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Moduli of plane curve singularities with \({\mathbb{C}}^*\)-action. (English) Zbl 0691.14018

Singularities, Banach Cent. Publ. 20, 255-278 (1988).
[For the entire collection see Zbl 0653.00009.]
This paper describes the moduli space of plane curve singularities with a fixed topological type, i.e. of the topological type of (X,0) defined by the zero set of a weighted homogeneous polynomial over the field of complex numbers. The moduli are described in the following way. Let \(Z\to B\) be a good representative of the versal \(\mu\)-constant deformation, \(\mu\) the Milnor number of (X,0), V the kernel of the Kodaira-Spencer map of the family \(Z\to B\), where Z is a hypersurface in \(B\times {\mathbb{C}}^ 2\). To obtain a moduli space the authors look for the quotient of B by the group \(G=\exp (V)\) and investigate when X is reducible by the action of a finite group. It turns out that the base B has a stratification \(S_{\tau}\) defined by fixing the Tyurina number \(\tau\), i.e. the dimension of the base of the versal deformation of the singularity in the corresponding fibre of the family. The quotient \(S_{\tau}/G\) always exists in the analytic category and \(S_{\tau_{\min}}/G\) is a quasi- smooth algebraic variety, where \(\mu \geq \tau \geq \tau_{\min}.\)
An example showing that this is not true for surface singularities in \({\mathbb{C}}^ 3\) is given, i.e. \(S_{\tau}\) may be empty for some \(\tau\), \(\tau_{\min}\leq \tau \leq \mu\).
Reviewer: V.Vedernikov

MSC:

14H20 Singularities of curves, local rings
14H10 Families, moduli of curves (algebraic)

Citations:

Zbl 0653.00009