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

http://mathoverflow.net/questions/178168/g-w-invariants-pt-pt-0-a-neq-0-such-that-there-exists-ample-l-with

 but phrased in a classical way.