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

Abstract

In this paper we prove that if C C is a reduced curve which is invariant by a foliation F \mathcal F in the complex projective plane then one has ∂ ∘ _ C ≤ ∂ ∘ _ F + 2 + a \partial ^{\underline {\circ }} C\leq \partial ^{\underline {\circ }} \mathcal F+2+a where a a is an integer obtained from a concrete problem of imposing singularities to projective plane curves. If F \mathcal F is nondicritical or if C C has only nodes as singularities, then one gets a = 0 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.