2024/12/09 by Giusi Capobianco, Yoav Len, Capobianco, Giusi +1
Social Sciences · #14H40 #14T20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Historical Geography and Cartography
paper · pdf · doi:10.48550/arxiv.2412.06971
openalex publication_date 2024/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that the tropical Abel--Prym map Ψ\colon \widetildeΓ→ Prym(\widetildeΓ/Γ) associated with a free double cover π\colon \widetildeΓ→ Γ of hyperelliptic metric graphs is harmonic of degree 2 in accordance with the already established algebraic result. We then prove a partial converse. Contrary to the analogous algebraic result, when the source graph of the double cover is not hyperelliptic, the Abel--Prym map is often not injective. When the source graph is hyperelliptic, we show that the Abel--Prym graph Ψ(\widetildeΓ) is a hyperelliptic metric graph of genus gΓ-1 whose Jacobian is isomorphic, as pptav, to the Prym variety of the cover. En route, we count the number of distinct free double covers by hyperelliptic metric graphs.