Rocky Mountain Journal of Mathematics

Intersections on tropical moduli spaces

Johannes Rau

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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).

Article information

Rocky Mountain J. Math., Volume 46, Number 2 (2016), 581-662.

First available in Project Euclid: 26 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14T05: Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]
Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45] 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]

Tropical Gromov-Witten theory tropical intersection theory tropical geometry


Rau, Johannes. Intersections on tropical moduli spaces. Rocky Mountain J. Math. 46 (2016), no. 2, 581--662. doi:10.1216/RMJ-2016-46-2-581.

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