Parameterizing tropical curves, I: Curves of genus zero and one

Speyer, David

doi:10.2140/ant.2014.8.963

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Parameterizing tropical curves I: Curves of genus zero and one

David E. Speyer

Vol. 8 (2014), No. 4, 963–998

DOI: 10.2140/ant.2014.8.963

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

Vol. 8, No. 4, 2014