Fano 4-folds with rational fibrations

We study (smooth, complex) Fano 4-folds X having a rational contraction of fiber type, that is, a rational map X Y that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to the existence of a nonzero, nonbig movable divisor on X. Our main result is that if Y is not1 or2, then the Picard number ρX of X is at most 18, with equality only if X is a product of surfaces. We also show that if a Fano 4-fold X has a dominant rational map X Z, regular and proper on an open subset of X, with dim(Z) = 3, then either X is a product of surfaces, or ρX is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry.