Quasi-elementary contractions of Fano manifolds

Let X be a smooth complex Fano variety. We study quasi-elementary' contractions of fiber type of X, which are a natural generalization of elementary contractions of fiber type. If f : X -> Y is such a contraction, then the Picard numbers satisfy rho X <= rho Y + rho F, where F is a general fiber of f. We show that, if dim Y <= 3 and rho Y >= 4, then Y is smooth and Fano; if moreover rho Y >= 6, then X is a product. This yields sharp bounds on rho X when dim X = 4 and X has a quasi-elementary contraction of fiber type, and other applications in higher dimensions.