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Numerische Charakterisierung quasihomogener Gorenstein- Kurvensingularitäten. (German) Zbl 0587.14016

Let (X,O) be a complex reduced curve singularity of any embedding dimension. Generalizing earlier results of Saito for hypersurfaces and of Scheja-Wiebe for complete intersections, E. Kunz and W. Ruppert [Manuscr. Math. 22, 47-62 (1972; Zbl 0364.14006)] have shown that (X,O) is quasi-homogeneous iff there exists a surjective \({\mathcal O}_{X,O}\)-homomorphism from \(\Omega^ 1_{X,0}\) to \({\mathfrak m}_{X,O}\). Using this criterion, the authors are able to show the following result. Suppose that (X,O) is Gorenstein. Then the following conditions are equivalent: \((i)\quad (X,O)\) is quasi-homogeneous. \((ii)\quad \Omega^ 1_{X,O}\) has maximal torsion. \((iii)\quad \mu (X,O)=\dim (E,O),\) where \(\mu\) (X,O) is the Milnor number and E is a smoothing component of a basis of a semi-universal deformation of (X,O). That was known before only for irreducible smoothable curves by the first author [Habilitationsschrift (Bonn 1979)] and R. Waldi [Math. Ann. 242, 201-208 (1979; Zbl 0426.14004)]. The paper concludes with some open problems on the relationship between \(\mu\) (X,0) and dim(E,0) in the general case.
Reviewer: Ngo Viet Trung

MSC:

14H20 Singularities of curves, local rings
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14B05 Singularities in algebraic geometry
14H45 Special algebraic curves and curves of low genus
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References:

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