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Intersections on tropical moduli spaces. (English) Zbl 1379.14035
Summary: This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a “boundary” divisor, and we prove general tropical versions of the WDVV, respectively, topological recursion equations (under some assumptions). As a direct application, we prove that, for the toric varieties $$\mathbb P^1$$, $$\mathbb P^2$$, $$\mathbb P^1 \times \mathbb P^1$$ and with $$\Psi$$-conditions only in combination with point conditions, the tropical and classical descendant Gromov-Witten invariants coincide (which extends the result for $$\mathbb P^2$$ in [H. Markwig and the author, Manuscr. Math. 129, No. 3, 293–335 (2009; Zbl 1171.14039)]. Our approach uses tropical intersection theory and unifies and simplifies some parts of the existing tropical enumerative geometry (for rational curves).

##### MSC:
 14T05 Tropical geometry (MSC2010) 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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