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arXiv · 2501.06083

Refined Brill-Noether Theory for Complete Graphs

Haruku Aono·Eric Burkholder·Owen Craig·Ketsile Dikobe·David Jensen·Ella Norris
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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 $K_n$.

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Haruku Aono, Eric Burkholder, Owen Craig, Ketsile Dikobe, David Jensen, Ella Norris. 2025-01-10. Refined Brill-Noether Theory for Complete Graphs. https://arxiv.org/abs/2501.06083

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