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)).


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


Zbl 0521.14014
Full Text: DOI


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