Parameterizing tropical curves, I: Curves of genus zero and one
David Speyer · Algebra & Number Theory · 2014
In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space.We study the problem of reversing this process for curves of genus zero and one.Our methods focus on describing curves by parameterizations, not by their defining equations; we give parameterizations by rational functions in the genus-zero case and by nonarchimedean elliptic functions in the genus-one case.For genus-zero curves, those graphs which can be lifted can be characterized in a completely combinatorial manner.For genus-one curves, we show that certain conditions identified by Mikhalkin are sufficient and we also identify a new necessary condition.1. Curves in toric varieties 965 2. Basic tropical background 967 3. Statement of results 971 4. The Bruhat-Tits tree 974 5. Lemmas on zero-tension curves 976 6. Tropical curves of genus zero 977 7. Tropical curves of genus one 981 8. Superabundant curves 991 9.The necessity of well-spacedness and the j-invariant 994