Geometry of graph varieties

Author:
Jeremy L. Martin

Journal:
Trans. Amer. Math. Soc. **355** (2003), 4151-4169

MSC (2000):
Primary 05C10, 14N20; Secondary 05B35

Published electronically:
May 15, 2003

MathSciNet review:
1990580

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Abstract | References | Similar Articles | Additional Information

Abstract: A *picture* of a graph consists of a point for each vertex and a line for each edge , all lying in the projective plane over a field and subject to containment conditions corresponding to incidence in . A *graph variety* is an algebraic set whose points parametrize pictures of . We consider three kinds of graph varieties: the *picture space* of all pictures; the *picture variety* , an irreducible component of of dimension , defined as the closure of the set of pictures on which all the are distinct; and the *slope variety* , obtained by forgetting all data except the slopes of the lines . 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 ;
- (3)
- a proof that and are Cohen-Macaulay when satisfies a sparsity condition,
*rigidity independence*.

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Additional Information

**Jeremy L. Martin**

Affiliation:
Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112

Address at time of publication:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
martin@math.umn.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-03-03321-X

Keywords:
Graphs,
graph varieties,
configuration varieties

Received by editor(s):
June 27, 2002

Received by editor(s) in revised form:
January 28, 2003

Published electronically:
May 15, 2003

Article copyright:
© Copyright 2003
American Mathematical Society