Journal of Differential Geometry· 2026Q1
Realization of tropical curves in Abelian surfaces
- 2citations
- Q1SCImago
- 2026year
Short summary
A new theorem establishes a direct correspondence between tropical curves in real tori and algebraic curves in abelian surfaces, enabling combinatorial counting of algebraic curves.
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Key points
- Introduces an algebraic-tropical correspondence theorem for abelian surfaces.
- Provides a necessary and sufficient condition for realizing tropical curves in real tori as algebraic curves in abelian surfaces.
- Enables combinatorial computation of the number of algebraic curves through generic points.
- The theorem is analogous to Mikhalkin's correspondence theorem for toric surfaces.
AI-generated from the title and abstract; the full text is not read.
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.
The authors' abstract, as published at the source. Journal of Differential Geometry, 2026 · DOI ↗
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Field: Computational Theory and Mathematics
Computational Theory and MathematicsComputer Science