The Baum-Bott indexes are important invariants of singular holomorphic foliations by curves with isolated singularities. If a foliation has a non-degenerate singularity, its indexes at that point can be easily calculated using the eigenvalues of the linear part of a germ of a vector field, which defines the foliation at a neighborhood of the singular point. The Baum-Bott map is defined on the space of one-dimensional holomorphic foliations on a compact complex manifold with a fixed cotangent bundle to the foliation. This map associates to a foliation its Baum-Bott indexes at each singular point. We concentrate on foliations on the complex projective space. The generic rank of this map on the space of one-dimensional foliations on the projective plane is already known. We give an upper bound of the generic rank of the Baum-Bott map for foliations on projective spaces, the number depends on the degree of the foliation and the dimension of the projective space. Moreover, we extend the known results for the projective plane and determine the generic rank for degree-two foliations on even-dimensional projective spaces, as well as for degree up to eight on the three-dimensional projective space. Additionally, we study the rank at the Jouanolou foliation.