Vol. 8, No. 4, 2014

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Parameterizing tropical curves I: Curves of genus zero and one

David E. Speyer

Vol. 8 (2014), No. 4, 963–998
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 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.

Keywords
tropical geometry, curves, nonarchimedean, Tate curve
Mathematical Subject Classification 2010
Primary: 14T05
Milestones
Received: 12 June 2013
Revised: 20 December 2013
Accepted: 29 January 2014
Published: 10 August 2014
Authors
David E. Speyer
Department of Mathematics
University of Michigan
2844 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States