  In this paper we study smooth, complex Fano 4-folds X with large Picard
number rho(X), with techniques from birational geometry. Our main result is
that if X is isomorphic in codimension one to the blow-up of a smooth
projective 4-fold Y at a point, then rho(X) is at most 12. We give examples of
such X with Picard number up to 9. The main theme (and tool) is the study of
fixed prime divisors in Fano 4-folds, especially in the case rho(X)>6, in which
we give some general results of independent interest.
