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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We classify analytically isolated plane curve singularities defined by weighted homogeneous polynomials , 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 either satisfies the same property as the above does or is homogeneous, then we prove easily that the weights of the above determine the topological type of and conversely. So, this gives another easy proof for the topological classification theorem of quasihomogenous singularities in , which was already known. Also, as an application, it can be shown that for a given , where is a quasihomogeneous holomorphic function with an isolated singularity at the origin or with a positive integer , analytic types of isolated hypersurface singularities defined by are easily classified where is defined just as above.

**[A]**V.I. Arnold,*Normal forms of functions in neighborhoods of degenerate critical points*, Russian Math. Surveys, 29, 10-50 (1974). MR**58:24324****[Br]**Brauner, K.,*Zur Geometrie der Funktionen Zweier Komplexen Vesänderlicken*, Abh. Math. Sem. Hamburg, 6, 1-54 (1928).**[Bu]**Burau, W.,*Kennzeichung der Schlauchknoten*, Abh. Math. Sem. Hamburg, 9, 125-133 (1932).**[K1]**Kang, C.,*Analytic classification of plane curve singularities defined by some homogeneous polynomials*, J. Korean Math. Soc. 30, No.2, 385-397 (1993). MR**94g:14015****[K2]**Kang, C.,*Topological classification of irreducible plane curve singularities in terms of Weierstrass polynomials*, Proc. A.M.S. 123, No.5, 1363-1371 (1995). MR**95f:14048****[L]**Lejeune, M.,*Sur l'équivalence des singularité des courbes algébroïdes planes*, Coefficients de Newton, Centre de Math. del 1'Ecole Polytechnique, 1969.**[S]**Saito, K.,*Quasihomogene isolierte Singularitäten von Hyperflächen*, Invent. Math. 14, 123-142 (1971). MR**45:3767****[Ya]**Yau, S. S.-T,*A remark on moduli of complex hypersurfaces*, Amer. J. Math.**113**, 287-292 (1991). MR**92g:32063****[Yo-Su]**Yoshinaga, E. and Suzuki, M.,*Topological types of quasihomogeneous singularities in*, Topology**18**, 113-116 (1979). MR**80k:32017****[Z1]**Zariski, O,*On the topology of algebroid singularities*, Amer. J. Math. 54, 433-465 (1932).**[Z2]**Zariski, O,*Studies in equisingularity I: Equivalent singularities of plane algebroid curves*, Amer. J. Math. 87, 507-536 (1965). MR**31:2243**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
32S15,
14E15

Retrieve articles in all journals with MSC (2000): 32S15, 14E15

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