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

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 such foliations by curves up to degree 3, also describing the possible singular schemes. In particular, we prove that foliations by curves of degree 1 or 2 are either contained on a pencil of planes or legendrian, and are given by the complete intersection of two codimension one distributions. 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.