On codimension one holomorphic distributions on compact toric orbifolds

Abstract

The number of singularities, counted with multiplicity, of a generic codimension one holomorphic distribution on a compact toric orbifold is determined. As a consequence, we give the classification of regular distributions on rational normal scrolls and weighted projective spaces. Under certain conditions, we also prove that the singular set of a codimension one holomorphic foliation on a toric orbifold admits at least one irreducible component of codimension two, and we give a Darboux-Jouanolou type integrability theorem for codimension one holomorphic foliations. We illustrate our results with several examples.