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Jump of Milnor numbers. (English) Zbl 1131.32015
In this article, the author studies the behaviour of the Milnor number in a 1-parameter deformation of an analytic germ of an isolated plane curve singularity. More precisely, he is interested in the smallest possible non-zero difference $$\lambda(f_0)=\mu(f_0)-\mu(f_s)$$ for all possible deformations, where $$s$$ denotes the deformation parameter. In particular, he shows (in a different way than S. Gusein-Zade) that considering 1-parameter deformations of an irreducible, Newton-nondegenerate plane curve singularity, this invariant is 1. To this end, he first considers only deformations in which all fibres (except $$f_0$$) are Newton-nondegenerate and establishes a characterization of the smallest difference of Milnor numbers for these deformations.

##### MSC:
 32S30 Deformations of complex singularities; vanishing cycles 32S20 Global theory of complex singularities; cohomological properties 14H20 Singularities of curves, local rings
##### Keywords:
plane curve singularities; deformation; Milnor number
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