Combinatorics of Hurwitz degenerations and tropical realizability

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. While the resulting criteria are typically inexplicit, we develop combinatorial techniques to derive explicit, verifiable conditions. We further introduce a dimensional reduction technique to deduce statements about maps to $\mathbb{R}^r$ from corresponding statements 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 interpreted as a comparison result for properness statements in moduli spaces of relative maps and admissible covers.