Clifford representatives via the uniform algebraic rank

TL;DR

This paper introduces the uniform algebraic rank for divisors on finite graphs, establishing its properties, proving a Riemann-Roch theorem, and demonstrating the existence of Clifford representatives with explicit descriptions for many graphs.

Contribution

It defines the uniform algebraic rank, proves foundational theorems, and shows the existence and explicit forms of Clifford representatives on graphs.

Findings

Uniform algebraic rank lies between algebraic and combinatorial ranks.

Riemann-Roch theorem holds for the uniform algebraic rank.

Clifford representatives always exist for the studied class of graphs.

Abstract

In this paper, we introduce the uniform algebraic rank of a divisor class on a finite graph. We show that it lies between Caporaso's algebraic rank and the combinatorial rank of Baker and Norine. We prove the Riemann-Roch theorem for the uniform algebraic rank, and show that both the algebraic and the uniform algebraic rank are realized on effective divisors. As an application, we use the uniform algebraic rank to show that Clifford representatives always exist. We conclude with an explicit description of such Clifford representatives for a large class of graphs.