Constant Milnor number implies constant multiplicity for quasihomogeneous singularities
The author studies that the multiplicity does not change for certain topologically trivial deformations of an isolated hypersurface singularity. His work applies to all \(\mu\)-constant first order deformations and to all \(\mu\)-constant deformations of a quasihomogeneous singularity, i.e., an isolated hypersurface singularity with \({\mathbb{C}}^*\)-action. His argument uses a valuation test of Lê and Saito. He can apply this to arbitrary isolated singularities. It is known that constant Milnor number implies constant multiplicity in some special cases. In the case of quasihomogeneous (semi-quasihomogeneous) singularities, the author gives a positive answer to Zariski's question whether for a hypersurface singularity the multiplicity is an invariant of the topological type. The statement of his result is the following: Let f be a quasi-homogeneous polynomial with isolated singularity and \(F: ({\mathbb{C}}^ n\times {\mathbb{C}},0)\to ({\mathbb{C}},0)\) a \(\mu\)-constant unfolding of f. Then, \(mult(f_ t)=mult(f)\) for small values of t, where \(f_ t(x)=F(x,t)\) for \((x,y)\in {\mathbb{C}}^ n\times {\mathbb{C}}\).
scientific article; zbMATH DE number 3955400