Uniformizing Tropical Curves I: Genus Zero and One

TL;DR

This paper investigates the reverse problem of tropical curves in toric varieties for genus zero and one, providing parameterizations and combinatorial characterizations, and refining conditions for liftability.

Contribution

It introduces parameterizations for genus zero and one tropical curves and refines criteria for lifting these curves, including new necessary conditions.

Findings

Genus zero curves can be characterized combinatorially for liftability.

Genus one curves satisfy certain sufficient conditions, with new necessary conditions identified.

Parameterizations involve rational functions and non-archimedean elliptic functions.

Abstract

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 non-archimedean 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, show that certain conditions identified by Mikhalkin are sufficient and we also identify a new necessary condition.