Abstract
We show that a holomorphic map germ \({f : (\mathbb{C}^n,0)\to(\mathbb{C}^{2n-1},0)}\) is finitely determined if and only if the double point scheme D(f) is a reduced curve. If n ≥ 3, we have that μ(D 2(f)) = 2μ(D 2(f)/S 2)+C(f)−1, where D 2(f) is the lifting of the double point curve in \({(\mathbb{C}^n\times \mathbb{C}^n,0)}\) , μ(X) denotes the Milnor number of X and C(f) is the number of cross-caps that appear in a stable deformation of f. Moreover, we consider an unfolding F(t, x) = (t, f t (x)) of f and show that if F is μ-constant, then it is excellent in the sense of Gaffney. Finally, we find a minimal set of invariants whose constancy in the family f t is equivalent to the Whitney equisingularity of F. We also give an example of an unfolding which is topologically trivial, but it is not Whitney equisingular.
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Jorge Pérez, V.H., Nuño-Ballesteros, J.J. Finite determinacy and Whitney equisingularity of map germs from \({\mathbb{C}^n}\) to \({\mathbb{C}^{2n-1}}\) . manuscripta math. 128, 389–410 (2009). https://doi.org/10.1007/s00229-008-0240-5
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DOI: https://doi.org/10.1007/s00229-008-0240-5