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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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