Topological types of quasi-ordinary singularities.(English)Zbl 0791.32018

If $$(X,x)$$ is the germ of an irreducible hypersurface of dimension $$d$$ in $$(\mathbb{C}^{d+1},0)$$ then there is a finite map of analytic germs $$\pi:(X,x) \to (\mathbb{C}^ d,0)$$. If the locus of the discriminant of this map has a normal crossing at $$0 \in \mathbb{C}^ d$$ then the germ $$(X,x)$$ is called a quasi-ordinary singularity. Every plane curve singularity is quasi-ordinary.
An irreducible quasi-ordinary singularity $$(X,x)$$ can be represented as the image of an open neighborhood of 0 in $$\mathbb{C}^ d$$ by the map $$(s_ 1,\dots,s_ d) \to (s^ n_ 1,\dots,s^ n_ d,\zeta (s_ 1,\dots,s_ d))$$, $$n>0$$, where $$\zeta$$ is a convergent power series. The author shows that certain characteristic monomials in the expansion $$\zeta (t_ 1^{{1 \over n}},\dots,t_ d^{{1 \over n}})$$ determine the embedded topology of the pair $$(X,x) \subset (\mathbb{C}^{d+1},0)$$. Vector field methods are used to construct the required homeomorphisms, rather than a saturation theorem of Zariski (for a discussion of the latter approach in the plane case see J. Lipman, Proc. Symp. Pure Math. 40, Part 2, 161-171 (1983; Zbl 0521.14014)).

MSC:

 32S45 Modifications; resolution of singularities (complex-analytic aspects) 14B05 Singularities in algebraic geometry

Zbl 0521.14014
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References:

 [1] Shreeram Abhyankar, On the ramification of algebraic functions, Amer. J. Math. 77 (1955), 575 – 592. · Zbl 0064.27501 [2] Dan Burghelea and Andrei Verona, Local homological properties of analytic sets, Manuscripta Math. 7 (1972), 55 – 66. · Zbl 0246.32007 [3] 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. · Zbl 0658.14004 [4] Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. · Zbl 0356.57001 [5] 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. · JFM 39.0493.01 [6] Joseph Lipman, Topological invariants of quasi-ordinary singularities, Mem. Amer. Math. Soc. 74 (1988), no. 388, 1 – 107. · Zbl 0658.14003 [7] -, 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. [8] -, Quasi-ordinary singularities of embedded surfaces, Thesis, Harvard Univ., 1965. [9] Oscar Zariski, Studies in equisingularity. II. Equisingularity in codimension 1 (and characteristic zero), Amer. J. Math. 87 (1965), 972 – 1006. · Zbl 0146.42502 [10] 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). · Zbl 0168.18903
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