Varieties with a reducible hyperplane section whose two components are hypersurfaces

We classify smooth complex projective varieties X ⊂ P^N of dimension n ≥ 2 admitting a divisor of the form A+B among their hyperplane sections, both A and B of codimension ≤ 1 in their respective linear spans. In this setting, one of the following holds: 1) X is either the Veronese surface in P^5 or its general projection to P^4, 2) n ≤ 3 and X ⊂ P^n+2 is contained in a quadric cone of rank 3 or 4, 3) n = 2 and X ⊂ P^3