Weakly quasihomogeneous hypersurface singularities in positive characteristic
\textit{K. Saito} in Invent. Math. 14, 123-142 (1971; Zbl 0224.32011) had given a characterisation of hypersurface singularities over the complex numbers admitting a good \(\mathbb{C}^*\)-action. This characterisation fails in positive characteristic. In the paper under review the authors prove a suitable version of this theorem in positive characteristic. They define a notion of weakly quasihomogeneous local rings and then they prove the following theorem: Let \(A=kx/(f)\) be the complete local ring of an isolated singularity and \({\mathfrak M}\) its maximal ideal. Then the following two conditions are equivalent: (1) \(A\) is weakly homogeneous. (2) There is a derivation \(\delta\in\text{Der}_ k{\mathfrak M}\) which is not nilpotent. The authors apply this theorem to the case of plane curves to derive various corollaries on the relations between the Milnor number, Tjurina number and other invariants of the plane curve.
scientific article; zbMATH DE number 54907