Foliations by curves on threefolds

We study the conormal sheaves and singular schemes of one‐dimensional foliations on smooth projective varieties X of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is μ‐stable whenever the tangent bundle TX$TX$ is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on P3$\mathbb {P}^3$ and on a smooth quadric hypersurface Q3⊂P4$Q_3\subset \mathbb {P}^4$. Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on Q3.