Geometry of graph varieties

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.