Classification of the invariants of foliations by curves of low degree on the three-dimensional projective space

TL;DR

This paper classifies the topological and algebraic invariants of foliations by curves of low degree on three-dimensional projective space, revealing their structure and moduli for degrees up to 3.

Contribution

It provides a comprehensive classification of invariants and structures of such foliations, including explicit descriptions for degrees 1 to 3 and special classes for higher degrees.

Findings

Degree 1 or 2 foliations are contained in a pencil of planes or are Legendrian.

Degree 3 foliations have conormal sheaves that split or are instanton bundles.

For higher degrees, Legendrian and null-correlation foliations are characterized and their moduli spaces described.

Abstract

We study foliations by curves on the three-dimensional projective space with no isolated singularities, which is equivalent to assuming that the conormal sheaf is locally free. We provide a classification of the topological and algebraic invariants of the conormal sheaves and singular schemes for such foliations by curves, up to degree 3. In particular, we prove that foliations by curves of degree 1 or 2 are contained in a pencil of planes or are Legendrian, and are given by the complete intersection of two codimension one distributions. Furthermore, we prove that the conormal sheaf of a foliation by curves of degree 3 with reduced singular scheme either splits as a sum of line bundles or is an instanton bundle. For degree larger than 3, we focus on two classes of foliations by curves, namely Legendrian foliations and those whose conormal sheaf is a twisted null-correlation bundle. We give characterizations of such foliations, describe their singular schemes and their moduli spaces.