Department of Mathematics, Aula Magna.
Let $X$ be a smooth, complex Fano 4-fold, and $b_2$ its second Betti number. We will discuss the following result: if $b_2>12$, then X is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions $f: X\to Y$ such that the image S of the exceptional divisor is a surface, together with my previous work on Fano 4-folds. In particular, given $f: X\to Y$ as above, under suitable assumptions we show that S is a smooth del Pezzo surface with $-K_S$ given by the restriction of $-K_Y$.
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