2004•arXiv (Cornell University)Open access

Toric degenerations of toric varieties and tropical curves

Takeo Nishinou, Bernd Siebert

Abstract

We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on degeneration techniques and log deformation theory. This generalizes results of Mikhalkin obtained by different methods in the surface case to arbitrary dimensions.

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Key concepts: Toric variety, Mathematics, Prime (order theory), Tropical geometry, Algebraic geometry, Pure mathematics, Variety (cybernetics), Algebraic surface