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.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Abhyankar, S. S.: A criterion of equisingularity, Amer. J. Math 90 (1968), 342–345.
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.
Briançon, J., Henry, J. P. G.: Équisingularité générique des familles de surfaces a singularité isolée, Bull. Soc. Math. France 108 (1980), 259–281.
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.
van den Dries, L., Miller, C.: Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497–540.
Encinas, S., Villamayor, O.: Good points and constructive resolution of singularities, Acta Math. 181 (1998), 109–158.
Gau, Y.-N.: Embedded topological classification of quasi-ordinary singularities, Memoirs Amer. Math. Soc. 388, 1988.
Gibson, G. C., Looijenga, E., du Plessis, A., Wirthmüller, K.: Topological Stability of Smooth Maps, Lecture Notes in Math. 552, Springer-Verlag, 1976.
Gaffney, T., Massey, D.: Trends in equisingularity theory, to appear in the Proceedings of the Liverpool Conference in Honor of C.T.C. Wall.
Goresky, M., MacPherson, R.: Stratified Morse Theory, Springer-Verlag, 1988.
Hironaka, H.: Normal cones in analytic Whitney stratifications, Publ. Math. IHES 36 (1969), 127–138.
Hironaka, H.: On Zariski dimensionality type, Amer. J. Math. 101 (1979), 384–419.
Kleiman, S. L.: Equisingularity, multiplicity, and dependence, to appear in the Proceedings of a Conference in Honor of M. Fiorentini, publ. Marcel Dekker.
Laufer, H: Strong simultaneous resolution for surface singularities, in “Complex Analytic Singularities,” North-Holland, Amsterdam-New York, 1987, pp. 207–214.
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.
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.
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.
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.
Lipman, J.: Topological invariants of quasi-ordinary singularities, Memoirs Amer. Math. Soc. 388, 1988.
Luengo, I.: A counterexample to a conjecture of Zariski, Math. Ann. 267, (1984), 487–494.
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.
Mather, J.: Stratifications and Mappings, in Dynamical Systems, Ed. M. M. Peixoto, Academic Press, New York, 1973, pp. 195–232.
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.
Orbanz, U.: Embedded resolution of algebraic surfaces, after Abhyankar (characteristic 0), in “Resolution of Surface Singularities,” Springer Lecture Notes 1101 (1984), pp. 1–50.
Parusiúski, A.: Lipschitz stratification of subanalytic sets, Ann. Scient. Éc. Norm. Sup. 27 (1994), 661–696.
Speder, J.-P.: Equisingularité et conditions de Whitney, Amer. J. Math. 97 (1975)
Teissier, B.: Résolution Simultanée—II, in “Séminaire sur les Singularités des Surfaces,” Lecture Notes in Math. 777, Springer-Verlag, 1980, pp. 82–146.
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.
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.
Varchenko, A. N.: The relation between topological and algebro-geometric equisingularities according to Zariski, Funct. Anal. Appl. 7 (1973), 87–90.
Verdier, J.-L.: Stratifications de Whitney et théorème de Bertini-Sard, Inventiones math. 36 (1976), 295–312.
Villamayor, O.: On equiresolution and a question of Zariski, preprint.
Wahl, J.: Equisingular deformations of normal surface singularities, Annals of Math. 104 (1976), 325–356.
Whitney, H.: Tangents to an analytic variety, Annals of Math. 81 (1965), 496–549.
Zariski, O.: Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481–491. (Reprinted in [Z3], pp. 238–248.)
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].)
Zariski, O.: Collected Papers, vol. IV, MIT Press, Cambridge, Mass., 1979.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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