Deformation of Rational Curves Along Foliations

TL;DR

This paper investigates how rational curves deform along foliations, revealing that certain codimension one foliations with non-algebraic leaves are transversely homogeneous, and applies this to classify degree three foliations on projective 3-space.

Contribution

It establishes a link between the tangential foliation structure and transversely homogeneous foliations with non-algebraic leaves, and provides a classification result for degree three foliations on P^3.

Findings

Codimension one foliations with non-algebraic leaves are transversely homogeneous.

The structure group is determined by the codimension of the non-algebraic leaf.

A classification theorem for degree three foliations on P^3.

Abstract

Deformation of morphisms along leaves of foliations define the tangential foliation on the corresponding space of morphisms. We prove that codimension one fo-liations having a tangential foliation with at least one non-algebraic leaf are transversely homogeneous with structure group determined by the codimension of the non-algebraic leaf in its Zariski closure. As an application, we provide a structure theorem for degree three foliations on $P^3$ .