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    Geometry of graph varieties

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    Geometry of graph varieties.pdf (268.5Kb)
    Issue Date
    2003
    Author
    Martin, Jeremy L.
    Publisher
    American Mathematical Society
    Type
    Article
    Article Version
    Scholarly/refereed, publisher version
    Published Version
    http://www.ams.org/journals/tran/2003-355-10/S0002-9947-03-03321-X/S0002-9947-03-03321-X.pdf
    Version
    http://arxiv.org/abs/math.CO/0302089
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    Abstract
    A picture P of a graph G = (V,E) consists of a point P(v) for each vertex v ∈ V and a line P(e) for each edge e ∈ E, all lying in the projective plane over a field k and subject to containment conditions corresponding to incidence in G. A graph variety is an algebraic set whose points parametrize pictures of G. We consider three kinds of graph varieties: the picture space X(G) of all pictures; the picture variety V(G), an irreducible component of X(G) of dimension 2|V |, defined as the closure of the set of pictures on which all the P(v) are distinct; and the slope variety S(G), obtained by forgetting all data except the slopes of the lines P(e). We use combinatorial techniques (in particular, the theory of combinatorial rigidity) to obtain the following geometric and algebraic information on these varieties: (1) a description and combinatorial interpretation of equations defining each variety set-theoretically; (2) a description of the irreducible components of X(G); (3) a proof that V(G) and S(G) are Cohen-Macaulay when G satisfies a sparsity condition, rigidity independence. In addition, our techniques yield a new proof of the equality of two matroids studied in rigidity theory.
    Description
    First published in Transactions of the American Mathematical Society in volume 355 (2003), 4151--4169, published by the American Mathematical Society.
    URI
    http://hdl.handle.net/1808/6207
    Collections
    • Mathematics Scholarly Works [282]
    Citation
    Geometry of graph varieties, Transactions of the American Mathematical Society, 355 (2003), 4151--4169.

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    KU Libraries
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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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