Non-degenerate jumps of Milnor numbers of quasihomogeneous singularities
Volume 123 / 2019
Annales Polonici Mathematici 123 (2019), 369-386
MSC: Primary 32S05; Secondary 32S30.
DOI: 10.4064/ap180914-15-4
Published online: 23 September 2019
Abstract
Let be a holomorphic function germ having an isolated critical point at 0\in \mathbb {C}^n and let [f_0] be the singularity generated by f_0, i.e. the equivalence class of f_0 with respect to right-left holomorphic equivalence. The non-degenerate jump of Milnor number \lambda ^{ {\rm nd}}(f_0) of f_0 is the minimal non-zero difference between the Milnor number of f_0 and the Milnor number of a generic element of (f_t) among all holomorphic non-degenerate deformations (f_t) of f_0. For the class [f_0] we define \lambda ^{ {\rm nd}}([f_0]) as the minimum of \lambda ^{ {\rm nd}}(g_0) over g_0\in [f_0]. We give a formula for \lambda ^{ {\rm nd}}([f_0]) when f_0 is quasihomogeneous in two variables.
