On varieties whose general surface section has negative Kodaira dimension

Abstract: In this paper, inspired by work of Fano, Morin and Campana--Flenner, we give a full projective classification of (however singular) varieties of dimension 3 whose general hyperplane sections have negative Kodaira dimension, and we partly extend such a classification to varieties of dimension $n\geq 4$ whose general surface sections have negative Kodaira dimension. In particular we prove that a variety of dimension $n\geq 3$ 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.