Abstract
We show that up to automorphism of \({\mathbb {P}}^{2}_{{\mathbb {C}}}\) there are 5 homogeneous convex foliations of degree four on \({\mathbb {P}}^{2}_{{\mathbb {C}}}.\) Using this result, we give a partial answer to a question posed in 2013 by Marín and Pereira about the classification of reduced convex foliations on \({\mathbb {P}}^{2}_{{\mathbb {C}}}.\)
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Artebani, M., Dolgachev, I.V.: The Hesse pencil of plane cubic curves. Enseign. Math. (2) 55(3–4), 235–273 (2009)
Baum, P., Bott, R.: Singularities of holomorphic foliations. J. Differ. Geom. 7, 279–342 (1972)
Brunella, M.: Birational Geometry of Foliations. IMPA Monographs, vol. 1, pp. xiv+130. Springer, Cham (2015)
Bedrouni, S., Marín, D.: Tissus plats et feuilletages homogènes sur le plan projectif complexe. Bull. Soc. Math. Fr. 146(3), 479–516 (2018)
Bedrouni, S.: Feuilletages de degré trois du plan projectif complexe ayant une transformée de Legendre plate. PhD thesis, University of Sciences and Technology Houari Boumediene (2017). arXiv:abs/1712.03895
Camacho, C., Sad, P.: Invariant varieties through singularities of holomorphic vector fields. Ann. Math. (2) 115(3), 579–595 (1982)
Cerveau, D., Déserti, J., Belko, D.Garba, Meziani, R.: Géométrie classique de certains feuilletages de degré deux. Bull. Braz. Math. Soc. (N.S.) 41(2), 161–198 (2010)
Cordwell, K., Gilbertson, S., Nuechterlein, N., Pilgrim, K.M., Pinella, S.: On the classification of critically fixed rational maps. Conform. Geom. Dyn. 19, 51–94 (2015)
Favre, C., Pereira, J.V.: Webs invariant by rational maps on surfaces. Rend. Circ. Mat. Palermo (2) 64(3), 403–431 (2015)
Marín, D., Pereira, J.V.: Rigid flat webs on the projective plane. Asian J. Math. 17(1), 163–191 (2013)
Pereira, J.V.: Vector fields, invariant varieties and linear systems. Ann. Inst. Fourier (Grenoble) 51(5), 1385–1405 (2001)
Pereira, J.V., Pirio, L.: An Invitation to Web Geometry. IMPA Monographs, 2, pp. xvii+213. Springer, Cham (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
D. Marín acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through grant MTM2015-66165-P and the “María de Maeztu” Programme for Units of Excellence in R&D (MDM-2014-0445).
Rights and permissions
About this article
Cite this article
Bedrouni, S., Marín, D. Convex foliations of degree 4 on the complex projective plane. Math. Z. 295, 381–394 (2020). https://doi.org/10.1007/s00209-019-02356-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-019-02356-z