Radial foliations in dimension three
Felipe Cano, Beatriz Molina-Samper
TL;DR
The paper classifies radial foliated spaces in dimension three as exactly open-book foliations, realized after a controlled sequence of blow-ups. It develops a comprehensive framework for reduction of singularities in three dimensions, introducing almost radial spaces, Hirzebruch tubes, and a detailed analysis of monoidal and quadratic blow-ups, leading to the open-book model $\omega=ydz-zdy$ with $E^0\subset (xyz=0)$. The authors extend the 2D radial theory to 3D, establish a two-dimensional groundwork via Cart-wheel and Hirzebruch surface results, and provide a thorough treatment of the general case, including the persistence of invariant hypersurfaces in almost radial settings. The work thus provides a robust structural classification and tools for understanding desingularization behavior, with implications for local Brunella-type alternatives and the existence of invariant surfaces.
Abstract
Radial germs of holomorphic foliations in dimension two have a characteristic property: they are the only singular foliations whose reduction of singularities has no singular points. We also know that they are desingularized by a single dicritical blowing-up. Let us say that a foliated space ((C3, 0),E,F) is almost radial when it has a reduction of singularities without singular points; it will be "radial" under a certain additional condition on the morphism of reduction of singularities. We show that the radial condition corresponds to the "open book" situation. We end the paper with a discussion on the general almost radial case.