Equisingularity of families of map germs between curves
The authors generalise their earlier results [Bull. Lond. Math. Soc. 40, No. 1, 129--138 (2008; Zbl 1171.32018)] on functions on curves to the case of maps between curves. Let \(F: (\mathcal{X},0)\to (\mathcal{Y},0)\) be a deformation of the map \(f: (X,0)\to (Y,0)\), such that \(X_t\setminus \{0\}\) and \(Y_t\setminus \{0\}\) are smooth. Then \(F\) is topologically trivial if and only if \(\mu(X_t)\), \(\mu(Y_t)\) and \(\deg(f_t)\) are constant, and \(F\) is Whitney equisingular if and only if in addition the multiplicities \(m(X_t)\) and \(m(Y_t)\) are constant. As an application some known equisingularity results for map germs are proved: an unfolding of a finitely determined map germ \(f: (\mathbb{C}^2,0)\to (\mathbb{C}^2,0)\) is Whitney equisingular if and only if the Milnor number of the discriminant curve \(\Delta_t\) is constant; an unfolding of a germ \(f: (\mathbb{C}^2,0)\to (\mathbb{C}^3,0)\) is topologically trivial if and only if the Milnor number of the double point curve \(D^2_t\) is constant.
scientific article; zbMATH DE number 6097130
scientific article