A metric graph satisfying $w^1_4=1$ that cannot be lifted to a curve satisfying $\dim (W^1_4)=1$

TL;DR

The paper constructs a specific metric graph with certain properties that cannot be lifted to a curve with a corresponding linear system dimension, highlighting limitations in translating classical algebraic geometry results to tropical geometry.

Contribution

It demonstrates the existence of metric graphs with $w^1_4=1$ and Clifford index 2 that cannot be lifted to genus 1 curves via degree 2 harmonic morphisms, challenging direct analogies.

Findings

Existence of such metric graphs for all $g \,\geq\, 6$

Counterexamples to direct translation of dimension theorems

Limitations in tropical lifting of linear systems

Abstract

For all integers $g \geq 6$ we prove the existence of a metric graph $G$ with $w^1_4=1$ such that $G$ has Clifford index 2 and there is no tropical modification $G'$ of $G$ such that there exists a finite harmonic morphism of degree 2 from $G'$ to a metric graph of genus 1. Those examples show that dimension theorems on the space classifying special linear systems for curves do not all of them have immediate translation to the theory of divisors on metric graphs.