A construction of smooth varieties admitting small contractions

Given a smooth variety together with two smooth subvarieties, we construct, via two successive blowups, smooth varieties admitting small contractions.

This generalizes Kawamata's example of small contraction in dimension 4.

We also construct the flip of the contraction explicitly.

As an application, from products of two del Pezzo surfaces we obtain smooth weak Fano fourfolds that admit small contractions with Picard number up to 10.