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Isolated invariant curves of a foliation


Authors: Nuria Corral and Percy Fernández-Sánchez
Journal: Proc. Amer. Math. Soc. 134 (2006), 1125-1132
MSC (2000): Primary 32S65; Secondary 37F75
DOI: https://doi.org/10.1090/S0002-9939-05-08044-5
Published electronically: August 29, 2005
MathSciNet review: 2196047
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Abstract: We bound the equisingularity type of the set of isolated separatrices of a holomorphic foliation $\mathcal F$ of $({\mathbb C}^2,0)$ in terms of the Milnor number of $\mathcal F$. This result gives a bound for the degree of an algebraic invariant curve $C \subset {\mathbb P}^{2}_{\mathbb C}$ of a foliation $\mathcal G$ of ${\mathbb P}^{2}_{\mathbb C}$ in terms of the degree of $\mathcal G$, provided that all the branches of $C$ are isolated separatrices.


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Additional Information

Nuria Corral
Affiliation: Équipe de Géométrie et Dynamique, Institut de Mathématiques de Jussieu, 175, Université Paris 7, Rue du Chevaleret, 75013 Paris, France
Address at time of publication: Departamento Matemática Aplicada I, Universidad de Vigo, E.U.E.T. Forestal, Campus A Xunqueira, 36005 Pontevedra, España
Email: ncorral@uvigo.es

Percy Fernández-Sánchez
Affiliation: Instituto de Matemática y Ciencias Afines, Universidad de Ingeniería, Pontificia Universidad Católica del Perú, Casa de las Trece Monedas, Jr. Ancash, 536 Lima 1, Perú
Email: pefernan@pucp.edu.pe

DOI: https://doi.org/10.1090/S0002-9939-05-08044-5
Keywords: Singular foliation, isolated separatrix, algebraic invariant curve
Received by editor(s): July 22, 2004
Received by editor(s) in revised form: September 24, 2004, and November 2, 2004
Published electronically: August 29, 2005
Additional Notes: The first author was supported by a Postdoctoral Grant of the Spanish Ministry of Education and partially supported by the research project BFM 2001-2010 and by the Junta de Castilla y León (VA123/04). The second author was supported by a Postdoctoral Grant of the AECI, Spanish Ministry of Foreign Affairs and partially supported by the research project by Concytec (n. 823-2001), Perú
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2005 American Mathematical Society