Combinatorics of Hurwitz degenerations and tropical realizability

TL;DR

This paper explores the realizability of functions on tropical curves, providing explicit combinatorial criteria for genus two curves and connecting modifiability with realizability, advancing understanding of tropical admissible covers.

Contribution

It develops explicit combinatorial criteria for realizability of functions on genus two tropical curves and links modifiability with realizability, extending genus one results.

Findings

Explicit criteria for superabundant functions on genus two curves.

Equivalence of modifiability and realizability in genus one.

New proof that well-spaced maps are realizable.

Abstract

We investigate the realizability of balanced functions on tropical curves, establishing new sufficient criteria for superabundant functions on genus two curves, analogous to the well-spacedness condition in genus one. We find that realizability is sensitive to the precise locations of conjugate and Weierstrass points on the tropical curve. The key input is a combinatorial comparison of semistable limit theorems for maps of curves. Amini-Baker-Brugall\'e-Rabinoff previously showed that realizability of functions is equivalent to ``modifiability'' to a tropical admissible cover. The resulting criteria are typically inexplicit; we develop combinatorial techniques to derive explicit, verifiable criteria from these. We then develop a dimensional reduction technique to deduce statements about maps to $\mathbb{R}^r$ from ones about maps to $\mathbb{R}$. By proving directly that modifiability and well-spacedness are equivalent in genus one, we obtain a new proof that well-spaced maps are realizable. Along the way, we explain how the modifiability criterion can be viewed as a comparison result for properness statements for moduli of relative maps and admissible covers.