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no. 2
The tropical $n$-gonal construction
Röhrle, Felix1; Zakharov, Dmitry2
1 Universität Tübingen Fachbereich Mathematik 72076 Tübingen, Germany
2 Central Michigan University Department of Mathematics Mount Pleasant, MI 48859, USA
Algebraic Combinatorics, Volume 8 (2025) no. 2, pp. 319-378.

We give a purely tropical analogue of Donagi’s $n$-gonal construction and investigate its combinatorial properties. The input of the construction is a harmonic double cover of an $n$-gonal tropical curve. For $n = 2$ and a dilated double cover, the output is a tower of the same type, and we show that the Prym varieties of the two double covers are dual tropical abelian varieties. For $n=3$ and a free double cover, the output is a tetragonal tropical curve with dilation profile nowhere $(2,2)$ or $(4)$, and we show that the construction can be reversed. Furthermore, the Prym variety of the double cover and the Jacobian of the tetragonal curve are isomorphic as principally polarized tropical abelian varieties. Our main tool is tropical homology theory, and our proofs closely follow the algebraic versions.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.410
Classification: 14T20, 14H40
Keywords: tropical curve, gonality, abelian variety, Jacobian, Prym variety

Röhrle, Felix 1; Zakharov, Dmitry 2

1 Universität Tübingen Fachbereich Mathematik 72076 Tübingen, Germany
2 Central Michigan University Department of Mathematics Mount Pleasant, MI 48859, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Röhrle, Felix; Zakharov, Dmitry. The tropical $n$-gonal construction. Algebraic Combinatorics, Volume 8 (2025) no. 2, pp. 319-378. doi : 10.5802/alco.410. https://alco.centre-mersenne.org/articles/10.5802/alco.410/

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