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
$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$
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.
Hilbert series of modules of GK-dimension two over elliptic algebras
