Do there exist interesting examples of projective algebraic varieties such that the two-point genus 0 Gromov Witten invariants in homology class $[A]$, $GW<pt,pt>_{0,[A]}$, 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.
Edit: I would also be interested in learning about examples where the three-point invariants $GW<pt,pt,N>_{0,[A]}$ is non-zero, where $N$ is a third homology class and $[A]$ is as above.
I'm most interested in the Fano case, since that is where Gromov-Witten invariants are easiest to define using symplectic geometry. However, other examples are welcome too.
