Abstract
For a finite morphism \(\varphi =(f,g)\) from the plane to the plane we describe the topology of the image of a branch in the source by the use of iterated pencils of analytic functions, constructed inductively in a natural way starting from the components of the map. In the case of the study of the topology of the discriminant curve, image by \(\varphi \) of the critical locus of the map, we show that the special fibres of the pencil \( \langle f,g\rangle \) suffice to determine the topological type of each branch of the discriminant curve. This is due to the known relations that exist between the branches of the critical locus of \(\varphi \) and the special fibres.
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References
Casas-Alvero E.: Singularities of plane curves. In: London Mathematical Society Lecture Note Series, vol. 276. Cambridge University Press, Cambridge (2000)
Casas-Alvero, E.: Local geometry of planar morphisms. Asian J. Math. 11(3), 373–426 (2007)
Delgado, F.: An arithmetical factorization for the critical point set of some maps from \({\mathbb{C}}^2\) to \({\mathbb{C}}^2\). In: Jean-Paul B. (ed.) Singularities-Lille 1991. London Mathematical Society Lecture Note Series, vol. 201, pp. 61–100. Cambridge University Press, Cambridge (1994)
Delgado, F., Maugendre, H.: Special fibres and critical locus for a pencil of plane curve singularities. Compositio Math. 136, 69–87 (2003)
Fulton, W.: Intersection theory. Springer, New York (1984)
García Barroso, E., Ploski, A.: An approach to plane algebroid branches. Preprint ArXiv 1208.0913 (2012)
Lê-Dũng Tràng., Weber, C.: Équisingularité dans les pinceaux de germes de courbes planes et \(C^0\)-suffisance. L’Enseignement Mathématique 43, 355–380 (1997)
MacLane, S.: A construction for absolute values in polynomial rings. Trans. Am. Math. Soc. 40(3), 363–395 (1936)
Maugendre, H.: Discriminant of a germ \(\Phi : (^2,0)\rightarrow (^2,0)\) and Seifert fibred manifolds. J. London Math. Soc. 59(1), 207–226 (1999)
Maugendre, H.: Topological invariants of higher order for a pair of plane curve germs. Topol. App. 123, 297–312 (2002)
Zariski, O.: Le problème des modules pour les branches planes (Appendice par Teissier, B.). Hermann, Paris, (1986)
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F. Delgado is partially supported by the Grants MTM2007-64704 and MTM2012-36917-C03-01 (both Grants with the help of FEDER Program). The author is thankful to the Institut Fourier, Université de Grenoble I for hospitality.
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Delgado, F., Maugendre, H. On the topology of the image by a morphism of plane curve singularities. Rev Mat Complut 27, 369–384 (2014). https://doi.org/10.1007/s13163-013-0141-3
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DOI: https://doi.org/10.1007/s13163-013-0141-3
Keywords
- Iterated pencils
- Topological type
- Analytic morphisms
- Maximal contact values
- Discriminant curve
- Critical locus