  This paper examines an algebraic variety that controls an important part of
the structure and representation theory of the algebra $Q_{n,k}(E,\tau)$
introduced by Feigin and Odesskii. The $Q_{n,k}(E,\tau)$'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 $\tau\in E$. It is already known that the
structure and representation theory of $Q_{n,1}(E,\tau)$ 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 $z\mapsto z+\tau$ interacts with that geometry. For $k\ge 2$ a
similar phenomenon occurs: $(E,\tau)$ is replaced by $(X_{n/k},\sigma)$ where
$X_{n/k}\subseteq\mathbb P^{n-1}$ is the characteristic variety of the title
and $\sigma$ is an automorphism of it that is determined by the negative
continued fraction for $\frac{n}{k}$. There is a surjective morphism $\Phi:E^g
\to X_{n/k}$ 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 $\Phi$ 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 $Q_{n,k}(E,\tau)$. The backward $\sigma$-orbit of each point on
$X_{n/k}$ determines a point module for $Q_{n,k}(E,\tau)$.
