Global Invariant Branches of Non-degenerate Foliations on Projective Toric Surfaces

Abstract: We show that the isolated invariant branches globalize to algebraic curves, when we consider weak toric type complex hyperbolic foliations on projective toric ambient surfaces. To do it, we pass through a characterization of weak toric type foliations in terms of "non-degeneracy" conditions, associated to Newton polygons. We also give a description of the relationship between invariant algebraic curves and isolated invariant branches, valid for the case of toric type, by means of the following dichotomy. Either there is a rational first integral and there are no isolated invariant branches or we have only finitely many global invariant curves, all of them extending isolated invariant branches.