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 $n>kge 1$, an elliptic curve $E$, and a point $tauin E$. This paper examines a canonical homomorphism from $Q_{n,k}(E,tau)$ to the twisted homogeneous coordinate ring $B(X_{n/k},sigma,mathcal{L}_{n/k})$ 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 $S^gE$ we show the homomorphism $Q_{n,k}(E,tau) to B(X_{n/k},sigma,mathcal{L}_{n/k})$ is surjective, that the relations for $B(X_{n/k},sigma,mathcal{L}_{n/k})$ 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 $S^gE$, respectively. When $X_{n/k}=E^g$ and $tau=0$, the results about $B(X_{n/k},sigma,mathcal{L}_{n/k})$ 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.
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