Proximity inequalities and bounds for the degree of invariant curves by foliations of \(\mathbb P^ 2_{\mathbb C}\). (English) Zbl 0873.32030
Summary: 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.
MSC:
| 32S65 | Singularities of holomorphic vector fields and foliations |
Keywords:
invariant curves by foliations; reduced curve; complex projective plane; proximity formulae; local invariantsReferences:
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