# Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

## TL;DR

This paper explores the relationship between elliptic algebras and twisted homogeneous coordinate rings, establishing conditions for surjectivity, relation degrees, and geometric embeddings of associated varieties.

## Contribution

It provides new results on the surjectivity of homomorphisms from elliptic algebras to coordinate rings and describes geometric properties of the characteristic varieties.

## Key findings

- Homomorphism from $Q_{n,k}(E, au)$ to $B(X_{n/k},\sigma',\mathcal{L}'_{n/k})$ is surjective when $X_{n/k}$ is isomorphic to $E^g$ or $S^gE$.
- Relations for $B(X_{n/k},\sigma',\mathcal{L}'_{n/k})$ are generated in degrees ≤ 3.
- When $X_{n/k}=E^g$ and $ au=0$, $E^g$ embeds as a projectively normal subvariety in projective space.

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