The characteristic variety for Feigin and Odesskiis elliptic algebras

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 $n>kge 1$, an elliptic curve $E$, and a point $tauin E$. 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 $zmapsto z+tau$ interacts with that geometry. For $kge 2$ a similar phenomenon occurs: $(E,tau)$ is replaced by $(X_{n/k},sigma)$ where $X_{n/k}subseteqmathbb P^{n-1}$ 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 $Q_{n,k}(E,tau)$.