Sheaves and Duality in the Two-Vertex Graph Riemann-Roch Theorem

TL;DR

This paper extends the Baker-Norine Graph Riemann-Roch theorem for two-vertex graphs by introducing sheaves and duality principles, providing a sheaf-theoretic perspective and new duality results.

Contribution

It introduces a sheaf-theoretic framework and duality theorems for the Baker-Norine Graph Riemann-Roch theorem on two-vertex graphs.

Findings

Sheaf of vector spaces models the divisor rank.

Duality theorems generalize the classical Riemann-Roch.

Zeroth Betti number relates to divisor rank plus one.

Abstract

For each graph on two vertices, and each divisor on the graph in the sense of Baker-Norine, we describe a sheaf of vector spaces on a finite category whose zeroth Betti number is the Baker-Norine "Graph Riemann-Roch" rank of the divisor plus one. We prove duality theorems that generalize the Baker-Norine "Graph Riemann-Roch" Theorem.