The generic rank of the Baum–Bott map for foliations of the projective plane

Abstract

Our main result says that the generic rank of the Baum–Bott map for foliations of degree $d,\ d\ge 2$, of the projective plane is $d^2+d$. This answers a question of Gómez-Mont and Luengo and shows that are no other universal relations between the Baum–Bott indexes of a foliation of $\mathbb P^2$ besides the Baum–Bott formula. We also define the Camacho–Sad field for foliations on surfaces and prove its invariance under the pull-back by meromorphic maps. As an application we prove that a generic foliation of degree $d\ge 2$ is not the pull-back of a foliation of smaller degree. In Appendix A we show that the monodromy of the singular set of the universal foliation with very ample cotangent bundle is the full symmetric group.