Open Access
2016 Intersections on tropical moduli spaces
Johannes Rau
Rocky Mountain J. Math. 46(2): 581-662 (2016). DOI: 10.1216/RMJ-2016-46-2-581


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 $\PP ^1$, $\PP ^2$, $\PP ^1 \times \PP ^1$ and with $\Psi $-conditions only in combination with point conditions, the tropical and classical descendant Gromov-Witten invariants coincide (which extends the result for $\PP ^2$ in \cite {MR08}). Our approach uses tropical intersection theory and unifies and simplifies some parts of the existing tropical enumerative geometry (for rational curves).


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Johannes Rau. "Intersections on tropical moduli spaces." Rocky Mountain J. Math. 46 (2) 581 - 662, 2016.


Published: 2016
First available in Project Euclid: 26 July 2016

zbMATH: 1379.14035
MathSciNet: MR3529085
Digital Object Identifier: 10.1216/RMJ-2016-46-2-581

Primary: 14T05
Secondary: 14N35 , 52B20

Keywords: Tropical geometry , Tropical Gromov-Witten theory , tropical intersection theory

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.46 • No. 2 • 2016
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