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Relating Different Definitions of Linear Series on Tropical Curves

Eric Burkholder

TL;DR

This work unifies two tropical theories of linear series by proving that strongly recursive tropical linear series of rank $r$ always yield combinatorial limit linear series of rank $r$, while providing counterexamples to the converse for $r\ge 3$. It develops a robust local-array framework that links global submodule structure to local permutation-arrays, employs slope-structure notions, and analyzes Baker–Norine rank properties via slope limits. Extensions include relaxing locality conditions, establishing structured TLS equivalence with CLLS, and exploring realizability of permutation arrays as local arrays under both frameworks. The results illuminate when tropical linear series can be realized combinatorially and offer a fertile ground for further questions on higher-rank realizability and the full TLS vs. CLLS equivalence spectrum.

Abstract

We compare strongly recursive tropical linear series as defined by Farkas, Jensen, and Payne with combinatorial limit linear series as defined by Amini and Gierczak. We show that strongly recursive tropical linear series of rank $r$ are combinatorial limit linear series of rank $r$ and construct a counterexample to the converse. We also present several extensions of our main result, including a simplification of the definition of combinatorial limit linear series and an investigation of their relationship with tropical linear series in the sense of Chang et al. Finally, we address the realizability of permutation arrays as local arrays of linear series on tropical curves. Finally, discuss the realizability of permutation arrays as local arrays of linear series on tropical curves.

Relating Different Definitions of Linear Series on Tropical Curves

TL;DR

This work unifies two tropical theories of linear series by proving that strongly recursive tropical linear series of rank always yield combinatorial limit linear series of rank , while providing counterexamples to the converse for . It develops a robust local-array framework that links global submodule structure to local permutation-arrays, employs slope-structure notions, and analyzes Baker–Norine rank properties via slope limits. Extensions include relaxing locality conditions, establishing structured TLS equivalence with CLLS, and exploring realizability of permutation arrays as local arrays under both frameworks. The results illuminate when tropical linear series can be realized combinatorially and offer a fertile ground for further questions on higher-rank realizability and the full TLS vs. CLLS equivalence spectrum.

Abstract

We compare strongly recursive tropical linear series as defined by Farkas, Jensen, and Payne with combinatorial limit linear series as defined by Amini and Gierczak. We show that strongly recursive tropical linear series of rank are combinatorial limit linear series of rank and construct a counterexample to the converse. We also present several extensions of our main result, including a simplification of the definition of combinatorial limit linear series and an investigation of their relationship with tropical linear series in the sense of Chang et al. Finally, we address the realizability of permutation arrays as local arrays of linear series on tropical curves. Finally, discuss the realizability of permutation arrays as local arrays of linear series on tropical curves.

Paper Structure

This paper contains 14 sections, 38 theorems, 25 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General Preliminaries
  4. Strongly Recursive Tropical Linear Series
  5. Permutation Arrays
  6. Combinatorial Limit Linear Series
  7. Local Arrays of Strongly Recursive Tropical Linear Series
  8. Local Rank and Baker-Norine Rank Properties
  9. Proof and Extensions of Theorem \ref{['Thm:SRTLS-are-CLLS']} and Counterexamples to its Converse
  10. Extending Theorem \ref{['Thm:SRTLS-are-CLLS']}
  11. Counterexamples to the Converse of Theorem \ref{['Thm:SRTLS-are-CLLS']}
  12. Realizability of Permutation Arrays as Local Arrays of Linear Series on Tropical Curves
  13. Realizability as Local Arrays of Strongly Recursive Tropical Linear Series
  14. Realizability as Local Arrays of Combinatorial Limit Linear Series

Key Result

Theorem 1.1

Let $D$ be an effective divisor on a metric graph $\Gamma$ and let $\Sigma \subseteq R(D)$. If $(D,\Sigma)$ is a strongly recursive tropical linear series of rank $r$, then $(D,\Sigma)$ is a combinatorial limit linear series of rank $r$.

Figures (7)

  • Figure 1: Generators of a Tropical Linear Series of Rank 1 over an Interval
  • Figure 2: Non-Rankable Dot Array $P$
  • Figure 3: Totally Rankable Dot Array and Its Rank Array
  • Figure 4: Redundant Closure of Figure \ref{['Fig:totally-rankable-dot-array']}
  • Figure 5: All $[1]^1$ and $[1]^2$ Local Arrays of Strongly Recursive Tropical Linear Series
  • ...and 2 more figures

Theorems & Definitions (82)

  • Theorem 1.1
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Lemma 2.8
  • ...and 72 more