On varieties whose general surface section has negative Kodaira dimension

In this paper, inspired by work of Fano, Morin, and Campana–Flenner, we give a projective classification of varieties of dimension 3 whose general hyperplane sections have negative Kodaira dimension, and we partly extend such a classification to varieties of dimension $n⩾4$ whose general surface sections have negative Kodaira dimension. In particular, we prove that a variety of dimension $n⩾3$ whose general surface sections have negative Kodaira dimension is birationally equivalent to the product of a general surface section times $P^{n-2}$ unless (possibly) if the variety is a cubic hypersurface.