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

We study smooth, complex Fano 4-folds X with large Picard number ρX, 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 ρX≤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 ρX>6, in which we give some general results of independent interest.