Riemann-Roch for Toric Rank Functions Öffentlichkeit
Bidleman, Dalton Ethan Monroe (2015)
Published
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
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