Classifying Fano 4-folds with a rational fibration onto a 3-fold

We study smooth, complex Fano 4-folds X with a rational contraction onto a 3-fold, namely a rational map X-->Y that factors as a sequence of flips X-->X' followed by a surjective morphism X'->Y with connected fibers, where Y is normal, projective, and dim Y=3. We show that if X has a rational contraction onto a 3-fold and X is not a product of del Pezzo surfaces, then the Picard number rho (X) of X is at most 9; this bound is sharp. As an application, we show that every Fano 4-fold X with rho (X) =12 is isomorphic to a product of surfaces, thus improving the result by the first named author that shows the same for rho (X) >12. We also give a classification result for Fano 4-folds X, not products of surfaces, having a "special" rational contraction X-->Y with dim Y=3, rho (X) -rho (Y) =2, and rho (X) >6; we show that there are only three possible families. Then we prove that the first family exists if rho (X) =7, and that the second family exists if and only if rho (X) =7. This provides the first examples of Fano 4-folds with rho (X) >6 different from products of del Pezzo surfaces and from the Fano models of the blow-up of P⁴ in points. We also construct three new families with rho (X) =6. Finally we show that if a Fano 4-fold X has Lefschetz defect delta (X) =2, then rho (X) is at most 6; this bound is again sharp.