Invariant Surfaces for Toric Type Foliations in Dimension Three

TL;DR

This paper proves that toric type foliations in three dimensions without saddle-nodes always have an invariant surface, extending previous results to include dicritical components and using global invariant curves.

Contribution

It extends the existence of invariant surfaces to three-dimensional toric type foliations with dicritical components, generalizing prior work to a broader class.

Findings

Every toric type foliation on (C3, 0) without saddle-nodes has an invariant surface.

Invariant surfaces are constructed as germs along the singular locus and global invariant curves.

The proof extends Cano-Cerveau's argument to dicritical components, utilizing Ortiz-Rosales-Voronin's results.

Abstract

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, done 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 global curves. We build the invariant surface as a germ along the singular locus and those global invariant curves. The result of Ortiz-Rosales-Voronin, about the distribution of invariant curves in dimension two, is a key argument in our proof.