Description:
Let F be a codimension one foliation on Pn : to each point p 2 Pn we set J (F; p) = the order of the first non-zero jet jk p (!) of a holomorphic 1-form ! defining F at p. The singular set of F is sing(F) = fp 2 Pn jJ (F; p) ≥ 1g. We prove (main theorem 2) that a foliation F satisfying J (F; p) ≤ 1 for all p 2 Pn has a non-constant rational first integral. Using this fact, we are able to prove that any foliation of degree three on Pn, n ≥ 3, is either the pull-back of a foliation on P2, or has a transverse affine structure with poles. This extends previous results for foliations of degree ≤ 2.