Refined Brill-Noether Theory for Complete Graphs

TL;DR

This paper explores the divisor theory of complete graphs, showing that their splitting types align with those of line bundles on smooth plane curves, extending previous results on divisor ranks.

Contribution

It computes all splitting types of divisors on complete graphs and demonstrates their correspondence with line bundles on plane curves, generalizing earlier divisor rank results.

Findings

Splitting types of divisors on $K_n$ match those on smooth plane curves of degree $n$

Generalizes previous divisor rank computations for $K_n$

Provides a comprehensive classification of divisor splitting types on complete graphs

Abstract

The divisor theory of the complete graph $K_n$ is in many ways similar to that of a plane curve of degree $n$. We compute the splitting types of all divisors on the complete graph $K_n$. We see that the possible splitting types of divisors on $K_n$ exactly match the possible splitting types of line bundles on a smooth plane curve of degree $n$. This generalizes the earlier result of Cori and Le Borgne computing the ranks of all divisors on $K_n$, and the earlier work of Cools and Panizzut analyzing the possible ranks of divisors of fixed degree on $K_n$.