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The link of {f(x, y) + zn = 0} and Zariski's conjecture

Published online by Cambridge University Press:  10 February 2005

Robert Mendris
Affiliation:
Department of Mathematical Sciences, Shawnee State University, 940 Second Street, Portsmouth, OH 45662, [email protected]
András Némethi
Affiliation:
Department of Mathematics, Ohio State University, Columbus, OH 43210, [email protected], [email protected] Rényi Institute of Mathematics, Budapest, POB 127, H-1364, Hungary
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Abstract

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We consider suspension hypersurface singularities of type g = f(x, y) + zn, where f is an irreducible plane curve singularity. For such germs, we prove that the link of g determines completely the Newton pairs of f and the integer n except for two pathological cases, which can be completely described. Even in the pathological cases, the link and the Milnor number of g determine uniquely the Newton pairs of f and n. In particular, for such g, we verify Zariski's conjecture about the multiplicity. The result also supports the following conjecture formulated in this paper: if the link of an isolated hypersurface singularity is a rational homology 3-sphere, then it determines the equisingularity type, the embedded topological type, the equivariant Hodge numbers and the multiplicity of the singularity. The conjecture is verified for weighted homogeneous singularities too.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2005