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Abstract: The moduli space of algebraic curves of genus g admits a natural stratification by gonality, defined as the smallest positive integer d for which a curve admits a degree d morphism to the projective line, or equivalently, a linear series of degree d and dimension one. After motivating the study of the tropical analogue of the moduli space of trigonal curves—namely, curves of gonality three—we will recall the definitions of gonality in tropical geometry. Unlike in the classical setting, the various notions of gonality do not, in general, coincide for tropical curves. We will present first the hyperelliptic case (d=2), where the definitions coincide, as shown by Melody Chan. In contrast, for the trigonal case (d=3), equivalence among the different tropical notions holds under the assumption of 3-edge-connectivity, up to tropical modifications. This allows for the construction of the moduli space of 3-edge-connected tropical trigonal curves, whose dimension matches that of the classical trigonal locus.In the general, non–3-edge-connected case, these definitions diverge; nonetheless, we can explicitly characterize the tropical curves for which the notions of gonality may not coincide.
This work is in collaboration with Margarida Melo.