematics Research Notices13 (1997), 595–611.

    Article  MathSciNet  Google Scholar 

  12. G.-M. Greuel, C. Lossen and E. Shustin,Castelnuovo function, zero-dimensional schemes and singular plane curves, Journal of Algebraic Geometry9 (2000), 663–710.

    MATH  MathSciNet  Google Scholar 

  13. G.-M. Greuel and E. Shustin,Geometry of Equisingular Families of Curves, inSingularity Theory (B. Bruce and D. Mond, eds.), Proceedings of the European Singularities Conference, Liverpool, August 1996. Dedicated to C. T. C. Wall on the occasion of his 60th birthday, London Mathematical Society Lecture Note Series263, Cambridge University Press, Cambridge, 1999, pp. 79–108.

    Google Scholar 

  14. A. Hirschowitz,Le methode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Mathematica50 (1985), 337–388.

    Article  MATH  MathSciNet  Google Scholar 

  15. I. Itenberg and E. Shustin,Complexification of the Viro theorem and topology of real combinatorial hypersurfaces, Preprint no. 111, Max-Planck-Institut für Mathematik, 1999.

  16. J. N. Mather,Stability of C ∞-mappings, III: Finitely determined map-germs, Publications Mathématiques de l’Institut des Hautes Études Scientifiques35 (1968), 127–156.

    Article  MATH  Google Scholar 

  17. J. N. Mather,Stability of C ∞ mappings. II. Infinitesimal stability implies stability, Annals of Mathematics (2)89 (1969), 254–291.

    Article  MathSciNet  Google Scholar 

  18. J.-J. Risler,Construction d’hypersurfaces réelles [d’après Viro], Séminaire N. Bourbaki, no. 763, Vol. 1992–93, Novembre 1992.

  19. I. Scherbak and A. Szpirglas,Boundary singularities: topology and duality, Advances in Soviet Mathematics21 (1994), 213–223.

    Google Scholar 

  20. E. Shustin,New M-curve of the 8th degree, Mathematical Notes of the Academy of Sciences of the USSR42 (1987), 606–610.

    Article  MathSciNet  MATH  Google Scholar 

  21. E. Shustin,Lower deformations of isolated hypersurface singularities, Algebra i Analiz10 (1999), 221–249.

    Google Scholar 

  22. E. Shustin and I. Tyomkin,Versal deformations of algebraic hypersurfaces with isolated singularities, Mathematische Annalen313 (1999), 297–314.

    Article  MATH  MathSciNet  Google Scholar 

  23. J. C. Tougeron,Ideaux des fonctions différentiables, Annales de l’Institut Fourier (Grenoble)18 (1968), 177–240.

    MATH  MathSciNet  Google Scholar 

  24. O. Ya. Viro,Gluing of algebraic hypersurfaces, smoothing of singularities and construction of curves, inProceedings of the Leningrad International Topological Conference, Leningrad, August 1982, Nauka, Leningrad, 1983, pp. 149–197 (Russian).

  25. O. Ya. Viro,Gluing of plane real algebraic curves and construction of curves of degrees 6 and 7, Lecture Notes in Mathematics1060, Springer, Berlin, 1984, pp. 187–200.

    Google Scholar 

  26. O.Ya. Viro,Real algebraic plane curves: constructions with controlled topology, Leningrad Mathematical Journal1 (1990), 1059–1134.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University Ramat Aviv, 69978, Tel Aviv, Israel

    I. Scherback & E. Shustin

Authors
  1. I. Scherback
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Shustin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. Scherback.

Additional information

This work was partially supported by Grant No.6836-1-9 of the Israeli Ministry of Sciences. The second author thanks the Max-Planck Institut (Bonn) for hospitality and financial support.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Scherback, I., Shustin, E. Equimultiple deformations of isolated singularities. Isr. J. Math. 125, 293–315 (2001). https://doi.org/10.1007/BF02773384

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02773384

Keywords

Search

Navigation