## Geometry of graph varieties

##### 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

##### Metadata

Show full item record##### 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.

##### Collections

##### Citation

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

Items in KU ScholarWorks are protected by copyright, with all rights reserved, unless otherwise indicated.

We want to hear from you! Please share your stories about how Open Access to this item benefits YOU.