Realization of tropical curves in Abelian surfaces
Abstract
We construct algebraic curves in abelian surfaces by utilizing tropical curves in real tori. We give a necessary and sufficient condition for a tropical curve in a real torus to be realizable by an algebraic curve in an abelian surface. When this condition is satisfied, the number of algebraic curves can be computed by a combinatorial formula. This provides the algebraic-tropical correspondence theorem for abelian surfaces analogous to Mikhalkin's correspondence theorem for toric surfaces. Thus, the number of algebraic curves passing through generic points in an abelian surface can be computed purely combinatorially via tropical curves.
Fields of Study & Subjects
Polynomial and algebraic computation
Algebraic Geometry and Number Theory
Cryptography and Residue Arithmetic
Publication details
- Journal / source
- Journal of Differential Geometry
- Publisher
- International Press of Boston
- Volume
- 134
- Issue
- 1
- Pages
- 1–95
- Publication date
- 1 September 2026
- Language
- EN