Refined Brill-Noether Theory for Complete Graphs

TL;DR

The paper studies refined Brill-Noether data for divisors on the complete graph $K_n$ by relating graph degenerations to plane curves of degree $n$. It develops canonical representatives (concentrated, sorted divisors) and a direct splitting-type formula $e_i=a_i-i+1$ to classify all splitting types, proving that $\mathrm{rk}(D+kL)=\sum_{i=1}^n \max\{0,e_i+k+1\}-1$ for all integers $k$. The main contributions are a complete classification of splitting types on $K_n$, an algorithm to compute concentrated representatives for any divisor, and a precise equivalence between graph-splitting data and plane-curve splitting types, thereby generalizing previous rank-based results. This work strengthens the graph-theoretic analogue of Brill-Noether theory and provides a concrete bridge to classical algebraic geometry through degeneration to $K_n$.

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 $K_n$.

Figures (2)

• Figure 1: Finding a concentrated divisor.
• Figure 2: A divisor $D$ (left) and its associated concentrated divisor $D'$ (right).

Theorems & Definitions (14)

• Definition 1.1
• Proposition 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5
• proof : Proof of Proposition \ref{['Prop:Concentrated']}