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

Abstract

In this paper we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a reduced curve which is invariant by a foliation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the complex projective plane then one has <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Superscript ModifyingBelow ring With bar Baseline upper C less-than-or-equal-to partial-differential Superscript ModifyingBelow ring With bar Baseline script upper F plus 2 plus a"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:munder> <mml:mo>∘</mml:mo> <mml:mo>_</mml:mo> </mml:munder> </mml:mrow> </mml:msup> <mml:mi>C</mml:mi> <mml:mo>≤</mml:mo> <mml:msup> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:munder> <mml:mo>∘</mml:mo> <mml:mo>_</mml:mo> </mml:munder> </mml:mrow> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial ^{\underline {\circ }} C\leq \partial ^{\underline {\circ }} \mathcal F+2+a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an integer obtained from a concrete problem of imposing singularities to projective plane curves. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is nondicritical or if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has only nodes as singularities, then one gets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a equals 0"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">a=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.