ON THE PICARD NUMBER OF DIVISORS IN FANO MANIFOLDS

Let X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in X. We consider the image N-1(D, X) of N-1(D) in N-1(X) under the natural push-forward of 1-cycles. We show that rho(X) - rho(D) <= codim N-1(D, X) <= 8. Moreover if codim N-1(D, X) >= 3, then either X congruent to S x T where S is a Del Pezzo surface, or codim N-1(D, X) = 3 and X has a fibration in Del Pezzo surfaces onto a Fano manifold T such that rho(X) - rho(T) = 4.