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Tropical trigonal curves
Melo, Margarida1; Zheng, Angelina2
1Università di Roma Tre, Dipartimento di Matematica e Fisica, Largo San Leonardo Murialdo 1, 00146 Rome (Italy)
2Universität Tübingen, Fachbereich Mathematik, Auf der Morgenstelle 10, 72076 Tübingen (Germany)
Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 95-130

We prove that the existence of a divisor of degree $3$ and Baker-Norine rank at least $1$ on a $3$-edge connected tropical curve is equivalent to the existence of a non-degenerate harmonic morphism of degree $3$ from a tropical modification of it to a tropical rational curve. Using the second description, we define the moduli spaces of $3$-edge connected tropical trigonal covers and of $3$-edge connected tropical trigonal curves, the latter as a locus in the moduli space of tropical curves. Finally, we prove that the moduli space of $3$-edge connected genus $g$ tropical trigonal curves has the same dimension as the moduli space of genus $g$ algebraic trigonal curves.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.465
Classification: 14T05
Keywords: gonality, trigonal curve, tropical curve, harmonic morphism, divisor, rank, moduli space

Melo, Margarida  1 ; Zheng, Angelina  2

1 Università di Roma Tre, Dipartimento di Matematica e Fisica, Largo San Leonardo Murialdo 1, 00146 Rome (Italy)
2 Universität Tübingen, Fachbereich Mathematik, Auf der Morgenstelle 10, 72076 Tübingen (Germany)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Melo, Margarida; Zheng, Angelina. Tropical trigonal curves. Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 95-130. doi: 10.5802/alco.465

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