Kontsevichs formula is a recursion that calculates the number of rational degree $d$ curves in $mathbb{P}_{mathbb{C}}^2$ passing through $3d-1$ general positioned points. Kontsevich proved it by considering curves that satisfy extra conditions besides the given point conditions. These crucial extra conditions are two line conditions and a condition called cross-ratio. This paper addresses the question whether there is a general Kontsevichs formula which holds for more than one cross-ratio. Using tropical geometry, we obtain such a recursive formula. For that we use a correspondence theorem arXiv:1509.07453 that relates the algebro-geometric numbers in question to tropical ones. It turns out that the general tropical Kontsevichs formula we obtain is capable of not only computing the algebro-geometric numbers we are looking for, but also of computing further tropical numbers for which there is no correspondence theorem yet. We show that our recursive general Kontsevichs formula implies the original Kontsevichs formula and that the initial values are the numbers Kontsevichs fomula provides and purely combinatorial numbers, so-called cross-ratio multiplicities.
Download