On dicritical foliations and Halphen pencils

Luís Gustavo Mendes; Paulo Sad

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 1, page 93-109
  • ISSN: 0391-173X

Abstract

top
The aim of this article is to provide information on the number and on the geometrical position of singularities of holomorphic foliations of the projective plane. As an application it is shown that certain foliations are in fact Halphen pencils of elliptic curves. The results follow from Miyaoka’s semipositivity theorem, combined with recent developments on the birational geometry of foliations.

How to cite

top

Mendes, Luís Gustavo, and Sad, Paulo. "On dicritical foliations and Halphen pencils." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 1.1 (2002): 93-109. <http://eudml.org/doc/84469>.

@article{Mendes2002,
abstract = {The aim of this article is to provide information on the number and on the geometrical position of singularities of holomorphic foliations of the projective plane. As an application it is shown that certain foliations are in fact Halphen pencils of elliptic curves. The results follow from Miyaoka’s semipositivity theorem, combined with recent developments on the birational geometry of foliations.},
author = {Mendes, Luís Gustavo, Sad, Paulo},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {holomorphic foliation; singularity; complex projective plane},
language = {eng},
number = {1},
pages = {93-109},
publisher = {Scuola normale superiore},
title = {On dicritical foliations and Halphen pencils},
url = {http://eudml.org/doc/84469},
volume = {1},
year = {2002},
}

TY - JOUR
AU - Mendes, Luís Gustavo
AU - Sad, Paulo
TI - On dicritical foliations and Halphen pencils
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2002
PB - Scuola normale superiore
VL - 1
IS - 1
SP - 93
EP - 109
AB - The aim of this article is to provide information on the number and on the geometrical position of singularities of holomorphic foliations of the projective plane. As an application it is shown that certain foliations are in fact Halphen pencils of elliptic curves. The results follow from Miyaoka’s semipositivity theorem, combined with recent developments on the birational geometry of foliations.
LA - eng
KW - holomorphic foliation; singularity; complex projective plane
UR - http://eudml.org/doc/84469
ER -

References

top
  1. [B-B] P. Baum – R. Bott, On the zeroes of meromorphic vector fields, Essays on Topology and Related Topics (Mémoires dédiés à Georges De Rham), Springer, New York (1970), 29-74. Zbl0193.52201MR261635
  2. [B-P-V] W. Barth – C. Peters – A. Van de Ven, “Compact complex surfaces”, Springer-Verlag, 1984. Zbl0718.14023MR749574
  3. [Br1] M. Brunella, Feuilletages holomorphes sur les surfaces complexes compactes, Ann. Sci. École Norm. Sup. 30 (1997), 569-594. Zbl0893.32019MR1474805
  4. [Br2] M. Brunella, Birational geometry of foliations, First Latin-American Congress of Mathematicians, IMPA, Brazil, 2000. Zbl1073.14022MR1948251
  5. [C-D] F. Cossec – I. Dolgachev, “Enriques surfaces I”, Progress in Mathematics 76, Birkhauser, 1989. Zbl0665.14017MR986969
  6. [F-M] R. Friedman – J. Morgan, “Smooth Four-Manifolds and Complex Surfaces”, Springer-Verlag, 1994. Zbl0817.14017MR1288304
  7. [G-S-V] X. Gomez-Mont – J. Seade – A. Verjovsky, The index of a holomorphic flow with an isolated singularity, Math. Ann. 291 (1991), 737-751. Zbl0725.32012MR1135541
  8. [K] M. Klughertz, Existence d’une intégrale première méromorphe pour des germes de feuilletages à feuilles fermées du plan complexe, Topology 31 (1992), 255-269. Zbl0776.57014MR1167168
  9. [LN] A. Lins Neto, Some examples for the Poincaré and Painlevé problems, to appear in Ann. Sc. Ec. Norm. Sup., 2001. Zbl1130.34301MR1914932
  10. [M-M] J. F. Mattei – R. Moussu, Holonomie et intégrales premières, Ann. Sci. École Norm. Sup. 13 (1980), 469-523. Zbl0458.32005MR608290
  11. [McQ1] M. McQuillan, Diophantine approximations and foliations, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 121-174. Zbl1006.32020MR1659270
  12. [McQ2] M. McQuillan, Non-commutative Mori theory, preprint IHES 2000. 
  13. [Men] L. G. Mendes, Kodaira dimension of holomorphic singular foliations, Bull. Braz. Math. Soc. 31-2 (2000), 127-143. Zbl0979.32017MR1785264
  14. [Mir] R. Miranda, “The basic theory of elliptic surfaces”, ETS Editrice, 1989. Zbl0744.14026MR1078016
  15. [Miy] Y. Miyaoka, Theme and variations - Inequalities between Chern numbers, Sugaku Expositions 4 (1991), 154-176. Zbl0749.14006MR1139548
  16. [Po] H. Poincaré, Sur l’intégration algébrique des équations différentielles du 1er ordre et du 1er degré, Rend. Circ. Mat. Palermo 5 (1891), 161-191. JFM23.0319.01
  17. [S] P. Sad, Regular foliations along curves, Ann. Fac. Sci. Toulouse 8 (1999), 661-675. Zbl0983.32033MR1815160
  18. [Se] A. Seidenberg, Reduction of singularities of the differentiable equation A d x = B d y , Amer. J. Math. 90 (1968), 248-269. Zbl0159.33303MR220710
  19. [Su] M. Suzuki, “Sur les intégrales premières de certains feuilletages analytiques complexes”, Springer Lecture Notes in Math. 670, 53-79. Zbl0391.32017MR521913

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.