The characteristic variety for Feigin and Odesskii's elliptic algebras

TL;DR

This paper studies the geometric structure called the characteristic variety associated with Feigin and Odesskii's elliptic algebras, revealing it as a quotient of a product of elliptic curves and exploring its implications for algebra representations.

Contribution

It characterizes the characteristic variety $X_{n/k}$ as a quotient of $E^g$ by a finite group and connects it to sheaf theory and module classification, extending prior work on related algebras.

Findings

$X_{n/k}$ is a quotient of $E^g$ by a finite group.

The morphism $ ext{E}^g o X_{n/k}$ is associated with a specific invertible sheaf.

The generalized Fourier-Mukai transform relates invertible sheaves to indecomposable modules.

Abstract

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 $Q_{n,k}(E,\tau)$.