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    Enumerative and Algebraic Aspects of Slope Varieties

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    Enkosky_ku_0099D_11634_DATA_1.pdf (417.1Kb)
    Issue Date
    2011-05-18
    Author
    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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    Abstract
    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.
    URI
    http://hdl.handle.net/1808/8062
    Collections
    • Dissertations [4660]
    • Mathematics Dissertations and Theses [179]

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    KU Libraries
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    785-864-8983

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    Lawrence, KS 66045
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    Contact KU ScholarWorks
    785-864-8983
    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    785-864-8983

    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    Image Credits
     

     

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