2021 Invariant surfaces for toric type foliations in dimension three
Felipe Cano, Beatriz Molina-Samper
Publ. Mat. 65(1): 291-307 (2021). DOI: 10.5565/PUBLMAT6512109


A foliation is of toric type when it has a combinatorial reduction of singularities. We show that every toric type foliation on $(\mathbb{C}^3,0)$ without saddle-nodes has invariant surface. We extend the argument of Cano-Cerveau for the nondicritical case to the compact dicritical components of the exceptional divisor. These components are projective toric surfaces and the isolated invariant branches of the induced foliation extend to closed irreducible curves. We build the invariant surface as a germ along the singular locus and those closed irreducible invariant curves. The result of Ortiz-Bobadilla-Rosales-Gonzalez-Voronin about the distribution of invariant branches in dimension two is a key argument in our proof.

Funding Statement

Both authors are supported by the Ministerio de Economía y Competitividad from Spain, under the Project “Algebra y geometría en sistemas dinámicos y foliaciones singulares” (Ref.: MTM2016-77642-C2-1-P). The second author is also supported by the Ministerio de Educación, Cultura y Deporte of Spain (FPU14/02653 grant).


Download Citation

Felipe Cano. Beatriz Molina-Samper. "Invariant surfaces for toric type foliations in dimension three." Publ. Mat. 65 (1) 291 - 307, 2021. https://doi.org/10.5565/PUBLMAT6512109


Received: 26 September 2019; Revised: 18 May 2020; Published: 2021
First available in Project Euclid: 11 December 2020

MathSciNet: MR4185834
Digital Object Identifier: 10.5565/PUBLMAT6512109

Primary: 32S65
Secondary: 14E15 , 14M25

Keywords: combinatorial blowing-ups , invariant surfaces , singular foliations , toric varieties

Rights: Copyright © 2021 Universitat Autònoma de Barcelona, Departament de Matemàtiques


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Vol.65 • No. 1 • 2021
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