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Degree of the exceptional component of foliations of degree two and codimension one in \({\mathbb {P}}^{3}\)

  • A. RossiniEmail author
  • I. Vainsencher
Original Paper
  • 10 Downloads

Abstract

The purpose of this work is to obtain the degree of the exceptional component of the space of holomorphic foliations of degree two and codimension one in \({\mathbb {P}}^3\). This component is the closure of the orbit of the foliation defined by the differential form
$$\begin{aligned} \omega =(3fdg-2gdf)/x_0,\quad \text{ where } f=x_0^2x_3-x_0x_1x_2+\dfrac{x_1^3}{3},\ g=x_0x_2-\dfrac{x_1^2}{2} \end{aligned}$$
under the natural action of the group of automorphisms of \({\mathbb {P}}^3\). Our first task is to unravel a geometric characterization of the pair gf. This leads us to the construction of a parameter space as an explicit fiber bundle over the variety of complete flags. Using tools from equivariant intersection theory, especially Bott’s formula, the degree is expressed as an integral over our parameter space.

Keywords

Exceptional component Foliations Degree Bott’s formula 

Mathematics Subject Classification

14N99 57R30 

Notes

Acknowledgements

Our hearts and minds go with Flaviano Bahia P. Vieira (1984–2011). He has worked on this subject, especially during a visit at the UVa. His untimely death has deprived us of the company of a beloved friend. Thanks are due to the referees for several suggestions/corrections.

References

  1. 1.
    Calvo Andrade, O.: Irreducible components of the space of holomorphic foliations. Math. Ann. 229, 751–767 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Calvo-Andrade, O., Cukierman, F.: A note on the \(\jmath \) invariant and foliations. In: Actas del Congreso Latinoamericano de Matematica (2006). arXiv:0611595v1.pdf [math]
  3. 3.
    Calvo Andrade, O., Cerveau, D., Giraldo, L., Lins Neto, A.: Irreducible components of the space of foliations associated to the affine Lie algebra. Ergod. Theory Dyn. Syst. 24, 987–1014 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cerveau, D., Lins Neto, A.: Irreducible components of the space of holomorphic foliations of degree two in CP(n), n \(\ge \) 3. Ann. Math. 143(3), 577–612 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cerveau, D., Lins Neto, A., Edixhoven, S.J.: Pull-back components of the space of holomorphic foliations on CP(n), n \(\ge \) 3. J. Algebraic Geom. 10(4), 695–711 (2011)zbMATHGoogle Scholar
  6. 6.
    Cukierman, F., Pereira, J.V., Vainsencher, I.: Stability of foliations induced by rational maps. Annales de la Faculté des Sciences de Toulouse XVIII, 685–715 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ellingsrud, G., Strømme, S.A.: Bott’s formula and enumerative geometry. J. Am. Math. Soc. 9(1), 175–193 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fulton, W.: Intersection Theory. Springer, Berlin (1997)zbMATHGoogle Scholar
  9. 9.
    Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
  10. 10.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics. Springer, New York (2013)zbMATHGoogle Scholar
  11. 11.
    Jouanolou, J.P.: Equations de Pfaff Algebriques. Lectures Notes in Mathematics. Springer, Berlin (1979)CrossRefzbMATHGoogle Scholar
  12. 12.
    Leite, D., Vainsencher, I.: Degrees of spaces of holomorphic foliations of codimension one in \({\mathbb{P}}^n\). J. Pure Appl. Algebra 221, 2791–2804 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
  14. 14.
    Meurer, P.: The number of rational quartics on Calabi–Yau hypersurfaces in weighted projective space p(2, 1\(^4\)). Math. Scand. 78, 63–83 (1996). arXiv:alg-geom/9409001 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Rossini, A.: Degree of the exceptional component of the space of holomorphic foliations of degree two and codimension one in \({\mathbb{P}}^{3}\). arXiv:1806.11340

Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Instituto Federal do Sudeste de Minas GeraisJuiz de ForaBrazil
  2. 2.Universidade Federal de Minas GeraisBelo HorizonteBrazil

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