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

Abstract

The elliptic algebras in the title are connected graded $\mathbb{C}$-algebras, denoted $Q_{n,k}(E,\tau )$, depending on a pair of relatively prime integers $n>k\ge 1$, an elliptic curve $E$, and a point $\tau \in E$. This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},{\sigma }^{\prime },{\mathcal{L}}_{n/k}^{\prime })$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$. When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^{g}E$ we show the homomorphism $Q_{n,k}(E,\tau )\to B(X_{n/k},{\sigma }^{\prime },{\mathcal{L}}_{n/k}^{\prime })$ is surjective, that the relations for $B(X_{n/k},{\sigma }^{\prime },{\mathcal{L}}_{n/k}^{\prime })$ are generated in degrees $\le 3$, and the non-commutative scheme ${\mathrm{Proj}}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^{g}E$, respectively. When $X_{n/k}=E^g$ and $\tau =0$, the results about $B(X_{n/k},{\sigma }^{\prime },{\mathcal{L}}_{n/k}^{\prime })$ show that the morphism ${\Phi }_{|{\mathcal{L}}_{n/k}|}:E^g\to {\mathbb{P}}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

42 pages + references; a number of minor changes