Intersections on Tropical Moduli Spaces

This article tries to answer the question: How far can the algebro-geometric theory of rational descendant Gromov-Witten invariants be carried over to the tropical world? Given the fact that our moduli spaces are non-compact, the answer is surprisingly positive: We discuss universal families and the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a “boundary” divisor and we give two criteria that suffice to prove the tropical version of a particular WDVV or topological recursion equation. Discussing these criteria in the case of curves in R1 or R2, we prove, for example, that for the toric varieties P1, P2, P1 × P1, F1, Bl2(P), Bl3(P) and with Psi-conditions only in combination with point conditions, the tropical and conventional descendant Gromov-Witten invariants coincide. In particular, we can unify and simplify the proofs of the previous tropical enumerative results.