Classification of Fano fourfolds with large anticanonical base locus

We give a classification of smooth Fano fourfolds such that the base scheme of the anticanonical system is a smooth surface. As a consequence we show that there are exactly 22 deformation families of such manifolds and they are all obtained by the same geometric construction. These 22 families are closely related to the list of smooth Fano threefolds that admit a $\mathbb P^1$-bundle structure.