Combinatorics of Hurwitz degenerations and tropical realizability
| Autor: | Lam, Mia, Ng, Chi Kin, Ranganathan, Dhruv |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 38 pages, 16 figures |
| Databáze: | arXiv |
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