Skip to main content
SpringerLink
Log in
Book cover

Resolution of Singularities pp 485–505Cite as

Birkhäuser

Equisingularity and Simultaneous Resolution of Singularities

  • Chapter

Part of the book series: Progress in Mathematics ((PM,volume 181))

Abstract

Zariski defined equisingularity on an n-dimensional hypersurface V via stratification by “dimensionality type,” an integer associated to a point by means of a generic local projection to affine n-space. A possibly more intuitive concept of equisingularity can be based on stratification by simultaneous resolvability of singularities. The two approaches are known to be equivalent for families of plane curve singularities. In higher dimension we ask whether constancy of dimensionality type along a smooth subvariety W of V implies the existence of a simultaneous resolution of the singularities of V along W. (The converse is false.)

The underlying idea is to follow the classical inductive strategy of Jung —begin by desingularizing the discriminant of a generic projection — to reduce to asking if there is a canonical resolution process which when applied to quasi-ordinary singularities depends only on their characteristic monomials. This appears to be so in dimension 2. In higher dimensions the question is quite open.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abhyankar, S. S.: A criterion of equisingularity, Amer. J. Math 90 (1968), 342–345.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bierstone, E., Milman, P.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), 207–302.

    MathSciNet  MATH  Google Scholar 

  3. Briançon, J., Henry, J. P. G.: Équisingularité générique des familles de surfaces a singularité isolée, Bull. Soc. Math. France 108 (1980), 259–281.

    MathSciNet  MATH  Google Scholar 

  4. Briançon, J., Speder, J.-P.: La trivialité topologique n’implique pas les conditions de Whitney, C. R. Acad. Sci. Paris, Sér. A-B 280 (1975), no. 6, Aiii, A365–A367.

    Google Scholar 

  5. van den Dries, L., Miller, C.: Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497–540.

    Article  MathSciNet  MATH  Google Scholar 

  6. Encinas, S., Villamayor, O.: Good points and constructive resolution of singularities, Acta Math. 181 (1998), 109–158.

    Article  MathSciNet  MATH  Google Scholar 

  7. Gau, Y.-N.: Embedded topological classification of quasi-ordinary singularities, Memoirs Amer. Math. Soc. 388, 1988.

    Google Scholar 

  8. Gibson, G. C., Looijenga, E., du Plessis, A., Wirthmüller, K.: Topological Stability of Smooth Maps, Lecture Notes in Math. 552, Springer-Verlag, 1976.

    Google Scholar 

  9. Gaffney, T., Massey, D.: Trends in equisingularity theory, to appear in the Proceedings of the Liverpool Conference in Honor of C.T.C. Wall.

    Google Scholar 

  10. Goresky, M., MacPherson, R.: Stratified Morse Theory, Springer-Verlag, 1988.

    Book  Google Scholar 

  11. Hironaka, H.: Normal cones in analytic Whitney stratifications, Publ. Math. IHES 36 (1969), 127–138.

    MathSciNet  MATH  Google Scholar 

  12. Hironaka, H.: On Zariski dimensionality type, Amer. J. Math. 101 (1979), 384–419.

    Article  MathSciNet  MATH  Google Scholar 

  13. Kleiman, S. L.: Equisingularity, multiplicity, and dependence, to appear in the Proceedings of a Conference in Honor of M. Fiorentini, publ. Marcel Dekker.

    Google Scholar 

  14. Laufer, H: Strong simultaneous resolution for surface singularities, in “Complex Analytic Singularities,” North-Holland, Amsterdam-New York, 1987, pp. 207–214.

    Google Scholar 

  15. Lê, D. T., Mebkhout, Z.: Introduction to linear differential systems, in “Singularities,” Proc. Sympos. Pure Math., vol.40, Part 2, Amer. Math. Soc., Providence, 1983, pp. 31–63.

    Google Scholar 

  16. Lê, D. T., Teissier, B.: Cycles évanescents, Sections Planes, et conditions de Whitney. II. in “Singularities,” Proc. Sympos. Pure Math., vol. 40, Part 2, Amer. Math. Soc., Providence, 1983, pp. 65–103.

    Google Scholar 

  17. Lipman, J.: Introduction to resolution of singularities, in “Algebraic Geometry,” Proc. Sympos. Pure Math., vol. 29, Amer. Math. Soc., Providence, 1975, Lecture 3, pp. 218–228.

