Tropical Curves

Abstract

A tropical curve is a graph with specified edge lengths, some of which may be infinite. Various facts and attributes about algebraic curves have analogs for tropical curves. In this article, we focus on divisors and linear series, and prove the Riemann-Roch formula for divisors on tropical curves. We describe two ways in which algebraic curves may be transformed into tropical curves: by amoebas and by specialization on arithmetic surfaces. We discuss how the study of linear series on tropical curves can be used to obtain results about linear series on algebraic curves, and summarize several recent applications.

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