Counting tropical elliptic plane curves with fixed j-invariant

Details Counting tropical elliptic plane curves with fixed j-invariant Counting tropical elliptic plane curves with fixed j-invariant

2006-08-18

arxiv.org/abs/math/0608472v2

Comment. Math. Helv. 84 (2) 2009, 387-427

Mathematics

Algebraic Geometry

34 pages; minor changes to match the published version

Scientific paper

In complex algebraic geometry, the problem of enumerating plane elliptic curves of given degree with fixed complex structure has been solved by R.Pandharipande using Gromov-Witten theory. In this article we treat the tropical analogue of this problem, the determination of the number of tropical elliptic plane curves of degree d and fixed ``tropical j-invariant'' interpolating an appropriate number of points in general position. We show that this number is independent of the position of the points and the value of the j-invariant and that it coincides with the number of complex elliptic curves. The result can be used to simplify Mikhalkin's algorithm to count curves via lattice paths in the case of rational plane curves.

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