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Proximity inequalities and bounds for the degree of invariant curves by foliations of $\mathbb P^2_{\mathbb C}$


Authors: Antonio Campillo and Manuel M. Carnicer
Journal: Trans. Amer. Math. Soc. 349 (1997), 2211-2228
MSC (1991): Primary 32L30
MathSciNet review: 1407696
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Abstract: In this paper we prove that if $C$ is a reduced curve which is invariant by a foliation $\mathcal F$ in the complex projective plane then one has $\partial ^{\underline {\circ }} C\leq \partial^{\underline {\circ }} \mathcal F+2+a$ where $a$ is an integer obtained from a concrete problem of imposing singularities to projective plane curves. If $\mathcal F$ is nondicritical or if $C$ has only nodes as singularities, then one gets $a=0$ and we recover known bounds. We also prove proximity formulae for foliations and we use these formulae to give relations between local invariants of the curve and the foliation.


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Additional Information

Antonio Campillo
Affiliation: Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Valladolid, 47005 Valladolid. Spain
Address at time of publication: Laboratoire de Mathematiques Emile Picard, UMR CNRS 5580, Univ. Paul Sabatier, U.F.R.-M.I.G., 118 Route de Narbonne, 31062 Toulouse Cedex, France
Email: campillo@cpd.uva.es

Manuel M. Carnicer
Affiliation: Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Valladolid, 47005 Valladolid. Spain
Address at time of publication: Laboratoire de Mathematiques Emile Picard, UMR CNRS 5580, Univ. Paul Sabatier, U.F.R.-M.I.G., 118 Route de Narbonne, 31062 Toulouse Cedex, France
Email: mcarnicer@cpd.uva.es

DOI: https://doi.org/10.1090/S0002-9947-97-01898-9
Received by editor(s): August 22, 1995
Additional Notes: The first author was partially supported by the D.G.I.C. y T. (PB-91-0210-C02-01); the second author was partially supported by the D.G.I.C. y T. (PB-91-0195)
Article copyright: © Copyright 1997 American Mathematical Society