Foliations on $\mathbb{CP}^3$ of degree $2$ that have a line as singular set

TL;DR

This paper classifies degree 2 codimension 1 foliations on complex projective 3-space with a line as the singular set, describing their components and boundary structures, and constructs examples for higher degrees.

Contribution

It provides a complete classification of such foliations, describes the boundary of the exceptional component, and constructs new examples for degrees greater than or equal to 3.

Findings

Boundary of the exceptional component has only 3 foliations up to coordinate change.

The boundary is contained in a logarithmic component.

Constructed examples of higher degree foliations with specific properties.

Abstract

In this work we classify foliations on $\mathbb{CP}^3$ of codimension 1 and degree $2$ that have a line as singular set. To achieve this, we do a complete description of the components. We prove that the boundary of the exceptional component has only 3 foliations up to change of coordinates, and this boundary is contained in a logarithmic component. Finally we construct examples of foliations on $\mathbb{CP}^3$ of codimension 1 and degree $s \geq 3$ that have a line as singular set and such that they form a family with a rational first integral of degree $s+1$ or they are logarithmic foliations where some of them have a minimal rational first integral of degree not bounded.