Is there a smooth, rationally connected, projective variety $M$, which is not a projective space and is equipped with an ample line bundle $L$, such that the homology class of the curves $[A]$ which connect any two points satisfies $c_1(L)([A])=1$?

This is closely related (but not equivalent) to my other question

G.W. invariants $<[pt],[pt]>_{0,[A]} \neq 0$ such that there exists ample L with $c_1(L)([A])=1$

but phrased in a classical way.