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F. Cukierman; J. V. Pereira; I. Vainsencher
Stability of foliations induced by rational maps
Annales de la faculté des sciences de Toulouse Sér. 6, 18 no. 4 (2009), p. 685-715, doi: 10.5802/afst.1221
Article PDF | Reviews MR 2590385 | Zbl 1208.32029 | 1 citation in Cedram

Résumé - 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.

Bibliography

[1] Calvo-Andrade (O.).— Deformations of branched Lefschetz pencils. Bol. Soc. Brasil. Mat. (N.S.) 26, no. 1, p. 67-83 (1995).  MR 1339179 |  Zbl 0843.58001
[2] Cerveau (D.) and Lins Neto (A.).— Irreducible components of the space of holomorphic foliations of degree two in CP(n). Ann. of Math., 143, p. 577-612 (1996).  MR 1394970 |  Zbl 0855.32015
[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).  MR 2248154 |  Zbl 1116.32023
[4] Cukierman (F.) and Pereira (J. V.).— Stability of Holomorphic Foliations with Split Tangent Sheaf, preprint (Arxiv). To appear in American J. of Math. arXiv |  MR 2405162 |  Zbl pre05278857
[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).  MR 1644323 |  Zbl 0885.14002
[7] Gómez-Mont (X.) and Lins Neto (A.).— Structural stability of singular holomorphic foliations having a meromorphic first integral. Topology 30, no. 3, p. 315-334 (1991).  MR 1113681 |  Zbl 0735.57014
[8] Grauert (H.) and Remmert (R.).— Theory of Stein spaces. Springer-Verlag (1979).  MR 580152 |  Zbl 0433.32007
[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.  Zbl 0902.14040
[10] Hartshorne (R.).— Algebraic Geometry, Springer-Verlarg, (1977).  MR 463157 |  Zbl 0367.14001
[11] Jouanolou (J. P.).— Équations de Pfaff algébriques. Lecture Notes in Mathematics, 708. Springer, Berlin, (1979).  MR 537038 |  Zbl 0477.58002
[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). Cedram |  MR 1842027 |  Zbl 0997.58001
[14] Muciño-Raymundo (J.).— Deformations of holomorphic foliations having a meromorphic first integral. J. Reine Angew. Math. 461, p. 189-219 (1995).  MR 1324214 |  Zbl 0816.32022
[15] Saito (K.).— On a generalization of de-Rham lemma. Ann. Inst.Fourier (Grenoble) 26, no. 2, vii, p. 165-170 (1976). Cedram |  MR 413155 |  Zbl 0338.13009
[16] Scárdua (B.).— Transversely affine and transversely projective holomorphic foliations. Ann. Sci. École Norm. Sup. (4) 30, no. 2, p. 169-204 (1997). Numdam |  MR 1432053 |  Zbl 0889.32031
[17] Vainsencher (I.).— http://www.mat.ufmg.br/$\widetilde{~}$israel/Publicacoes/Degsfol
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