Analytic types of plane curve singularities defined by weighted homogeneous polynomials

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.