Maps from Feigin and Odesskii’s elliptic algebras to twisted homogeneous coordinate rings

The elliptic algebras in the title are connected graded C-algebras, denoted Q_n,k (E, τ), depending on a pair of relatively prime integers n > k ≥ 1, an elliptic curve E and a point τ ∈ E. This paper examines a canonical homomorphism from Q_n,k (E, τ) to the twisted homogeneous coordinate ring B(X_n/k, σ^′, L_n^′_/k) on the characteristic variety X_n/_k for Q_n,k (E, τ). When X_n/_k is isomorphic to E^g or the symmetric power S^gE, we show that the homomorphism Q_n,k (E, τ) → B(X_n/k, σ^′, L_n^′_/k) is surjective, the relations for B(X_n/k, σ^′, L_n^′_/k) are generated in degrees ≤ 3 and the noncommutative scheme Proj_nc(Q_n,k (E, τ)) has a closed subvariety that is isomorphic to E^g or S^gE, respectively. When X_n/_k = E^g and τ = 0, the results about B(X_n/k, σ^′, L_n^′_/k) show that the morphism Φ_|Ln/_{k |} : E^g → Pⁿ⁻¹ embeds E^g as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.