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Toric degenerations of toric varieties and tropical curves. (English) Zbl 1105.14073
The paper under review concerns enumeration of rational curves on toric varieties. It extends to arbitrary dimension results of G. Mikhalkin valid for toric surfaces [J. Am. Math. Soc. 18, No. 2, 313–377 (2005; Zbl 1092.14068)].
In short, the main result of the article (Theorem 8.3) equates the number of torically transverse genus \(0\) stable maps to complete toric varieties, under incidence conditions, with a count of tropical curves weighed by combinatorial multiplicities. (The former geometric count is enumerative, no multiplicities are involved; it is in general not a Gromov-Witten invariant.) The paper gives in fact a detailed correspondence between the geometric and tropical counts. Both are shown to enumerate maximally degenerate curves, with log structures, in the central fiber of an explicit degeneration of the toric variety (the tropical curves arising as the dual intersection graphs of the maximally degenerate curves).

MSC:
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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