Equisingularity of families of map germs between curves
type: Item
instance of: scholarly article
title: Equisingularity of families of map germs between curves
zbMATH Open document ID: 1255.32012
author: J. N. Tomazella
author: J. J. Nuño-Ballesteros
published in: Mathematische Zeitschrift
publication date: 2012-10-22T00:00:00Z
review text: 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.
reviewed by: Jan Stevens
Mathematics Subject Classification ID: 32S30
Mathematics Subject Classification ID: 32S15
Mathematics Subject Classification ID: 58K05
zbMATH DE Number: 6097130
zbMATH Keywords: space curve singularities
zbMATH Keywords: Whitney equisingularity
zbMATH Keywords: topological triviality
zbMATH Keywords: Milnor number
MaRDI profile type: Publication
full work available at URL: s00209-011-0936-1
OpenAlex ID: W2031088225
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DOI: 10.1007/S00209-011-0936-1
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description: scientific article; zbMATH DE number 6097130