The tropical $n$-gonal construction

TL;DR

This paper develops a tropical analogue of Donagi's n-gonal construction, exploring its combinatorial properties and establishing duality and isomorphism results for Prym varieties and Jacobians in tropical geometry.

Contribution

It introduces a tropical n-gonal construction for harmonic double covers and proves duality and isomorphism properties of associated tropical abelian varieties.

Findings

For n=2, the Prym varieties are dual tropical abelian varieties.

For n=3, the construction produces a tetragonal tropical curve with specific dilation profiles.

The Prym variety and Jacobian are isomorphic as principally polarized tropical abelian varieties.

Abstract

We give a purely tropical analogue of Donagi's $n$-gonal construction and investigate its combinatorial properties. The input of the construction is a harmonic double cover of an $n$-gonal tropical curve. For $n = 2$ and a dilated double cover, the output is a tower of the same type, and we show that the Prym varieties of the two double covers are dual tropical abelian varieties. For $n=3$ and a free double cover, the output is a tetragonal tropical curve with dilation profile nowhere $(2,2)$ or $(4)$, and we show that the construction can be reversed. Furthermore, the Prym variety of the double cover and the Jacobian of the tetragonal curve are isomorphic as principally polarized tropical abelian varieties. Our main tool is tropical homology theory, and our proofs closely follow the algebraic versions.