Geometry of graph varieties

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:

1. a description and combinatorial interpretation of equations defining each variety set-theoretically;

2. a description of the irreducible components of $\mathcal {X}(G)$;

3. 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.