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

TL;DR

This paper employs tropical geometry techniques to compute the second moment of the tropical Prym variety for double covers of tropical curves of genus up to 4, revealing new connections to matroid theory and moduli space compactifications.

Contribution

It introduces a tropical trigonal construction to explicitly calculate the second moment of the tropical Prym variety, linking it to signed graphic matroids and moduli space extensions.

Findings

Explicit formula involving polynomial and piecewise-polynomial terms

Connection between the second moment and the Prym--Torelli map extension

Application to tropical curves of genus up to 4

Abstract

We use the tropical trigonal construction to calculate the second moment of the tropical Prym variety of all double covers $\pi:\widetilde{\Gamma}\to \Gamma$ of tropical curves of genus $g(\Gamma)\leq 4$. 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.