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...

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Bibliographic Details
Published in:Journal of algebraic combinatorics 2022-03, Vol.55 (2), p.559-589
Main Authors: Echavarria, Marino, Everett, Max, Huang, Shinyu, Jacoby, Liza, Morrison, Ralph, Tewari, Ayush K., Vlad, Raluca, Weber, Ben
Format: Article
Language:English
Subjects:
Algebra
Combinatorics
Computer Science
Convex and Discrete Geometry
Group Theory and Generalizations
Lattices
Linear algebra
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
Polygons
Topological manifolds
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Summary: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 .
ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-021-01062-6