Moduli dimensions of lattice polygons

TL;DR

This paper classifies the possible dimensions of moduli spaces of tropical curves associated with lattice polygons, revealing specific ranges and exceptions based on hyperellipticity and genus.

Contribution

It provides a complete classification of moduli space dimensions for tropical curves linked to lattice polygons, including hyperelliptic and non-hyperelliptic cases, with detailed genus-specific results.

Findings

Dimension ranges for non-hyperelliptic polygons are from g to 2g+1.

Dimension ranges for hyperelliptic polygons are from g to 2g-1.

Specific exceptions occur at genera 3, 4, and 7.

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 $P$.