Foliations by Curves with Curves as Singularities

TL;DR

This paper investigates the structure of holomorphic one-dimensional foliations on projective space, focusing on singular loci composed of curves and points, and establishes formulas and conditions relating these singularities to the foliation's invariants.

Contribution

It provides explicit formulas for counting point singularities based on invariants and characterizes when a foliation is uniquely determined by its singular locus.

Findings

Number of point singularities expressed via invariants

Conditions for foliation determination by singular locus

Results on foliations with a single nonzero dimensional singular component

Abstract

Let $\mathcal F$ be a holomorphic one-dimensional foliation on $\mathbb{P}^n$ such that the components of its singular locus $\Sigma$ are curves $C_i$ and points $p_j$. We determine the number of $p_j$, counted with multiplicities, in terms of invariants of $\mathcal F$ and $C_i$, assuming that $\mathcal F$ is special along the $C_i$. Allowing just one nonzero dimensional component on $\Sigma$, we also prove results on when the foliation happens to be determined by its singular locus.