  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
$B(X_{n/k},\sigma',\mathcal{L}'_{n/k})$ 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.