    Google Scholar 

  18. Lipman, J.: Quasi-ordinary singularities of surfaces in C3, in “Singularities,” Proc. Sympos. Pure Math., vol. 40, Part 2, Amer. Math. Soc., Providence, 1983, pp. 161–172.

    Google Scholar 

  19. Lipman, J.: Topological invariants of quasi-ordinary singularities, Memoirs Amer. Math. Soc. 388, 1988.

    Google Scholar 

  20. Luengo, I.: A counterexample to a conjecture of Zariski, Math. Ann. 267, (1984), 487–494.

    Article  MathSciNet  MATH  Google Scholar 

  21. MacPherson, R.: Global questions in the topology of singular spaces, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 213–236, PWN, Warsaw, 1984.

    MathSciNet  Google Scholar 

  22. Mather, J.: Stratifications and Mappings, in Dynamical Systems, Ed. M. M. Peixoto, Academic Press, New York, 1973, pp. 195–232.

    Google Scholar 

  23. Mather, J.: How to stratify mappings and jet spaces, in “Singularités d’Applications Différentiables, Lecture Notes in Math. 535, Springer-Verlag, 1976, pp. 128–176.

    Google Scholar 

  24. Orbanz, U.: Embedded resolution of algebraic surfaces, after Abhyankar (characteristic 0), in “Resolution of Surface Singularities,” Springer Lecture Notes 1101 (1984), pp. 1–50.

    MathSciNet  Google Scholar 

  25. Parusiúski, A.: Lipschitz stratification of subanalytic sets, Ann. Scient. Éc. Norm. Sup. 27 (1994), 661–696.

    Google Scholar 

  26. Speder, J.-P.: Equisingularité et conditions de Whitney, Amer. J. Math. 97 (1975)

    Google Scholar 

  27. Teissier, B.: Résolution Simultanée—II, in “Séminaire sur les Singularités des Surfaces,” Lecture Notes in Math. 777, Springer-Verlag, 1980, pp. 82–146.

    MathSciNet  Google Scholar 

  28. Teissier, B.: Variétés polaires II: multiplicités polaires, sections planes, et conditions de Whitney, in “Algebraic Geometry, Proceedings, La Rabida, 1981,” Lecture Notes in Math. 961, Springer-Verlag, 1982, pp. 314–491.

    Google Scholar 

  29. Teissier, B.: Sur la classification des singularités des espaces analytiques complexes, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 763–781, PWN, Warsaw, 1984.

    Google Scholar 

  30. Varchenko, A. N.: The relation between topological and algebro-geometric equisingularities according to Zariski, Funct. Anal. Appl. 7 (1973), 87–90.

    Google Scholar 

  31. Verdier, J.-L.: Stratifications de Whitney et théorème de Bertini-Sard, Inventiones math. 36 (1976), 295–312.

    Article  MathSciNet  MATH  Google Scholar 

  32. Villamayor, O.: On equiresolution and a question of Zariski, preprint.

    Google Scholar 

  33. Wahl, J.: Equisingular deformations of normal surface singularities, Annals of Math. 104 (1976), 325–356.

    Article  MathSciNet  MATH  Google Scholar 

  34. Whitney, H.: Tangents to an analytic variety, Annals of Math. 81 (1965), 496–549.

    Article  MathSciNet  MATH  Google Scholar 

  35. Zariski, O.: Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481–491. (Reprinted in [Z3], pp. 238–248.)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zariski, O.: Foundations of a general theory of equisingularity on r-dimensional algebroid and algebraic varieties, of embedding dimension r+1, Amer. J. Math. 101 (1979), 453–514. (Reprinted in [Z3], pp. 573–634; and summarized in [Z3, pp. 635–651].)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zariski, O.: Collected Papers, vol. IV, MIT Press, Cambridge, Mass., 1979.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Joseph Lipman

Authors
  1. Joseph Lipman
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Mathematik, Universität Innsbruck, Innsbruck, A-6020, Österreich

    Herwig Hauser

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Joseph Lipman

  3. Department of Mathematics, Universiteit Utrecht, Utrecht, 3508 TA, The Netherlands

    Frans Oort

  4. Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, 28049, Spain

    Adolfo Quirós

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Basel AG

About this chapter

Cite this chapter

Lipman, J. (2000). Equisingularity and Simultaneous Resolution of Singularities. In: Hauser, H., Lipman, J., Oort, F., Quirós, A. (eds) Resolution of Singularities. Progress in Mathematics, vol 181. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8399-3_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8399-3_17

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9550-7

  • Online ISBN: 978-3-0348-8399-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation