Fano 4-folds with b2>12 are products of surfaces

Let X be a smooth, complex Fano 4-fold, and ρ X its Picard number. We show that if ρ X > 12 , then X is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions f : X → Y such that dim f ( Exc ( f ) ) = 2 , together with the author’s previous work on Fano 4-folds. In particular, given f : X → Y as above, under suitable assumptions we show that S : = f ( Exc ( f ) ) is a smooth del Pezzo surface with − K S = ( − K Y ) | S .