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

Abstract

We prove that the isolated invariant branches of a weak toric type generalized curve de fined over a projective toric ambient sur- faces 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 nd that there is a dichotomy: Either it has rational fi rst 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.