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arXiv · 2312.12538

Constructions of superabundant tropical curves in higher genus

Sae Koyama
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Abstract

We construct qualitatively new examples of superabundant tropical curves which are non-realizable in genus $3$ and $4$. These curves are in $\mathbb{R}^3$ and $\mathbb{R}^4$ respectively, and have properties resembling canonical embeddings of genus $3$ and $4$ algebraic curves. In particular, the genus $3$ example is a degree $4$ planar tropical curve, and the genus $4$ example is contained in the product of a tropical line and a tropical conic. They have excess dimension of deformation space equal to $1$. Non-realizability follows by combining this with a dimension calculation for the corresponding space of logarithmic curves.

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Sae Koyama. 2024-12-19. Constructions of superabundant tropical curves in higher genus. https://doi.org/10.1112/blms.13209

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