Abstract
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.
Highlights
This paper examines a canonical homomorphism from, (, ) to the twisted homogeneous coordinate ring ( /, ′, L′ / ) on the characteristic variety / for, (, )
Tate and Van den Bergh showed that (, ) is a 3-dimensional regular algebra if and only if (, ) ∈ P2 − {12 points}. They introduced the notion of a twisted homogeneous coordinate ring [ATVdB90] (Odesskii and Feigin discovered this notion around the same time [FO89, p. 7] and [OF89, p. 208]), and showed that there is a surjective homomorphism
Corollaries 2.6 and 2.7, which appear to be new, give a criterion for -ampleness that is useful for the types of twisted homogeneous coordinate rings that appear in the study of, (, ). §3.1 records some results and notation from our earlier papers about, (, ) that are used in this paper
Summary
For a fixed and , the elliptic algebras , ( , ), defined by Feigin and Odesskii in 1989 [OF89], are noncommutative deformations of the polynomial ring on variables. Twisted homogeneous coordinate rings are noncommutative analogues (often deformations) of homogeneous coordinate rings (more precisely, section rings) for projective algebraic varieties. This paper uses the latter to study the former
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