Moduli dimensions of lattice polygons

Given a lattice polygon P with g interior lattice points, we can associate to P two moduli spaces: the moduli space of algebraic curves that are non-degenerate with respect to P and 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 in the tropical case. For non-hyperelliptic polygons, the dimension must be between g and 2 g + 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 2 g - 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 .