A matroidal perspective on the tropical Prym variety

Abstract

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

31 pages