Stability of foliations induced by rational maps

F. Cukierman[1]; J. V. Pereira[2]; I. Vainsencher[3]

  • [1] Depto. Matemática, FCEN-UBA, Ciudad Universitaria, 1428 Buenos Aires, Argentina
  • [2] IMPA, Estrada Dona Castorina 110, 22 460-320 Rio de Janeiro, Brasil
  • [3] Depto. Matemática, UFMG, Av. Antonio Carlos 6627, 31 270-901 Belo Horizonte, Brasil

Annales de la faculté des sciences de Toulouse Mathématiques (2009)

  • Volume: 18, Issue: 4, page 685-715
  • ISSN: 0240-2963

Abstract

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We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space ℱ q ( r , d ) of singular foliations of codimension q and degree d on the complex projective space ℙ r , when 1 ≤ q ≤ r - 2 . We study the geometry of these irreducible components. In particular we prove that they are all rational varieties and we compute their projective degrees in several cases.

How to cite

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Cukierman, F., Pereira, J. V., and Vainsencher, I.. "Stability of foliations induced by rational maps." Annales de la faculté des sciences de Toulouse Mathématiques 18.4 (2009): 685-715. <http://eudml.org/doc/10124>.

@article{Cukierman2009,
abstract = {We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space $\{\cal F\}_q(r, d)$ of singular foliations of codimension $q$ and degree $d$ on the complex projective space $\{\{\mathbb\{P\}\}\}^r$, when $1\le q \le r-2$. We study the geometry of these irreducible components. In particular we prove that they are all rational varieties and we compute their projective degrees in several cases.},
affiliation = {Depto. Matemática, FCEN-UBA, Ciudad Universitaria, 1428 Buenos Aires, Argentina; IMPA, Estrada Dona Castorina 110, 22 460-320 Rio de Janeiro, Brasil; Depto. Matemática, UFMG, Av. Antonio Carlos 6627, 31 270-901 Belo Horizonte, Brasil},
author = {Cukierman, F., Pereira, J. V., Vainsencher, I.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {holomorphic foliations; higher codimension; homogeneous forms; irreducible components},
language = {eng},
month = {10},
number = {4},
pages = {685-715},
publisher = {Université Paul Sabatier, Toulouse},
title = {Stability of foliations induced by rational maps},
url = {http://eudml.org/doc/10124},
volume = {18},
year = {2009},
}

TY - JOUR
AU - Cukierman, F.
AU - Pereira, J. V.
AU - Vainsencher, I.
TI - Stability of foliations induced by rational maps
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2009/10//
PB - Université Paul Sabatier, Toulouse
VL - 18
IS - 4
SP - 685
EP - 715
AB - We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space ${\cal F}_q(r, d)$ of singular foliations of codimension $q$ and degree $d$ on the complex projective space ${{\mathbb{P}}}^r$, when $1\le q \le r-2$. We study the geometry of these irreducible components. In particular we prove that they are all rational varieties and we compute their projective degrees in several cases.
LA - eng
KW - holomorphic foliations; higher codimension; homogeneous forms; irreducible components
UR - http://eudml.org/doc/10124
ER -

References

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  1. Calvo-Andrade (O.).— Deformations of branched Lefschetz pencils. Bol. Soc. Brasil. Mat. (N.S.) 26, no. 1, p. 67-83 (1995). Zbl0843.58001MR1339179
  2. Cerveau (D.) and LinsNeto (A.).— Irreducible components of the space of holomorphic foliations of degree two in CP(n). Ann. of Math., 143, p. 577-612 (1996). Zbl0855.32015MR1394970
  3. Coutinho (S. C.) and Pereira (J. V.).— On the density of algebraic foliations without algebraic invariant sets, Crelle’s J. reine angew. Math. 594, p. 117-135 (2006). Zbl1116.32023MR2248154
  4. Cukierman (F.) and Pereira (J. V.).— Stability of Holomorphic Foliations with Split Tangent Sheaf, preprint (Arxiv). To appear in American J. of Math. Zbl1189.32021MR2405162
  5. Cukierman (F.), Pereira (J. V.) and Vainsencher (I.).— preprint http://arxiv.org/abs/0709.4072 
  6. Fulton (W.).— Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag (1998). Zbl0885.14002MR1644323
  7. Gómez-Mont (X.) and LinsNeto (A.).— Structural stability of singular holomorphic foliations having a meromorphic first integral. Topology 30, no. 3, p. 315-334 (1991). Zbl0735.57014MR1113681
  8. Grauert (H.) and Remmert (R.).— Theory of Stein spaces. Springer-Verlag (1979). Zbl0433.32007MR580152
  9. Greuel (G.-M.), Pfister (G.), and Schönemann (H.).— Singular 3.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2005). http://www.singular.uni-kl.de. Zbl0902.14040
  10. Hartshorne (R.).— Algebraic Geometry, Springer-Verlarg, (1977). Zbl0367.14001MR463157
  11. Jouanolou (J. P.).— Équations de Pfaff algébriques. Lecture Notes in Mathematics, 708. Springer, Berlin, (1979). Zbl0477.58002MR537038
  12. Katz (S.) and Stromme (S.A.).— Schubert: a maple package for intersection theory, http://www.mi.uib.no/schubert/ 
  13. de Medeiros (A.).— Singular foliations and differential p -forms. Ann. Fac. Sci. Toulouse Math. (6) 9, no. 3, p. 451-466 (2000). Zbl0997.58001MR1842027
  14. Muciño-Raymundo (J.).— Deformations of holomorphic foliations having a meromorphic first integral. J. Reine Angew. Math. 461, p. 189-219 (1995). Zbl0816.32022MR1324214
  15. Saito (K.).— On a generalization of de-Rham lemma. Ann. Inst.Fourier (Grenoble) 26, no. 2, vii, p. 165-170 (1976). Zbl0338.13009MR413155
  16. Scárdua (B.).— Transversely affine and transversely projective holomorphic foliations. Ann. Sci. École Norm. Sup. (4) 30, no. 2, p. 169-204 (1997). Zbl0889.32031MR1432053
  17. Vainsencher (I.).— http://www.mat.ufmg.br/ ˜ israel/Publicacoes/Degsfol 

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