Enumerative and Algebraic Aspects of Slope Varieties
Issue Date
2011-05-18Author
Enkosky, Thomas
Publisher
University of Kansas
Format
67 pages
Type
Dissertation
Degree Level
Ph.D.
Discipline
Mathematics
Rights
This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
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Show full item recordAbstract
The slope variety of a graph G is an algebraic variety whose points correspond to the slopes arising from point-line configurations of G. We start by reviewing the background material necessary to understand the theory of slope varieties. We then move on to slope varieties over finite fields and determine the size of this set. We show that points in this variety correspond to graphs without an induced path on four vertices. We then establish a bijection between graphs without an induced path on four vertices and series-parallel networks. Next, we study the defining polynomials of the slope variety in more detail. The polynomials defining the slope variety are understood but we show that those of minimal degree suffice to define the slope variety set theoretically. We conclude with some remarks on how we would define the slope variety for point-line configurations in higher dimensions.
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- Mathematics Dissertations and Theses [179]
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