Tropical curves, their Jacobians and Theta functions

2006/12/11 by Grigory Mikhalkin, Mikhalkin, Grigory, Ilia Zharkov +1 · 7 citations
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Polynomial and algebraic computation #math.AG #math.CO

paper · pdf · doi:10.48550/arxiv.math/0612267

Journal version. Errors corrected, proofs improved, references added. 32 pages, 11 figures

openalex publication_date 2006/12/11 · arxiv created 2007/11/30 · arxiv updated 2011/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Cited by

• Topology and Euler characteristics of tropical varieties
• Tropical linear series and matroids
• Dubrovin-Natanzon divisors on MM-curves
• Tropical Donagi theorem
• Enumeration of geometric Weierstrass points of metric graphs
• Relating Different Definitions of Linear Series on Tropical Curves
• Metric Graph Kernels via the Tropical Torelli Map

Related

• Jacobian varieties of reduced tropical curves
• Tropical Theta Functions and Riemann-Roch Inequality for Tropical Abelian Surfaces
• Tropical Abel-Jacobi theory
• Tropical curves of hyperelliptic type
• Tropical methods in the moduli theory of algebraic curves