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A study of H. Martens' Theorem on chains of cycles

Marc Coppens

TL;DR

The paper investigates whether H. Martens' Theorem for smooth curves extends to chains of cycles, a class of metric graphs, by employing Brill-Noether ranks $w^r_d(\Gamma)$ and a combinatorial framework of representing divisors through $\langle\xi_i\rangle_i$ and displacement tableaux. It proves a graph analogue (Theorem A): if $1 \le r \le g-2$ and $2r \le d \le g-3+r$ with $w^r_d(\Gamma)=d-2r$, then $\Gamma$ is hyperelliptic, and shows the existence of non-hyperelliptic chains achieving $w^r_{g-2+r}(\Gamma)=g-2-r$ for $g \ge 2r+3$, highlighting a divergence from the curve case. To capture these phenomena, the authors define Martens-special chains of cycles (type $k$) and establish gonality bounds (at most $k+2$), yielding a weak Martens-type theorem for $w^r_{g-2+r}$ and a refined understanding of Brill-Noether behavior on chains of cycles. They demonstrate that Brill-Noether equalities that hold for curves need not hold verbatim for graphs and provide a framework (Theorems B and C) for classifying and studying these special counterexamples. The approach hinges on the exchange between tropical geometry (tropical Abel map) and combinatorial data encoded in $\underline{m}$-displacement tableaux, enabling explicit constructions of divisors with prescribed rank on chains of cycles.

Abstract

Let $Γ$ be a chain of cycles of genus $g$. Let $d$,$r$ be integers with $1 \leq r \leq g-2$ and $2r\leq d \leq g-3+r$. Then $w^r_d(Γ)=d-2r$ implies $Γ$ is hyperelliptic. For each $g \geq 2r+3$ there exist non-hyperelliptic chains of cycles satisfying $w^r_{g-2+r}(Γ)=g-2-r$. In the case of algebraic curves such equality implies the curve is hyperelliptic. In particular we obtain the existence of chains of cycles $Γ$ such that $w^r_{g-2+r}(Γ) \neq w^1_{g-r}(Γ)$ in case $r \geq 2$. In the case of algebraic curves such numbers are equal because of the Riemann-Roch Theorem.

A study of H. Martens' Theorem on chains of cycles

TL;DR

The paper investigates whether H. Martens' Theorem for smooth curves extends to chains of cycles, a class of metric graphs, by employing Brill-Noether ranks and a combinatorial framework of representing divisors through and displacement tableaux. It proves a graph analogue (Theorem A): if and with , then is hyperelliptic, and shows the existence of non-hyperelliptic chains achieving for , highlighting a divergence from the curve case. To capture these phenomena, the authors define Martens-special chains of cycles (type ) and establish gonality bounds (at most ), yielding a weak Martens-type theorem for and a refined understanding of Brill-Noether behavior on chains of cycles. They demonstrate that Brill-Noether equalities that hold for curves need not hold verbatim for graphs and provide a framework (Theorems B and C) for classifying and studying these special counterexamples. The approach hinges on the exchange between tropical geometry (tropical Abel map) and combinatorial data encoded in -displacement tableaux, enabling explicit constructions of divisors with prescribed rank on chains of cycles.

Abstract

Let be a chain of cycles of genus . Let , be integers with and . Then implies is hyperelliptic. For each there exist non-hyperelliptic chains of cycles satisfying . In the case of algebraic curves such equality implies the curve is hyperelliptic. In particular we obtain the existence of chains of cycles such that in case . In the case of algebraic curves such numbers are equal because of the Riemann-Roch Theorem.
Paper Structure (4 sections, 17 theorems, 5 figures)

This paper contains 4 sections, 17 theorems, 5 figures.

Table of Contents

  1. Introduction
  2. Generalities
  3. H. Martens' Theorem for chains of cycles
  4. Non-hyperelliptic chains of cycles satisfying $w^r_{g-2+r}=g-2-r$ for some $1 \leq r \leq g-2$

Key Result

Lemma 1

Let $D$ be any divisor of degree $d$ on a chain of cycles of genus $g$. Then $D$ is equivalent to a unique divisor of the form $\sum_{i=1}^g \langle\xi _i\rangle_i +(d-g)\cdot w_g$ (here $\xi _i \in \mathbb{R}$ and $\langle\xi _i\rangle_i \in C_i$ using Definition def2).

Figures (5)

  • Figure 1: a chain of cycles
  • Figure 2: rectangle $[12 \times 3]$
  • Figure 3: Example \ref{['example1']}
  • Figure 4: The hyperelliptic tableau of genus 6
  • Figure 5: Illustration of the proof of Lemma \ref{['lemma3']}

Theorems & Definitions (36)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Definition 4
  • Example 1
  • Theorem 1
  • Example 2
  • Proposition 1
  • proof
  • ...and 26 more