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Foliations on \(\mathbb {CP}^2\) with a unique singular point without invariant algebraic curves

  • Claudia R. AlcántaraEmail author
  • Rubí Pantaleón-Mondragón
Original Paper
  • 13 Downloads

Abstract

We prove that a foliation on \(\mathbb {CP}^2\) of degree d with a singular point of type saddle-node with Milnor number \(d^2+d+1\) does not have invariant algebraic curves. We give a family of this kind of foliations. We also present a family of foliations of degree d with a unique nilpotent singularity without invariant algebraic curves for d odd greater than 1. Finally we prove that the space of foliations on \(\mathbb {CP}^2\) of degree \(d \ge 2\) with a unique singular point has dimension at least \(3d+2\).

Keywords

Holomorphic foliation Saddle-node singularity Nilpotent singularity Invariant algebraic curve 

Mathematics Subject Classification (2010)

Primary 37F75 13P10 Secondary: 32S65 

Notes

Funding

Funding was provided by CONACYT (Grant No. 284424).

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de GuanajuatoGuanajuatoMexico
  2. 2.CimatGuanajuatoMexico

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