Equisingularity and Simultaneous Resolution of Singularities

  • Joseph Lipman
Part of the Progress in Mathematics book series (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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A]
    Abhyankar, S. S.: A criterion of equisingularity, Amer. J. Math 90 (1968), 342–345.MathSciNetMATHCrossRefGoogle Scholar
  2. [[BM]
    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.MathSciNetMATHGoogle Scholar
  3. [BH]
    Briançon, J., Henry, J. P. G.: Équisingularité générique des familles de surfaces a singularité isolée, Bull. Soc. Math. France 108 (1980), 259–281.MathSciNetMATHGoogle Scholar
  4. [BS]
    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. [DM]
    van den Dries, L., Miller, C.: Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497–540.MathSciNetMATHCrossRefGoogle Scholar
  6. [EV]
    Encinas, S., Villamayor, O.: Good points and constructive resolution of singularities, Acta Math. 181 (1998), 109–158.MathSciNetMATHCrossRefGoogle Scholar
  7. [G]
    Gau, Y.-N.: Embedded topological classification of quasi-ordinary singularities, Memoirs Amer. Math. Soc. 388, 1988.Google Scholar
  8. [GL]
    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. [GM]
    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. [GMc]
    Goresky, M., MacPherson, R.: Stratified Morse Theory, Springer-Verlag, 1988.CrossRefGoogle Scholar
  11. [H1]
    Hironaka, H.: Normal cones in analytic Whitney stratifications, Publ. Math. IHES 36 (1969), 127–138.MathSciNetMATHGoogle Scholar
  12. [H2]
    Hironaka, H.: On Zariski dimensionality type, Amer. J. Math. 101 (1979), 384–419.MathSciNetMATHCrossRefGoogle Scholar
  13. [K]
    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. [L]
    Laufer, H: Strong simultaneous resolution for surface singularities, in “Complex Analytic Singularities,” North-Holland, Amsterdam-New York, 1987, pp. 207–214.Google Scholar
  15. [LM]
    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. [LT]
    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. [L1]
    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. [L2]
    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. [L3]
    Lipman, J.: Topological invariants of quasi-ordinary singularities, Memoirs Amer. Math. Soc. 388, 1988.Google Scholar
  20. [Lu]
    Luengo, I.: A counterexample to a conjecture of Zariski, Math. Ann. 267, (1984), 487–494.MathSciNetMATHCrossRefGoogle Scholar
  21. [Mc]
    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.MathSciNetGoogle Scholar
  22. [Ml]
    Mather, J.: Stratifications and Mappings, in Dynamical Systems, Ed. M. M. Peixoto, Academic Press, New York, 1973, pp. 195–232.Google Scholar
  23. [M2]
    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. [O]
    Orbanz, U.: Embedded resolution of algebraic surfaces, after Abhyankar (characteristic 0), in “Resolution of Surface Singularities,” Springer Lecture Notes 1101 (1984), pp. 1–50.MathSciNetGoogle Scholar
  25. [P]
    Parusiúski, A.: Lipschitz stratification of subanalytic sets, Ann. Scient. Éc. Norm. Sup. 27 (1994), 661–696.Google Scholar
  26. [S]
    Speder, J.-P.: Equisingularité et conditions de Whitney, Amer. J. Math. 97 (1975)Google Scholar
  27. [T1]
    Teissier, B.: Résolution Simultanée—II, in “Séminaire sur les Singularités des Surfaces,” Lecture Notes in Math. 777, Springer-Verlag, 1980, pp. 82–146.MathSciNetGoogle Scholar
  28. [T2]
    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. [T3]
    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. [Va]
    Varchenko, A. N.: The relation between topological and algebro-geometric equisingularities according to Zariski, Funct. Anal. Appl. 7 (1973), 87–90.Google Scholar
  31. [Ve]
    Verdier, J.-L.: Stratifications de Whitney et théorème de Bertini-Sard, Inventiones math. 36 (1976), 295–312.MathSciNetMATHCrossRefGoogle Scholar
  32. [Vi]
    Villamayor, O.: On equiresolution and a question of Zariski, preprint.Google Scholar
  33. [Wa]
    Wahl, J.: Equisingular deformations of normal surface singularities, Annals of Math. 104 (1976), 325–356.MathSciNetMATHCrossRefGoogle Scholar
  34. [Wh]
    Whitney, H.: Tangents to an analytic variety, Annals of Math. 81 (1965), 496–549.MathSciNetMATHCrossRefGoogle Scholar
  35. [Z1]
    Zariski, O.: Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481–491. (Reprinted in [Z3], pp. 238–248.)MathSciNetMATHCrossRefGoogle Scholar
  36. [Z2]
    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].)MathSciNetMATHCrossRefGoogle Scholar
  37. [Z3]
    Zariski, O.: Collected Papers, vol. IV, MIT Press, Cambridge, Mass., 1979.MATHGoogle Scholar

Copyright information

© Springer Basel AG 2000

Authors and Affiliations

  • Joseph Lipman
    • 1
  1. 1.Dept. of MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations