Riemann-Roch for Toric Rank Functions

TL;DR

This thesis investigates the Riemann-Roch theorem for toric rank functions in chip firing games, providing proofs for specific cases, asymptotic results, and computational evidence supporting its validity.

Contribution

It presents new results confirming Riemann-Roch for trees, polygons, and large degree divisors, along with computational validation for toric rank functions.

Findings

Riemann-Roch holds for trees and polygons

Established Riemann-Roch for divisors of large degree

Computational evidence supports Riemann-Roch for toric rank functions

Abstract

In this thesis we study toric rank functions for chip firing games and prove special cases of a conjectural Riemann-Roch. The original motivation for an investigation into this area of study came for the adaptation (due to Matt Baker) of Riemann-Roch into a graph theoretic analogue through the use of chip-firing games. Here, we collect known results and present new observations that indicate Riemann--Roch holds for trees and polygons. We also prove an asymptotic case of Riemann--Roch (i.e.~Riemann--Roch for divisors of large degree). Finally, we also provide magma code and computational evidence that Riemann--Roch holds for the toric rank function.