Paper detail

The characteristic variety for Feigin and Odesskii's elliptic algebras

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