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Geometry of graph varieties
Martin, Jeremy L.
Martin, Jeremy L.
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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.
Date
2003
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American Mathematical Society
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Geometry of graph varieties, Transactions of the American Mathematical Society, 355 (2003), 4151--4169.