This paper examines an algebraic variety that controls an important part of the structure and representation theory of the algebra $Q_{n,k}(E,��)$ introduced by Feigin and Odesskii. The $Q_{n,k}(E,��)$'s are a family of quadratic algebras depending on a pair of coprime integers $n>k\ge 1$, an elliptic curve $E$, and a point $��\in E$. It is already known that the structure and representation theory of $Q_{n,1}(E,��)$ is controlled by the geometry associated to $E$ embedded as a degree $n$ normal curve in the projective space $\mathbb P^{n-1}$, and by the way in which the translation automorphism $z\mapsto z+��$ interacts with that geometry. For $k\ge 2$ a similar phenomenon occurs: $(E,��)$ is replaced by $(X_{n/k},��)$ where $X_{n/k}\subseteq\mathbb P^{n-1}$ is the characteristic variety of the title and $��$ is an automorphism of it that is determined by the negative continued fraction for $\frac{n}{k}$. There is a surjective morphism $��:E^g \to X_{n/k}$ where $g$ is the length of that continued fraction. The main result in this paper is that $X_{n/k}$ is a quotient of $E^g$ by the action of an explicit finite group. We also prove some assertions made by Feigin and Odesskii. The morphism $��$ is the natural one associated to a particular invertible sheaf $\mathcal L_{n/k}$ on $E^g$. The generalized Fourier-Mukai transform associated to $\mathcal L_{n/k}$ sends the set of isomorphism classes of degree-zero invertible $\mathcal O_E$-modules to the set of isomorphism classes of indecomposable locally free $\mathcal O_E$-modules of rank $k$ and degree $n$. Thus $X_{n/k}$ has an importance independent of the role it plays in relation to $Q_{n,k}(E,��)$. The backward $��$-orbit of each point on $X_{n/k}$ determines a point module for $Q_{n,k}(E,��)$.

43 pages + index + references; a number of minor changes