Invariant surfaces for toric type foliations in dimension three

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Felipe Cano Torres
Universidad de Valladolid. Departamento de Álgebra y Fundamentos.
Beatriz Molina-Samper
Universidad de Valladolid.Facultad de Ciencias. Departamento de Algebra, An´alisis Matem´atico, Geometr´ıa y Topolog´ıa
https://orcid.org/0000-0001-9105-0086

A foliation is of toric type when it has a combinatorial reduction of singularities. We show that every toric type foliation on (C3, 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 OrtizBobadilla–Rosales-Gonzalez–Voronin about the distribution of invariant branches in
dimension two is a key argument in our proof.

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

Article Details

How to Cite
Cano Torres, Felipe; and Molina-Samper, Beatriz. “Invariant surfaces for toric type foliations in dimension three”. Publicacions Matemàtiques, vol.VOL 65, no. 1, pp. 291–307, https://raco.cat/index.php/PublicacionsMatematiques/article/view/383986.
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