Riemann-Roch for Toric Rank Functions Open Access

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.

Table of Contents

1 Introduction 1

2 Background Information 3

2.1 Divisors on Graphs 3

2.2 Chip Firing Games 6

2.3 Divisors on the Smooth Curve and the Projective Plane 8

2.4 Divisors on Graph Curves 9

3 Toric Rank 11

3.1 Definition of Toric Rank 11

3.2 Results 14

4 Experimental Data 20

4.1 Data 20

4.2 Code 20

5 Bibliography 31

About this Master's Thesis

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School	Laney Graduate School
Department	Math and Computer Science
Degree	MS
Submission	Master's Thesis
Language	English
Research Field	Mathematics
Keyword	Riemann-Roch Chip Firing Toric Rank
Committee Chair / Thesis Advisor	Zureick-Brown, David M, Emory University
Committee Members	Ramirez, Shawn L, Emory University Gould, Ronald J, Emory University

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