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
$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ 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 that the homomorphism
$Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for
$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees
$\le 3$ and the noncommutative 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
$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ 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.