Proximity inequalities and bounds for the degree of invariant curves by foliations of ℙ_{ℂ}²

Campillo, Antonio; Carnicer, Manuel

doi:10.1090/S0002-9947-97-01898-9

Proximity inequalities and bounds for the degree of invariant curves by foliations of $\mathbb {P}_{\mathbb {C}}^2$
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by Antonio Campillo and Manuel M. Carnicer

Trans. Amer. Math. Soc. 349 (1997), 2211-2228

DOI: https://doi.org/10.1090/S0002-9947-97-01898-9

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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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