A structural theorem for codimension-one foliations on Pn, n ≥ 3, with an application to degree-three foliations

Let F be a codimension-one foliation on P-n: in for each point p is an element of P-n we define J (F, p) as the order of the first non-zero jet j(p)(k) (omega) of a holomorphic 1-form omega defining F at p. The singular set of F is sing(F) = {p is an element of P-n vertical bar J (F, p) >= 1}. We prove (main Theorem 1.2) that a foliation F satisfying J (F, p) <= 1 for all p is an element of P-n has a non-constant rational first integral. Using this fact we are able to prove that any foliation of degree-three on P-n, with n >= 3, is either the pull-back of a foliation on P-2, or has a transverse affine structure with poles. This extends previous results for foliations of degree at most two.