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arXiv · 2010.13135

Moduli dimensions of lattice polygons

Marino Echavarria·Max Everett·Shinyu Huang·Liza Jacoby·Ralph Morrison·Ayush Kumar Tewari·Raluca Vlad·Ben Weber
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Abstract

Given a lattice polygon $P$ with $g$ interior lattice points, we associate to it the moduli space of tropical curves of genus $g$ with Newton polygon $P$. We completely classify the possible dimensions such a moduli space can have. For non-hyperelliptic polygons the dimension must be between $g$ and $2g+1$, and can take on any integer value in this range, with exceptions only in the cases of genus $3$, $4$, and $7$. We provide a similar result for hyperelliptic polygons, for which the range of dimensions is from $g$ to $2g-1$. In the case of non-hyperelliptic polygons, our results also hold for the moduli space of algebraic curves that are non-degenerate with respect to $P$.

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Marino Echavarria, Max Everett, Shinyu Huang, Liza Jacoby, Ralph Morrison, Ayush Kumar Tewari, Raluca Vlad, Ben Weber. 2020-10-25. Moduli dimensions of lattice polygons. https://arxiv.org/abs/2010.13135

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