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no. 6
The tropical Abel–Prym map
Capobianco, Giusi1; Len, Yoav2
1 University of Roma Tor Vergata, Dept. of mathematics, Via Della Ricerca Scientifica 1, Rome, Italy, 00133 IT
2 University of St Andrews, Mathematical Institute, St Andrews KY16 9SS, UK
Algebraic Combinatorics, Volume 8 (2025) no. 6, pp. 1493-1527

We prove that, under mild assumptions, the tropical Abel–Prym map $\Psi \colon \widetilde{\Gamma }\rightarrow \operatorname{Prym}(\widetilde{\Gamma }/\Gamma )$ associated with a free double cover $\pi \colon \widetilde{\Gamma }\rightarrow \Gamma $ is harmonic of degree $2$ if and only if the source graph $\widetilde{\Gamma }$ is hyperelliptic. This is in accordance with the already established algebraic result. In this case, the Abel–Prym graph $\Psi (\widetilde{\Gamma })$ is hyperelliptic of genus $g_{\Gamma }-1$ 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.

Received:
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Accepted:
Published online:
DOI: 10.5802/alco.452
Classification: 14H40, 14H51, 14T20
Keywords: Algebraic geometry, tropical geometry, curves, divisors, abelian varieties, Pryms

Capobianco, Giusi 1; Len, Yoav 2

1 University of Roma Tor Vergata, Dept. of mathematics, Via Della Ricerca Scientifica 1, Rome, Italy, 00133 IT
2 University of St Andrews, Mathematical Institute, St Andrews KY16 9SS, UK
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Capobianco, Giusi; Len, Yoav. The tropical Abel–Prym map. Algebraic Combinatorics, Volume 8 (2025) no. 6, pp. 1493-1527. doi: 10.5802/alco.452

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