A famous theorem of Shokurov states that a general anticanonical divisor of a smooth Fano threefold is a smooth K3 surface. In a joint work with Andreas Höring we proved that for four-dimensional Fano manifolds the behaviour is completely opposite: if the anticanonical base locus is a normal surface, all the anticanonical divisors are singular.

In this talk I will present our follow-up result, namely the classification of smooth Fano fourfolds with scheme-theoretic base locus a smooth surface: they form 22 families. I will also mention a result on elliptic Calabi-Yau threefolds that we obtained as a technical step in our study.