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Topological types of quasi-ordinary singularities


Author: Kyungho Oh
Journal: Proc. Amer. Math. Soc. 117 (1993), 53-59
MSC: Primary 32S50; Secondary 32S05, 32S25
MathSciNet review: 1106181
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Abstract: A germ $ (X,x)$ of a complex analytic hypersurface in $ {\mathbb{C}^{d + 1}}$ is quasi-ordinary if it can be represented as the image of an open neighborhood of 0 in $ {\mathbb{C}^d}$ under the map $ ({s_1}, \ldots ,{s_d}) \mapsto (s_1^n, \ldots ,s_d^n,\zeta ({s_1}, \ldots ,{s_d})),\;n > 0$, where $ \zeta $ is a convergent power series. It is shown that the topological type of the singularity $ (X,x) \subset ({\mathbb{C}^{d + 1}},0)$ is determined by a certain set of fractional monomials, called the characteristic monomials, appearing in the fractional power series $ \zeta (t_1^{1/n}, \ldots ,t_d^{1/n})$.


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  • [A] Shreeram Abhyankar, On the ramification of algebraic functions, Amer. J. Math. 77 (1955), 575–592. MR 0071851
  • [BV] Dan Burghelea and Andrei Verona, Local homological properties of analytic sets, Manuscripta Math. 7 (1972), 55–66. MR 0310285
  • [G] Yih-Nan Gau, Embedded topological classification of quasi-ordinary singularities, Mem. Amer. Math. Soc. 74 (1988), no. 388, 109–129. With an appendix by Joseph Lipman. MR 954948
  • [H] Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. MR 0448362
  • [J] H. W. E. Jung, Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderliehen $ x,\;y$ in der Umbegung einer Stelle $ x = a,\;y = b$, J. Reine Angew. Math. 133 (1908), 289-314.
  • [L1] Joseph Lipman, Topological invariants of quasi-ordinary singularities, Mem. Amer. Math. Soc. 74 (1988), no. 388, 1–107. MR 954947, 10.1090/memo/0388
  • [L2] -, Quasi-ordinary singularities of surfaces in $ {\mathbb{C}^3}$, Proc. Sympos. Pure Math., vol. 40, part 2, Amer. Math. Soc., Providence, RI, 1983, pp. 161-171.
  • [L3] -, Quasi-ordinary singularities of embedded surfaces, Thesis, Harvard Univ., 1965.
  • [Z1] Oscar Zariski, Studies in equisingularity. II. Equisingularity in codimension 1 (and characteristic zero), Amer. J. Math. 87 (1965), 972–1006. MR 0191898
  • [Z2] Oscar Zariski, Exceptional singularities of an algebroid surface and their reduction, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 43 (1967), 135–146 (English, with Italian summary). MR 0229648

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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1106181-5
Article copyright: © Copyright 1993 American Mathematical Society