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Transactions of the American Mathematical Society

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Analytic types of plane curve singularities defined by weighted homogeneous polynomials


Author: Chunghyuk Kang
Journal: Trans. Amer. Math. Soc. 352 (2000), 3995-4006
MSC (2000): Primary 32S15, 14E15
DOI: https://doi.org/10.1090/S0002-9947-00-02478-8
Published electronically: February 25, 2000
MathSciNet review: 1661266
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Abstract: We classify analytically isolated plane curve singularities defined by weighted homogeneous polynomials $f(y,z)$, which are not topologically equivalent to homogeneous polynomials, in an elementary way. Moreover, in preparation for the proof of the above analytic classification theorem, assuming that $g(y,z)$ either satisfies the same property as the above $f$ does or is homogeneous, then we prove easily that the weights of the above $g$ determine the topological type of $g$ and conversely. So, this gives another easy proof for the topological classification theorem of quasihomogenous singularities in $\mathbb{C}^{2}$, which was already known. Also, as an application, it can be shown that for a given $h$, where $h(w_{1},\dots ,w_{n})$ is a quasihomogeneous holomorphic function with an isolated singularity at the origin or $h(w_{1})=w^{p}_{1}$ with a positive integer $p$, analytic types of isolated hypersurface singularities defined by $f+h$ are easily classified where $f$ is defined just as above.


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Additional Information

Chunghyuk Kang
Affiliation: Department of Mathematics, Seoul National University, Seoul 151–742, Korea
Email: chkang@math.snu.ac.kr

DOI: https://doi.org/10.1090/S0002-9947-00-02478-8
Received by editor(s): May 5, 1998
Published electronically: February 25, 2000
Additional Notes: Supported by MOE, Project No. BSRI-97-1413 and GARC-KOSEF, 1998. Also supported in part by the SNU-97-03-1061
Article copyright: © Copyright 2000 American Mathematical Society

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