Geometry of graph varieties

• Published in: Transactions of the American Mathematical Society 2003-10, Vol.355 (10), p.4151-4169
• Main Author: Martin, Jeremy L.
• Format: Article
• Language: English
• Subjects: Algebra; Algebraic geometry; Combinatorics; Combinatorics. Ordered structures; Coordinate systems; Designs and configurations; Exact sciences and technology; Fall lines; Geometric planes; Graph theory; Line graphs; Mathematical theorems; Mathematics; Matroids; Polynomials; Sciences and techniques of general use; Slope of a line; Vertices
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-03-03321-X

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: a description and combinatorial interpretation of equations defining each variety set-theoretically; a description of the irreducible components of \mathcal {X}(G); 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.