Semi-simple carrousels and the monodromy

Let U be an open neighborhood of the origin in Cn+1 and let f : (U, 0)->(C, 0) be complex analytic. Let z(0) be a generic linear forin on Cn+1. If the relative polar curve Gamma(1)(f,z0) at the origin is irreducible and the intersection number (Gamma V-1.(f,z0)(f))(0) is prime, then there are severe restrictions on the possible degree n cohomology of the Milnor fiber at the origin. We also obtain some interesting, weaker, results when (Gamma V-1.(f,z0)(f))(0) is not prime.