On Direct-Sum Decompositions of the Picard Group of a Graph Open Access

Abstract

The Picard group is an algebraic object that naturally arises in the study of a “chip- firing game,” a solitaire game played on the vertices of a graph. This Picard group has illuminated the study of multiple research areas, offering a link between disparate topics in combinatorics, graph theory, and algebra. Recently, algebraic geometers have taken interest in the game, as it pertains to the developing subfield “tropical geometry.” Such an exchange allows for the import of chip-firing observations into a geometric setting and has produced graph-theoretic analogues of classical algebraic geometry results. This thesis investigates the structure of the Picard group, particularly its behavior under a “coning” operation that we can iterate on its referent graph. Coning over a graph G entails adding a vertex to its vertex set which is adjacent to every other vertex in G. Recent papers demonstrate that coning over a graph produces a class of chip-firing game configurations that correspond to a subgroup of this Picard group. We investigate when the Picard group of the nth cone over G is the direct sum of this subgroup and another subgroup defined by the remaining game configurations. In particular, we show conditions on n for this decomposition to hold for a handful of different graphs, we show that this decomposition holds for infinitely many n on any graph, and we indicate a graph-theoretic property which necessitates this decomposition holds. This last result suggests there are graph properties that can elucidate the algebraic structure of the Picard group, which calls for further investigation. 

Table of Contents

1 Introduction 1

2 Background 7

2.1 Basic Definitions and Notation...................... 7

2.2 Combinatorial Foundations for Chip Firing . . . . . . . . . . . . . . . 18

2.3 The Geometry of a Divisor........................ 30

2.4 The Algebraic Turn............................ 40

3 Proof of Results 45

3.1 Theorem A ................................ 45

3.2 Theorem B ................................ 45

3.3 Theorem C ................................ 48

3.4 Theorem D ................................ 50

3.5 Theorem E ................................ 52

3.6 Theorem F ................................ 53

4 Applications and Further Speculation 55 

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Recreation Mathematics
Keyword	Algebraic Geometry Chip-Firing Graph Theory Combinatorics Algebra Mathematics
Committee Chair / Thesis Advisor	David Zureick-Brown, Emory University
Committee Members	Dwight Duffus, Emory University Susan Bredlau, Emory University

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