Geometry of graph varieties
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- by Jeremy L. Martin PDF
- Trans. Amer. Math. Soc. 355 (2003), 4151-4169 Request permission
Abstract:
A picture $\mathbf {P}$ of a graph $G=(V,E)$ consists of a point $\mathbf {P}(v)$ for each vertex $v \in V$ and a line $\mathbf {P}(e)$ for each edge $e \in E$, all lying in the projective plane over a field $\mathbf k$ 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 $\mathcal {X}(G)$ of all pictures; the picture variety $\mathcal {V}(G)$, an irreducible component of $\mathcal {X}(G)$ of dimension $2|V|$, defined as the closure of the set of pictures on which all the $\mathbf {P}(v)$ are distinct; and the slope variety $\mathcal {S}(G)$, obtained by forgetting all data except the slopes of the lines $\mathbf {P}(e)$. We use combinatorial techniques (in particular, the theory of combinatorial rigidity) to obtain the following geometric and algebraic information on these varieties:
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a description and combinatorial interpretation of equations defining each variety set-theoretically;
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a description of the irreducible components of $\mathcal {X}(G)$;
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a proof that $\mathcal {V}(G)$ and $\mathcal {S}(G)$ 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.
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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
- MR Author ID: 717661
- Email: martin@math.umn.edu
- Received by editor(s): June 27, 2002
- Received by editor(s) in revised form: January 28, 2003
- Published electronically: May 15, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 4151-4169
- MSC (2000): Primary 05C10, 14N20; Secondary 05B35
- DOI: https://doi.org/10.1090/S0002-9947-03-03321-X
- MathSciNet review: 1990580