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On the Milnor Fiber of a Quasi-ordinary Surface Singularity

Published online by Cambridge University Press:  20 November 2018

Chunsheng Ban ,
Lee J. McEwan  and
András Némethi
Chunsheng Ban
Affiliation:
Department of Mathematics Ohio State University 231 West 18th Avenue Columbus, Ohio U.S.A., email: [email protected]
Lee J. McEwan
Affiliation:
Department of Mathematics Ohio State University 231 West 18th Avenue Columbus, Ohio U.S.A., email: [email protected]
András Némethi
Affiliation:
Department of Mathematics Ohio State University 231 West 18th Avenue Columbus, Ohio U.S.A., email: [email protected]
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Abstract

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We verify a generalization of (3.3) from [Lê73] proving that the homotopy type of the Milnor fiber of a reduced hypersurface singularity depends only on the embedded topological type of the singularity. In particular, using [Zariski68, Lipman83, Oh93, Gau88] for irreducible quasi-ordinary germs, it depends only on the normalized distinguished pairs of the singularity. The main result of the paper provides an explicit formula for the Euler-characteristic of the Milnor fiber in the surface case.

Keywords

14B0514E1532S55
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 1 , 01 February 2002 , pp. 55 - 70
Copyright
Copyright © Canadian Mathematical Society 2002

References

[Abhyankar98] Abhyankar, S. S., Resolution of Singularities of Embedded Algebraic Surfaces. Springer Monographs in Math., 2nd Edition, 1998.Google Scholar
[A’Campo75] A’Campo, N., La fonction zeta d’une monodromie. Comment. Math. Helvetici, 50 (1975), 233–248.Google Scholar
[AGV88] Arnold, V. I., Gusein-Zade, S. M. and Varchenko, A. N., Singularities of differentiable maps. Vol. I and Vol. II, Birkhäuser, 1988.Google Scholar
[BMc00] Ban, C. and McEwan, L. J., Canonical resolution of a quasi-ordinary surface singularity. Canad. J. Math. (6) 52 (2000), 1149–1163.Google Scholar
[BM91] Bierstone, E. and Milman, P., A simple constructive proof canonical resolution of singularities. Effective Methods in Algebraic Geometry, Progr.Math. 94 (1991), 11–30.Google Scholar
[BK86] Brieskorn, E. and Knörrer, H., Plane Algebraic Curves. Birkhäuser, Boston 1986.Google Scholar
[Dimca92] Dimca, A., Singularities and Topology of Hypersurfaces. Universitext, Springer-Verlag, 1992.Google Scholar
[Gau88] Gau, Y.-N., Embedded topological classification of quasi-ordinary singularities. Mem. Amer. Math. Soc. 388(1988).Google Scholar
[GLM97] Gusein-Zade, S. M., Luengo, I. and Melle-Hernández, A., Partial resolutions and the zeta-function of a singularity. Comment. Math. Helv. 72 (1997), 244–256.Google Scholar
[Hironaka64] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero I, II. Ann. of Math. 79 (1964), 109–326.Google Scholar
[Lê73] Lê Duñg, Tráng , Topologie des singularités des hypersurfaces complexes. Astérisque 7–8 (1973), 171–182.Google Scholar
[Lipman65] Lipman, J., Quasi-ordinary singularities of embedded surfaces. Thesis, Harvard University, (1965).Google Scholar
[Lipman83] Lipman, J., Quasi-ordinary singularities of surfaces in C3. Proc. Symp. Pure Math. (2) 40 (1983), 161–172.Google Scholar
[Lipman88] Lipman, J., Topological invariants of quasi-ordinary singularities. Mem. Amer. Math. Soc. 388(1988).Google Scholar
[Milnor68] Milnor, J., Singular points of complex hypersurfaces. Annals of Math. Studies 61, Princeton Univ. Press, 1968.Google Scholar
[Oh93] Oh, K., Topological types of quasi-ordinary singularities. Proc. Amer. Math. Soc. (1) 117 (1993), 53–59.Google Scholar
[Teissier74] Teissier, B., Deformation á type topologique constant I, II. In Séminarie Douady-Verdier 1971–72, Astérisque, 16 (1974), 215–249.Google Scholar
[Villamayor89] Villamayor, O., Constructiveness of Hironaka's resolution. Ann. Scient. École Norm. Sup. (4) 22 (1989), 1–32.Google Scholar
[Yau89] Yau, S.-T. S., Topological types of isolated hypersurface singularities. Contemp. Math. 101 (1989), 303–321.Google Scholar
[Zariski68] Zariski, O., Studies in equisingularity, III. Saturation of local rings and equisingularity. Amer. J. Math. 90 (1968), 961–1023; reprinted in Collected papers 4, 96–158.Google Scholar