Abstract
Given a finite map germ f : (X, 0) → (Y, 0) between complex analytic reduced space curves, we look at invariants which control the topological triviality and the Whitney equisingularity in families of this type of map germs. In the case that (Y, 0) is smooth, the main invariant is the Milnor number of a function on a curve. We deduce some applications to the equisingularity of families of finitely determined map germs \({f : (\mathbb{C}^2, 0) \to (\mathbb{C}^2, 0)}\) and \({f : (\mathbb{C}^2, 0) \to (\mathbb{C}^3, 0)}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Calleja-Bedregal, R., Houston, K., Ruas, M.A.S.: Topological triviality of families of singular surfaces. preprint. arXiv:math/0611699
Briançon, J., Galligo, A., Granger, M.: Déformations équisingulières des germes de courbes gauches réduites. Mem. Soc. Math. France (N.S.) 1 (1980/81)
Buchweitz R.O., Greuel G.M.: The Milnor number and deformations of complex curve singularities. Invent. Math. 58, 241–281 (1980)
Fernández de Bobadilla J., Pe-Pereira M.: Equisingularity at the normalisation. J. Topol. 1(4), 879–909 (2008)
Gaffney T.: Polar multiplicities and equisingularity of map germs. Topology 32(1), 185–223 (1993)
Gaffney T., Mond D.: Cusps and double folds of germs of analytic maps \({\mathbb{C}^{2} \rightarrow \mathbb{C}^{2} }\) . J. Lond. Math. Soc. 42, 185–192 (1991)
Gibson, C.G., Wirthmüller, K., du Plessis, A.A., Looijenga, E.J.N.: Topological stability of smooth mappings. Lecture Notes in Mathematics, vol. 552. Springer, Berlin (1976)
Goryunov V.V.: Functions on space curves. J. Lond. Math. Soc. (2) 61(3), 807–822 (2000)
Matsumura H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1986)
Marar W.L., Mond D.: Multiple point schemes for corank 1 maps. J. Lond. Math. Soc. 39, 553–567 (1989)
Marar W.L., Mond D.: Real map-germs with good perturbations. Topology 35(1), 157–165 (1996)
Marar, W.L., Nuño-Ballesteros, J.J., Peñafort-Sanchis, G.: Double point curves for corank 2 map germs from \({\mathbb{C}^2}\) to \({\mathbb{C}^3}\) . Topology Appl. doi:10.1016/j.topol.2011.09.028
Mond D.: Some remarks on the geometry and classification of germs of maps from surfaces to 3-space. Topology 26, 361–383 (1987)
Mond, D., Pellikaan, R.: Fitting ideals and multiple points of analytic mappings. In: Algebraic geometry and complex analysis, Pátzcuaro (1987). Lecture Notes in Mathematics, vol. 1414, pp. 107–161
Mond D., van Straten D.: Milnor number equals Tjurina number for functions on space curves. J. Lond. Math. Soc. II. Ser. 63(1), 177–187 (2001)
Nuño-Ballesteros J.J., Tomazella J.N.: The Milnor number of a function on a space curve germ. Bull. Lond. Math. Soc. 40(1), 129–138 (2008)
Teissier, B.: The hunting of invariants in the geometry of discriminants. In: Real and Complex Singularities, Oslo 1976, pp. 565–678. Sijthoff & Noordhoff International Publishers, Germantown (1977)
Wall C.T.C.: Finite determinacy of smooth map-germs. Bull. Lond. Math. Soc. 13(6), 481–539 (1981)
Zariski O.: Studies in equisingularity. III. Saturation of local rings and equisingularity. Am. J. Math. 90, 961–1023 (1968)
Author information
Authors and Affiliations
Corresponding author
Additional information
J. J. Nuño-Ballesteros has been partially supported by DGICYT Grant MTM2009-08933.
Rights and permissions
About this article
Cite this article
Nuño-Ballesteros, J.J., Tomazella, J.N. Equisingularity of families of map germs between curves. Math. Z. 272, 349–360 (2012). https://doi.org/10.1007/s00209-011-0936-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-011-0936-1