In 1938 U. Morin, improving on earlier results by G. Fano (1918), stated a projective classification theorem for varieties of dimension $n\geq 3$ whose general surface sections are rational. Although Morin's result is correct, his proof is wrong. In the first part of this talk I will explain how to fix Morin's argument by using ideas from Mori's theory already exploited by F. Campana and H. Flenner to attack a quite similar problem. This part is joint work with C. Fontanari. In the second part of the talk I will make some application to rationality of Fano threefolds.
