A matroidal perspective on the tropical Prym variety

TL;DR

This paper introduces a matroidal framework to understand the tropical Prym variety associated with harmonic double covers of metric graphs, linking combinatorial matroid structures to tropical geometry.

Contribution

It establishes a new connection between matroids and tropical Prym varieties, showing how to reconstruct Prym varieties from associated matroids with additional data.

Findings

The matroid $M( ilde{ ame{Gamma}}/ ame{Gamma})$ encodes the structure of the double cover.

The tropical Prym variety can be reconstructed from the matroid and decorations.

Simplification of the matroid does not alter the Prym variety.

Abstract

We associate a matroid $M(\widetilde{\Gamma}/\Gamma)$ to a harmonic double cover $\pi:\widetilde{\Gamma}\to \Gamma$ of metric graphs. The matroid $M(\widetilde{\Gamma}/\Gamma)$ is a geometric interpretation of Zaslavsky's signed graphic matroid. We show that the principalization $\mathrm{Prym}_p(\widetilde{\Gamma}/\Gamma)$ of the tropical Prym variety of the double cover can be reconstructed from $M(\widetilde{\Gamma}/\Gamma)$, equipped with certain additional decorations. We describe the simplification of the matroid $M(\widetilde{\Gamma}/\Gamma)$ and show that the Prym variety does not change under simplification.