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Chip Firing, Riemann-Roch, and Brill-Noether Theory for Graphs

Vargas De León, A. J. (2016) Chip Firing, Riemann-Roch, and Brill-Noether Theory for Graphs. Master's Thesis / Essay, Mathematics.

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Abstract

We survey and motivate the theory behind a chip firing game defined on finite graphs. Configurations of this game can be represented as divisors, elements of the free abelian group on the set of vertices. The set of divisors one can reach via firing moves induces a linear equivalence class for each divisor. We study the problem of finding equivalent effective divisors, as it's done in algebraic curves, and other results in the area which are also otivated by analogy. The exposition will focus particularly in the notion of rank for divisors formulated by M. Baker and S. Norine. As research problem we will investigate a conjecture by Baker proposing a Brill-Noether theory for graphs. We will deal specifically with finding rank-1 divisors, the simplest non-trivial case. The existence of the rank-1 divisors with the degree predicted by the conjecture is an open question. We implement several algorithms in the literature, and with these programs generate a wealth of examples, contributing to the understanding of these divisors and their behaviour.

Item Type: Thesis (Master's Thesis / Essay)
Degree programme: Mathematics
Thesis type: Master's Thesis / Essay
Language: English
Date Deposited: 15 Feb 2018 08:24
Last Modified: 15 Feb 2018 08:24
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/14436

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