Algebraic and combinatorial rank of divisors on finite graphs

TL;DR

This paper investigates the algebraic rank of divisors on finite graphs, establishing key properties like Riemann-Roch, specialization, and Clifford inequalities, and comparing it to the combinatorial rank.

Contribution

It introduces the algebraic rank of divisors on graphs, proves its fundamental properties, and explores its relationship with the combinatorial rank, including cases of equality.

Findings

Algebraic rank satisfies Riemann-Roch, specialization, and Clifford inequalities.

Algebraic rank is at most equal to the combinatorial rank.

Equality between algebraic and combinatorial ranks holds in many cases, but not universally.

Abstract

We study the algebraic rank of a divisor on a graph, an invariant defined using divisors on algebraic curves dual to the graph. We prove it satisfies the Riemann-Roch formula, a specialization property, and the Clifford inequality. We prove that it is at most equal to the (usual) combinatorial rank, and that equality holds in many cases, though not in general.