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Foliations on ℂℙ2 of degree 2 with degenerate singularities

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Abstract

In this paper we construct non-singular, locally-closed, algebraic varieties which are sets of foliations on ℂℙ2 of degree 2 with a certain degenerate singularity. We obtain the dimension and closure of these varieties. To do that we construct a stratification (based on GIT, see [7]) of the space of foliations with respect to the action by change of coordinates. We prove that the set of unstable foliations has two irreducible components. We have the following corollary: a foliation of degree 2 defined by a pencil of conics is unstable if and only if the pencil is unstable. Finallywe give another proof of the fact that there are only 4 foliations of degree 2 with a unique singular point (see [5]).

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Guanajuato, Callejón Jalisco s/n A.P. 402, C.P. 36000, Guanajuato GTO., México

    Claudia R. Alcántara

Authors
  1. Claudia R. Alcántara
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Correspondence to Claudia R. Alcántara.

Additional information

Partially supported by CONACyT Grant 165891.

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Alcántara, C.R. Foliations on ℂℙ2 of degree 2 with degenerate singularities. Bull Braz Math Soc, New Series 44, 421–454 (2013). https://doi.org/10.1007/s00574-013-0020-z

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  • DOI: https://doi.org/10.1007/s00574-013-0020-z

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