Free divisors on metric graphs

TL;DR

This paper introduces the concept of free divisors on metric graphs, revealing differences from classical algebraic curves and highlighting new obstructions and classifications for divisors on graphs.

Contribution

It defines free divisors on metric graphs and demonstrates that the Clifford inequality is the main obstacle for very special free divisors, contrasting with classical curve theory.

Findings

Clifford inequality is the only obstruction for certain free divisors.

Many divisors on graphs cannot be lifted to curves with the same rank.

Classifications based on Clifford index for curves do not directly apply to graphs.

Abstract

On a metric graph we introduce the notion of a free divisor as a replacement for the notion of a base point free complete linear system on a curve. By means of an example we show that the Clifford inequality is the only obstruction for the existence of very special free divisors on a graph. This is different from the situation of base point free linear systems on curves. It gives rise to the existence of many types of divisors on graphs that cannot be lifted to curves maintaining the rank and it also shows that classifications made for linear systems of some fixed small positive Clifford index do not hold (exactly the same) on graphs.