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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 $n>kge 1$, an elliptic curve $E$, and a point $tauin E$. 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.
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We study the elliptic algebras $Q_{n,k}(E,tau)$ introduced by Feigin and Odesskii as a generalization of Sklyanin algebras. They form a family of quadratic algebras parametrized by coprime integers $n>kgeq 1$, an elliptic curve $E$, and a point $tauin E$. We consider and compare several different definitions of the algebras and provide proofs of various statements about them made by Feigin and Odesskii. For example, we show that $Q_{n,k}(E,0)$, and $Q_{n,n-1}(E,tau)$ are polynomial rings on $n$ variables. We also show that $Q_{n,k}(E,tau+zeta)$ is a twist of $Q_{n,k}(E,tau)$ when $zeta$ is an $n$-torsion point. This paper is the first of several we are writing about the algebras $Q_{n,k}(E,tau)$.
The algebras $Q_{n,k}(E,tau)$ introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers $n>kge 1$, a complex elliptic curve $E$, and a point $tauin E$. The main result in this paper is that $Q_{n,k}(E,tau)$ has the same Hilbert series as the polynomial ring on $n$ variables when $tau$ is not a torsion point. We also show that $Q_{n,k}(E,tau)$ is a Koszul algebra, hence of global dimension $n$ when $tau$ is not a torsion point, and, for all but countably many $tau$, it is Artin-Schelter regular. The proofs use the fact that the space of quadratic relations defining $Q_{n,k}(E,tau)$ is the image of an operator $R_{tau}(tau)$ that belongs to a family of operators $R_{tau}(z):mathbb{C}^notimesmathbb{C}^ntomathbb{C}^notimesmathbb{C}^n$, $zinmathbb{C}$, that (we will show) satisfy the quantum Yang-Baxter equation with spectral parameter.
This paper examines an algebraic variety that controls an important part of the structure and representation theory of the algebra $Q_{n,k}(E,tau)$ introduced by Feigin and Odesskii. The $Q_{n,k}(E,tau)$s are a family of quadratic algebras depending on a pair of coprime integers $n>kge 1$, an elliptic curve $E$, and a point $tauin E$. It is already known that the structure and representation theory of $Q_{n,1}(E,tau)$ is controlled by the geometry associated to $E$ embedded as a degree $n$ normal curve in the projective space $mathbb P^{n-1}$, and by the way in which the translation automorphism $zmapsto z+tau$ interacts with that geometry. For $kge 2$ a similar phenomenon occurs: $(E,tau)$ is replaced by $(X_{n/k},sigma)$ where $X_{n/k}subseteqmathbb P^{n-1}$ is the characteristic variety of the title and $sigma$ is an automorphism of it that is determined by the negative continued fraction for $frac{n}{k}$. There is a surjective morphism $Phi:E^g to X_{n/k}$ where $g$ is the length of that continued fraction. The main result in this paper is that $X_{n/k}$ is a quotient of $E^g$ by the action of an explicit finite group. We also prove some assertions made by Feigin and Odesskii. The morphism $Phi$ is the natural one associated to a particular invertible sheaf $mathcal L_{n/k}$ on $E^g$. The generalized Fourier-Mukai transform associated to $mathcal L_{n/k}$ sends the set of isomorphism classes of degree-zero invertible $mathcal O_E$-modules to the set of isomorphism classes of indecomposable locally free $mathcal O_E$-modules of rank $k$ and degree $n$. Thus $X_{n/k}$ has an importance independent of the role it plays in relation to $Q_{n,k}(E,tau)$. The backward $sigma$-orbit of each point on $X_{n/k}$ determines a point module for $Q_{n,k}(E,tau)$.
One of the difficulties in doing noncommutative projective geometry via explicitly presented graded algebras is that it is usually quite difficult to show flatness, as the Hilbert series is uncomputable in general. If the algebra has a regular central element, one can reduce to understanding the (hopefully more tractable) quotient. If the quotient is particularly nice, one can proceed in reverse and find all algebras of which it is the quotient by a regular central element (the filtered deformations of the quotient). We consider in detail the case that the quotient is an elliptic algebra (the homogeneous endomorphism ring of a vector bundle on an elliptic curve, possibly twisted by translation). We explicitly compute the family of filtered deformations in many cases and give a (conjecturally exhaustive) construction of such deformations from noncommutative del Pezzo surfaces. In the process, we also give a number of results on the classification of exceptional collections on del Pezzo surfaces, which are new even in the commutative case.
Let $E$ be an elliptic curve. When the symmetric group $Sigma_{g+1}$ of order $(g+1)!$ acts on $E^{g+1}$ in the natural way, the subgroup $E_0^{g+1}$, consisting of those $(g+1)$-tuples whose coordinates sum to zero, is stable under the action of $Sigma_{g+1}$. It is isomorphic to $E^g$. This paper concerns the structure of the quotient variety $E^g/Sigma$ when $Sigma$ is a subgroup of $Sigma_{g+1}$ generated by simple transpositions. In an earlier paper we observed that $E^g/Sigma$ is a bundle over a suitable power, $E^N$, with fibers that are products of projective spaces. This paper shows that $E^g/Sigma$ has an etale cover by a product of copies of $E$ and projective spaces with an abelian Galois group.
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