The trigonal construction and the second moment of the tropical Prym variety

2025/07/08 by Dmitry Zakharov, Zakharov, Dmitry
Computer Science · Mathematics · #05C22 #14H40 (primary) #14T20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Cryptography and Residue Arithmetic #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2507.06401

openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use the tropical trigonal construction to calculate the second moment of the tropical Prym variety of all double covers π:\widetildeΓ→ Γ of tropical curves of genus g(Γ)≤ 4. The answer is expressed in terms of the signed graphic matroid of the double cover and consists a polynomial and piecewise-polynomial term. We relate the latter term, which does not occur in the analogous formula for the tropical Jacobian, to the problem of extending the Prym--Torelli map from the moduli space of admissible double covers to the second Voronoi compactification of the moduli space of principally polarized abelian varieties.

Citations

• A matroidal perspective on the tropical Prym variety
• Counting two-forests and random cut size via potential theory
• The Prym variety of a dilated double cover of metric graphs
• The tropical n-gonal construction
• Tropical moments of tropical Jacobians
• On metric graphs with prescribed gonality
• Riemann-Roch and Abel-Jacobi theory on a finite graph
• Degenerations of Prym varieties

Related

• The tropical n-gonal construction
• Resolution of the Prym map in genus 4
• Tropical Donagi theorem
• A matroidal perspective on the tropical Prym variety
• Tropical split Jacobians of curves of genus 2 II