Distributions with Unstable Tangent Sheaf on ${\mathbb{P}}^3$

We study codimension one distributions on the projective three-space, focusing on cases where the tangent sheaf of the distribution is nonsplit and unstable. We relate the order of nonstability to the degree of the induced subfoliation by curves, showing that the order of nonstability is bounded. Moreover, we classify the tangent sheaf of the codimension one distributions that admit a subfoliation by curves of degree 1. In other words, assuming the sheaf is nonsplit, we classify the situations in which the tangent sheaf attains the maximal possible order of nonstability.