  The main result of this paper is a structural theorem for projective
Q-factorial toric varieties X in P^N, covered by lines. We prove that there
exists a toric fibration f: X -> Z, locally trivial in the Zariski topology,
with fiber a product of projective joins. All lines in X intersecting the open
subset isomorphic to the torus, are contained in some fiber of f. This
characterization has a geometrical application to dual defective toric
varieties, and a combinatorial application to discriminants of lattice subsets.
We prove that X has positive dual defect if and only if it has an elementary
extremal contraction of fiber type, whose general fiber is a projective join
with dual defect bigger than its codimension in X. Turning to combinatorics, we
characterize lattice subsets A with discriminant D_A equal to one, under
suitable assumptions on the polytope Conv(A).
