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A characterization of quasi-homogeneous Gorenstein surface singularities. (English) Zbl 0587.14024
A well known theorem of K. Saito states that an isolated hypersurface singularity admits a $${\mathbb{C}}^*$$ action if and only if the defining function belongs to its own Jacobian ideal. This is equivalent to equality of the Milnor and Tyurina numbers, $$\mu$$ and $$\tau$$ respectively. Here the result (in this form) is generalized to isolated complete intersection surface singularities. More generally, the difference $$\mu$$-$$\tau$$ is expressed in terms of other (nonnegative) analytic invariants. The theory is applied to obtain results about the irregularity and about equisingular deformations. The proof depends on choosing a component C of the exceptional divisor which either has positive genus or meets at least three other components, and extending a derivation from C to the whole divisor.
Reviewer: C.T.C.Wall

##### MSC:
 14J17 Singularities of surfaces or higher-dimensional varieties 32S45 Modifications; resolution of singularities (complex-analytic aspects) 14M10 Complete intersections 14B07 Deformations of singularities 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14B05 Singularities in algebraic geometry
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