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Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings
Forum of Mathematics, Sigma ( IF 1.389 ) Pub Date : 2021-01-12 , DOI: 10.1017/fms.2020.60
Alex Chirvasitu , Ryo Kanda , S. Paul Smith

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.

中文翻译:

从 Feigin 和 Odesskii 的椭圆代数到扭曲齐次坐标环的映射

题目中的椭圆代数是连通分级的 $\mathbb {C}$ -代数,表示 $Q_{n,k}(E,\tau )$ , 取决于一对相对素数的整数 $n>k\ge 1$ , 一条椭圆曲线和一点 $\tau \in E$ . 本文研究了一个典型的同态 $Q_{n,k}(E,\tau )$ 到扭曲齐次坐标环 $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ 关于特色品种 $X_{n/k}$ 为了 $Q_{n,k}(E,\tau )$ . 什么时候 $X_{n/k}$ 同构于 $E^g$ 或对称功率 $S^gE$ ,我们证明了同态 $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ 是满射的,关系为 $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ 以度数生成 $\le 3$ 和非交换方案 $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ 有一个封闭的子变体,它同构于 $E^g$ 要么 $S^gE$ , 分别。什么时候 $X_{n/k}=E^g$ $\tau =0$ , 结果关于 $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ 证明态射 $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ 嵌入 $E^g$ 作为一个投影正态子变体,它是二次和三次超曲面的方案理论交集。
更新日期:2021-01-12
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