The Milnor number of a function on a space curve germ

Abstract

Given a finite function germ f:(X, 0) → (ℂ, 0) on a reduced space curve singularity (X, 0), we show that μ(f) = μ(X, 0) + deg(f) − 1. Here, μ(f) and μ(X, 0) denote the Milnor numbers of the function and the curve, respectively, and deg(f) is the degree of f. We use this formula to obtain several consequences related to the topological triviality and Whitney equisingularity of families of curves and families of functions on curves.