Abstract
Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d, g) of complex plane curves of degree d and genus g through 3d + g − 1 general points with the help of relative Gromov-Witten invariants. Recently, Mikhalkin has found a way to reinterpret the numbers N(d, g) in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes. We relate these two results by defining an analogue of the relative Gromov-Witten invariants and rederiving the Caporaso–Harris formula in terms of both tropical geometry and lattice paths.
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References
Caporaso L., Harris J. (1998). Counting plane curves of any genus. Invent. Math. 131(2): 345–392
Gathmann A., Markwig H. (2007). The number of tropical plane curves through points in general position. Journal für die reine und angewandte Mathematik 602: 155–177
Mikhalkin G. (2005). Enumerative tropical geometry in \({\mathbb{R}}^2\). J. Am. Math. Soc. 18: 313–377
Nishinou, T., Siebert, B.: Toric degenerations of toric varieties and tropical curves. Preprint math.AG 0409060
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H. Markwig has been funded by the DFG grant Ga 636/2.
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Gathmann, A., Markwig, H. The Caporaso–Harris formula and plane relative Gromov-Witten invariants in tropical geometry. Math. Ann. 338, 845–868 (2007). https://doi.org/10.1007/s00208-007-0092-4
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DOI: https://doi.org/10.1007/s00208-007-0092-4