On foliations by curves with singularities of positive dimension

TL;DR

This paper provides new enumerative formulas for holomorphic foliations by curves on projective spaces with positive-dimensional singularities, improving previous results and bounding isolated singularities within invariant subvarieties.

Contribution

It introduces improved enumerative results for foliations with positive-dimensional singularities and bounds the number of isolated singularities in invariant subvarieties.

Findings

Enhanced enumerative formulas for foliations with positive-dimensional singularities.

Bounding the number of isolated singularities in invariant subvarieties.

Improved upon previous results by Corrêa et al.

Abstract

We present enumerative results for holomorphic foliations by curves on $\mathbb{P}^n$, $n\geq 3$, with singularities of positive dimension. Some of the results presented improve previous ones due to Corr\^ea--Fern\'andez-P\'erez--Nonato Costa--Vidal Martins and Nonato Costa. We also present an enumerative result bounding the number of isolated singularities in a projective subvariety invariant by a holomorphic foliation by curves on $\mathbb{P}^n$ with a singularity of positive dimension.