There is a famous recursion formula by Kontsevich to find the number of genus zero degree $d$ curves in $\mathbb{CP}^2$ through $3d-1$ points. My question is the following: Let $S$ be a complex surface and $A$ a fixed homology class in $H^2(S,\mathbb{Z})$. Is there any hope of answering this question:

``How many genus zero curves are there (through the right number of points) that represent the homology class $A$ in $S$?''

Does Kontsevich's argument rely heavily on the fact that $S$ is $\mathbb{P}^2$? Are there some large classes of $S$ for which his argument might go through?

Note that to find ``the right number of points'' you have to calculate the dimension of the moduli space (for which there is a formula).