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Combinatorics of Hurwitz degenerations and tropical realizability

Mia Lam, Chi Kin Ng, Dhruv Ranganathan

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-Brugallé-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.

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-Brugallé-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 from ones about maps to . 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.

Paper Structure

This paper contains 34 sections, 35 theorems, 43 equations, 16 figures.

Table of Contents

  1. Context
  2. Main results
  3. Genus two
  4. Hurwitz modifiability and combinatorial rederivations
  5. Outlook and techniques
  6. Tropical covers and modifications
  7. Tropical curves
  8. Balanced functions and harmonic maps
  9. Mildness conditions
  10. Tropicalization for families of curves, functions, and covers
  11. First for curves
  12. ...then with a function...
  13. ...and finally for covers
  14. Hurwitz modifiability
  15. Basic strategy
  16. ...and 19 more sections

Key Result

Theorem A

Let $F\colon \Gamma\to{\mathbb R}$ be a balanced function on an abstract tropical curve of $\Theta$ type, and assume that $F$ contracts the core. If then $F$ is realizable.

Figures (16)

  • Figure 1: A balanced map with a contracted $\Theta$ and three attaching points. It is realizable provided the lengths of the three segments $A_iB_i$ are all equal.
  • Figure 2: A balanced map with a contracted $\Theta$ and two attaching points that are conjugate.
  • Figure 3: A tropical map with a contracted cycle and a single non-well-spaceable critical path. To interpret the figure, note that the edge $AB$ and the cycle are both contracted. The map is linear of slope $s_1$ on the rays based at $A$, in the indicated direction.
  • Figure 4: A tropical map with a contracted cycle and a single critical path with three non-constant flags at the critical point. In the figure, the cycle and the edge $BC$ are contracted. The map is linear of slope $s_i$ on the three rays, in the indicated directions.
  • Figure 5: A tropical map with a contracted cycle with two critical paths. To interpret the figure, note that the cycle, $AB$, and $CD$ are all contracted. The map is linear of slope $s_1$ resp. $s_2$ on the upper resp. lower rays.
  • ...and 11 more figures

Theorems & Definitions (78)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 1.2.1
  • Definition 1.2.2: Harmonic maps
  • Definition 1.3.1
  • Definition 1.3.3
  • Proposition 1.4.1
  • proof
  • ...and 68 more