Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

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 \(3d+2\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alcántara, C.R.: Foliations on \(\mathbb{CP}^2\) of degree \(d\) with a singular point with Milnor number \(d^2+d+1\). Rev. Mat. Complut. 31(1), 187–199 (2018)

    Article  MathSciNet  Google Scholar 

  2. Brunella, M.: Birational Geometry of Foliations. IMPA Monographs 1. Springer, ISBN: 978-3-319-14309-5; 978-3-319-14310-1

  3. Briançon, J.: Description de \(Hilb^n {\mathbb{C}}\{x, y\}\). Invent. Math. 41(1), 45–89 (1977)

    Article  MathSciNet  Google Scholar 

  4. Camacho, C., Lins Neto, A., Sad, P.: Minimal sets of foliations on complex projective spaces. Publ. Math. 68, 187–203 (1988)

    Article  MathSciNet  Google Scholar 

  5. Campillo, A., Olivares, J.: A plane foliation of degree different from 1 is determined by its singular scheme. C. R. Acad. Sci. Paris Ser. I 328, 877–882 (1999)

    Article  MathSciNet  Google Scholar 

  6. Campillo, A., Olivares, J.: Polarity with respect to a foliation and Cayley–Bacharach theorems. J. Reine Angew. Math. 534, 95–118 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Cerveau, D., Déserti, J., Garba Belko, D., Meziani, R.: Géométrie classique de certains feuilletages de degré deux. Bull. Braz. Math. Soc. New Ser. 41(2), 161–198 (2010)

    Article  Google Scholar 

  8. Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math. 90, 511–521 (1968)

    Article  MathSciNet  Google Scholar 

  9. Gómez-Mont, X., Ortiz-Bobadilla, L.: Sistemas dinámicos holomorfos en superficies. Aportaciones Matemáticas: Notas de Investigación [Mathematical Contributions: Research Notes], 3. Sociedad Matemática Mexicana, México (1989)

  10. Gotzmann, G.: A stratification of of the Hilbert scheme of points in the projective plane. Math. Z. 199(4), 539–547 (1988)

    Article  MathSciNet  Google Scholar 

  11. Joanoulou, J.P.: Equations de Pfaff algébriques. (French) [Algebraic Pfaffian equations] Lecture Notes in Mathematics, vol. 708. Springer, Berlin (1979)

  12. Lins Neto, A., Soares, M.G.: Algebraic solutions of one-dimensional foliations. J. Differ. Geom. 43, 652–673 (1996)

    Article  MathSciNet  Google Scholar 

  13. Stròzyna, E.: The analytic and formal normal form for the nilpotent singularity. The case of generalized saddle-node. Bull. Sci. Math. 126(7), 555–579 (2002)

    Article  MathSciNet  Google Scholar 

  14. Takens, F.: Singularities of vector fields. Inst. Hautes Etudes Sci. Publ. Math. 43, 47–100 (1974)

    Article  MathSciNet  Google Scholar 

  15. Zoladek, H.: New examples of holomorphic foliations without algebraic leaves. Stud. Math. 131(2), 137–142 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Funding

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

Author information

Authors and Affiliations

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

    Claudia R. Alcántara

  2. Cimat, Callejón Jalisco s/n, A.P. 402, C.P. 36000, Guanajuato, Gto, Mexico

    Rubí Pantaleón-Mondragón

Authors
  1. Claudia R. Alcántara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rubí Pantaleón-Mondragón
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claudia R. Alcántara.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alcántara, C.R., Pantaleón-Mondragón, R. Foliations on \(\mathbb {CP}^2\) with a unique singular point without invariant algebraic curves. Geom Dedicata 207, 193–200 (2020). https://doi.org/10.1007/s10711-019-00492-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-019-00492-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation