{
  "id": "1908.06525",
  "version": "v1",
  "published": "2019-08-18T22:32:02.000Z",
  "updated": "2019-08-18T22:32:02.000Z",
  "title": "Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings",
  "authors": [
    "Alex Chirvasitu",
    "Ryo Kanda",
    "S. Paul Smith"
  ],
  "comment": "42 pages + references",
  "categories": [
    "math.AG",
    "math.QA",
    "math.RA"
  ],
  "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 $\\tau\\in E$. For fixed $n$ and $k$, they form a flat family of deformations of the polynomial ring $\\mathbb{C}[x_0,\\ldots,x_{n-1}]$. 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 $|\\mathcal{L}_{n/k}|:E^g \\to \\mathbb{P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-18T22:32:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14A22",
      "16S38",
      "16W50",
      "14H52",
      "14F05"
    ],
    "keywords": [
      "twisted homogeneous coordinate ring",
      "odesskiis elliptic algebras",
      "isomorphic",
      "paper examines",
      "elliptic curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}