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2014 Parameterizing tropical curves I: Curves of genus zero and one
David Speyer
Algebra Number Theory 8(4): 963-998 (2014). DOI: 10.2140/ant.2014.8.963


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.


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David Speyer. "Parameterizing tropical curves I: Curves of genus zero and one." Algebra Number Theory 8 (4) 963 - 998, 2014.


Received: 12 June 2013; Revised: 20 December 2013; Accepted: 29 January 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1301.14035
MathSciNet: MR3248991
Digital Object Identifier: 10.2140/ant.2014.8.963

Primary: 14T05

Rights: Copyright © 2014 Mathematical Sciences Publishers


Vol.8 • No. 4 • 2014
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