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 ≥ 1, an elliptic curve E, and a point τ ∈ 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 Pⁿ⁻¹, and by the way in which the translation automorphism [Formula Presented] interacts with that geometry. For k ≥ 2 a similar phenomenon occurs: (E, τ) is replaced by (X_n/k, σ) where [Formula Presented] is the characteristic variety of the title and σ is an automorphism of it that is determined by the negative continued fraction for [Formula Presented]. There is a surjective morphism Φ: E^g → 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 [Formula Presented] on E^g. The generalized Fourier-Mukai transform associated to [Formula Presented] sends the set of isomorphism classes of degree-zero invertible [Formula Presented]-modules to the set of isomorphism classes of indecomposable locally free [Formula Presented]-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).