{
  "id": "2204.09152",
  "version": "v1",
  "published": "2022-04-19T22:57:17.000Z",
  "updated": "2022-04-19T22:57:17.000Z",
  "title": "Feigin-Odesskii brackets, syzygies, and Cremona transformations",
  "authors": [
    "Alexander Polishchuk"
  ],
  "comment": "10 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "We identify Feigin-Odesskii brackets $q_{n,1}(C)$, associated with a normal elliptic curve of degree $n$, $C\\subset {\\mathbb P}^{n-1}$, with the skew-symmetric $n\\times n$ matrix of quadratic forms introduced by Fisher in arXiv:1510.04327 in connection with some minimal free resolutions related to the secant varieties of $C$. On the other hand, we show that for odd $n$, the generators of the ideal of the secant variety of $C$ of codimension $3$ give a Cremona transformation of ${\\mathbb P}^{n-1}$, generalizing the quadro-cubic Cremona transformation of ${\\mathbb P}^4$. We identify this transformation with the one considered in arXiv:alg-geom/9712022 and find explict formulas for the inverse transformation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-19T22:57:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "secant variety",
      "quadro-cubic cremona transformation",
      "minimal free resolutions",
      "normal elliptic curve",
      "identify feigin-odesskii brackets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}