Fano 4-folds with $b_{2}>12$ are products of surfaces

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