Riemann-Roch for Toric Rank Functions Open Access

Bidleman, Dalton Ethan Monroe (2015)

Permanent URL: https://etd.library.emory.edu/concern/etds/pg15bf34p?locale=en
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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

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

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