A matroidal perspective on the tropical Prym variety

Source abstract

We associate a matroid M ( Γ ~ / Γ ) M(\widetilde {\Gamma }/\Gamma ) to a harmonic double cover π : Γ ~ → Γ \pi :\widetilde {\Gamma }\to \Gamma of metric graphs. The matroid M ( Γ ~ / Γ ) M(\widetilde {\Gamma }/\Gamma ) is a geometric interpretation of Zaslavsky’s signed graphic matroid. We show that the principalization P r y m p ( Γ ~ / Γ ) Prym_p(\widetilde {\Gamma }/\Gamma ) of the tropical Prym variety of the double cover can be reconstructed from M ( Γ ~ / Γ ) M(\widetilde {\Gamma }/\Gamma ) , equipped with certain additional decorations. We describe the simplification of the matroid M ( Γ ~ / Γ ) M(\widetilde {\Gamma }/\Gamma ) and show that the Prym variety does not change under simplification.