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Combinatorial and inductive methods for the tropical maximal rank conjecture

David Jensen, Sam Payne

TL;DR

This work develops combinatorial, graph-based methods to study the tropical maximal rank conjecture (MRC) and proves substantial new cases. By introducing inductive procedures that raise genus while keeping $m$ fixed, the authors extend a base set of tropical independence results to broad families, and establish the conjecture for the canonical divisor in complete generality and for many $m=3$ cases. They formulate injective and surjective inductive steps and implement them via explicit constructions on chains of loops, along with detailed slope and degree analysis of associated divisors and functions. The results imply corresponding classical MRC outcomes through tropical lifting and specialization, and provide new, characteristic-free proofs for several previously known cases, as well as new base cases that seed broader inductions. Overall, the paper advances the tropical approach to MRC through a robust combinatorial toolkit and concrete inductive frameworks.

Abstract

We produce new combinatorial methods for approaching the tropical maximal rank conjecture, including inductive procedures for deducing new cases of the conjecture on graphs of increasing genus from any given case. Using explicit calculations in a range of base cases, we prove this conjecture for the canonical divisor, and in a wide range of cases for m=3, extending previous results for m=2.

Combinatorial and inductive methods for the tropical maximal rank conjecture

TL;DR

This work develops combinatorial, graph-based methods to study the tropical maximal rank conjecture (MRC) and proves substantial new cases. By introducing inductive procedures that raise genus while keeping fixed, the authors extend a base set of tropical independence results to broad families, and establish the conjecture for the canonical divisor in complete generality and for many cases. They formulate injective and surjective inductive steps and implement them via explicit constructions on chains of loops, along with detailed slope and degree analysis of associated divisors and functions. The results imply corresponding classical MRC outcomes through tropical lifting and specialization, and provide new, characteristic-free proofs for several previously known cases, as well as new base cases that seed broader inductions. Overall, the paper advances the tropical approach to MRC through a robust combinatorial toolkit and concrete inductive frameworks.

Abstract

We produce new combinatorial methods for approaching the tropical maximal rank conjecture, including inductive procedures for deducing new cases of the conjecture on graphs of increasing genus from any given case. Using explicit calculations in a range of base cases, we prove this conjecture for the canonical divisor, and in a wide range of cases for m=3, extending previous results for m=2.

Paper Structure

This paper contains 10 sections, 11 theorems, 48 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Combinatorics of the chain of loops
  3. Permissible functions
  4. Inductive Methods
  5. The Canonical Divisor
  6. Proof of Theorem \ref{['thm:mainthm']}
  7. Divisors of Small Rank
  8. Rank 3
  9. Rank 4
  10. Rank 5

Key Result

Theorem 1.1

Suppose $(r,s,\rho,m)$ is in the injective range. Then the tropical maximal rank conjecture for $(r,s,\rho,m)$ implies the tropical maximal rank conjecture for $(r,s,\rho + 1, m)$ and $(r,s+1, \rho, m)$.

Figures (11)

  • Figure 1: Cases of the maximal rank conjecture covered by Theorem \ref{['thm:mainthm']} for $\rho=0$ and $m=3$.
  • Figure 2: The graph $\Gamma$.
  • Figure 3: The graph $\Gamma$, with three blocks of four loops, when $g=12$, $r=2$, and $s=4$.
  • Figure 4: Decomposition of the graph $\Gamma$ into locally closed pieces $\{\gamma_k\}$.
  • Figure 5: The change in tableau when inducting on $s$ in Theorem \ref{['thm:injectiveinduction']}.
  • ...and 6 more figures

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 17 more