Geometry of rays-positive manifolds

Let M be a smooth complex projective variety and let L be a line bundle on it. Rays-positive manifolds, namely pairs (M,L) such that L is numerically effective and LR > 0 for all extremal rays R on M, are studied. Several illustrative examples and some applications are provided. In particular, projective varieties with crepant singularities and of small degree with respect to the codimension are classified, and the non-negativity of the sectional genus g(M,L) is proven, describing as well the pairs with g(M,L) = 0, 1.