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

Fano 4-folds, flips, and blow-ups of points