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

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 $B(X_{n\mathrm{/}k},{\sigma }^{\mathrm{\prime }},{\mathcal{L}}_{n\mathrm{/}k}^{\mathrm{\prime }})$ on the characteristic variety $X_{n\mathrm{/}k}$ for $Q_{n,k}(E,\tau )$. When $X_{n\mathrm{/}k}$ is isomorphic to $E^g$ or the symmetric power $S^{g}E$ we show the homomorphism $Q_{n,k}(E,\tau) \to B(X_{n/k},\sigma’,$\mathcal{L}$‘{n/k})$issurjective,thattherelationsfor$B(X{n/k},\sigma’,$\mathcal{L}$'_{n/k})$aregeneratedindegrees$\le 3$, and the non-commutative scheme ${\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^{g}E$, respectively. When $X_{n\mathrm{/}k}=E^g$ and $\tau =0$, the results about $B(X_{n\mathrm{/}k},{\sigma }^{\mathrm{\prime }},{\mathcal{L}}_{n\mathrm{/}k}^{\mathrm{\prime }})$ show that the morphism ${\mathrm{\Phi }}_{\mathrm{\mid }{\mathcal{L}}_{n\mathrm{/}k}\mathrm{\mid }}:E^g\to {\mathbb{P}}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.