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Moscow Mathematical Journal

Volume 22, Issue 3, July–September 2022  pp. 493–520.

Newton Non-Degenerate Foliations on Projective Toric Surfaces

Authors:  Beatriz Molina-Samper (1)
Author institution:(1) Dpto. Álgebra, Análisis Matemático, Geometría y Topología, Facultad de Ciencias, Universidad de Valladolid, Campus Miguel Delibes, Paseo de Belén, 7, 47011 Valladolid, Spain


We prove that the isolated invariant branches of a weak toric type generalized curve defined over a projective toric ambient surfaces extend to projective algebraic curves. To do it, we pass through the characterization of the weak toric type foliations in terms of “Newton non-degeneracy” conditions, in the classical sense of Kouchnirenko and Oka. Finally, under the strongest hypothesis of being a toric type foliation, we find that there is a dichotomy: Either it has rational first integral but does not have isolated invariant branches or it has finitely many global invariant curves and all of them are extending isolated invariant branches.

2020 Math. Subj. Class. 32S65, 14M25, 14E15.

Keywords: Singular foliations, invariant curves, Newton polygons, toric surfaces.

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