Here is the question written out in words. Do there exist interesting examples of Fano manifolds such that there are two-point genus 0 Gromov Witten invariants in homology class $[A]$ such that $GW<pt,pt>_{0,A}$ is non-zero, and there exists an ample line bundle $L$ such that $c_1(L)([A])=1$? The only example I can think of is projective space (or perhaps some blow-up) and I could imagine that these are the only examples, but I don't know how to prove that.