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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

Citations:

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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