The tropical Abel–Prym map

Giusi Capobianco; Yoav Len

doi:10.5802/alco.452

The tropical Abel–Prym map

Giusi Capobianco, Yoav Len

Pure Mathematics

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Abstract

We prove that, under mild assumptions, the tropical Abel–Prym map Ψ: Γ' → Prym(Γ'/Γ) associated with a free double cover π : Γ' → Γ is harmonic of degree 2 if and only if the source graph Γ' is hyperelliptic. This is in accordance with the already established algebraic result. In this case, the Abel–Prym graph Ψ(Γ') is hyperelliptic of genus g_Γ - 1 and its Jacobian is isomorphic, as a pptav, to the Prym variety of the cover. We further show that the Abel–Prym graph coincides with a connected component of the tropical bigonal construction. En route, we count the number of distinct free double covers by hyperelliptic metric graphs.

Original language	English
Pages (from-to)	1493-1527
Number of pages	35
Journal	Algebraic Combinatorics
Volume	8
Issue number	6
DOIs	https://doi.org/10.5802/alco.452
Publication status	Published - 6 Jan 2026

Keywords

Algebraic geometry
Tropical geometry
Curves
Divisors
Abelian varieties
Pryms

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10.5802/alco.452Licence: CC BY

Capobianco_2026_AC_Tropical-Abel-Prym-map_CC
© The author(s), 2025. This article is licensed under the Creative Commons Attribution (CC-BY) 4.0 License (https://creativecommons.org/licenses/by/4.0/).
Final published version, 1.16 MBLicence: CC BY

Tropical Geometry and the moduli space o: Tropical Geometry and the moduli space of Prym varieties
Len, Y. (PI)
EPSRC
1/05/23 → 31/10/25
Project: Standard

Cite this

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title = "The tropical Abel–Prym map",

abstract = "We prove that, under mild assumptions, the tropical Abel–Prym map Ψ: Γ' → Prym(Γ'/Γ) associated with a free double cover π : Γ' → Γ is harmonic of degree 2 if and only if the source graph Γ' is hyperelliptic. This is in accordance with the already established algebraic result. In this case, the Abel–Prym graph Ψ(Γ') is hyperelliptic of genus gΓ - 1 and its Jacobian is isomorphic, as a pptav, to the Prym variety of the cover. We further show that the Abel–Prym graph coincides with a connected component of the tropical bigonal construction. En route, we count the number of distinct free double covers by hyperelliptic metric graphs.",

keywords = "Algebraic geometry, Tropical geometry, Curves, Divisors, Abelian varieties, Pryms",

author = "Giusi Capobianco and Yoav Len",

note = "Funding: G.C. was supported by the University of Rome Tor Vergata, Y.L. was supported by the EPSRC New Investigator Award (grant number EP/X002004/1).",

year = "2026",

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doi = "10.5802/alco.452",

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TY - JOUR

T1 - The tropical Abel–Prym map

AU - Capobianco, Giusi

AU - Len, Yoav

N1 - Funding: G.C. was supported by the University of Rome Tor Vergata, Y.L. was supported by the EPSRC New Investigator Award (grant number EP/X002004/1).

PY - 2026/1/6

Y1 - 2026/1/6

N2 - We prove that, under mild assumptions, the tropical Abel–Prym map Ψ: Γ' → Prym(Γ'/Γ) associated with a free double cover π : Γ' → Γ is harmonic of degree 2 if and only if the source graph Γ' is hyperelliptic. This is in accordance with the already established algebraic result. In this case, the Abel–Prym graph Ψ(Γ') is hyperelliptic of genus gΓ - 1 and its Jacobian is isomorphic, as a pptav, to the Prym variety of the cover. We further show that the Abel–Prym graph coincides with a connected component of the tropical bigonal construction. En route, we count the number of distinct free double covers by hyperelliptic metric graphs.

AB - We prove that, under mild assumptions, the tropical Abel–Prym map Ψ: Γ' → Prym(Γ'/Γ) associated with a free double cover π : Γ' → Γ is harmonic of degree 2 if and only if the source graph Γ' is hyperelliptic. This is in accordance with the already established algebraic result. In this case, the Abel–Prym graph Ψ(Γ') is hyperelliptic of genus gΓ - 1 and its Jacobian is isomorphic, as a pptav, to the Prym variety of the cover. We further show that the Abel–Prym graph coincides with a connected component of the tropical bigonal construction. En route, we count the number of distinct free double covers by hyperelliptic metric graphs.

KW - Algebraic geometry

KW - Tropical geometry

KW - Curves

KW - Divisors

KW - Abelian varieties

KW - Pryms

U2 - 10.5802/alco.452

DO - 10.5802/alco.452

M3 - Article

SN - 2589-5486

VL - 8

SP - 1493

EP - 1527

JO - Algebraic Combinatorics

JF - Algebraic Combinatorics

IS - 6

ER -

The tropical Abel–Prym map

Abstract

Keywords

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Projects

Tropical Geometry and the moduli space o: Tropical Geometry and the moduli space of Prym varieties

Cite this