Fano 4-folds with \(b_{2}>12\) are products of surfaces

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