Tropical curves, their Jacobians and Theta functions

By G. Mikhalkin, I. Zharkov. 2006

379 citations recorded by the source corpus.

We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the Abel-Jacobi, Riemann-Roch and Riemann theta divisor theorems.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/514477ee36849b775794d4b40c1e5c37756bd567

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In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

A Riemann–Roch theorem in tropical geometry (2006, background): “Shortly after this manuscript had appeared on the e-print archive, Mikhalkin and Zharkov published a preprint that also includes a proof of the Riemann-Roch theorem for tropical curves [ MZ ].”
Tropical Geometry and its applications (2006, neutral): “map in this case (as well as all its translates in Pic 0 = Jac is called the Θ-divisor. There is a tropical counterpart of the Riemann theorem stating that the Θ-divisor is given by a θ-function, see [23]. Figure 6 sketches the Θ-divisor in the case of genus 2. In this case it is isomorphic to the c”
Specialization of linear systems from curves to graphs (2007, neutral): “613”
Metric properties of the tropical Abel–Jacobi map (2009, methodology): “This allows us to give a very simple proof of the tropical version of Abel’s theorem, first proved in [12].”
On Block-Goettsche multiplicities for planar tropical curves (2012, background): “[16, 18]) is a connected finite graph without 2-valent vertices whose edges are enhanced with lengths.”
Combinatorics of the tropical Torelli map (2010, background): “ g considered just as a topological space, and Caporaso and Viviani [8], who prove a tropical Torelli theorem stating that the tropical Torelli map is “mostly” injective, as originally conjectured in [19]. In laying the groundwork for the results we will present here, we ran into some inconsistencie”