  In this paper we study smooth complex projective polarized varieties (X,H) of
dimension n \ge 2 which admit a dominating family V of rational curves of
H-degree 3, such that two general points of X may be joined by a curve
parametrized by V, and such that there is a covering family of rational curves
of H-degree one. Our main result is that the Picard number of these manifolds
is at most three, and that, if equality holds, (X,H) has an adjuction theoretic
scroll structure over a smooth variety.
