{
  "id": "2103.04659",
  "version": "v1",
  "published": "2021-03-08T10:43:32.000Z",
  "updated": "2021-03-08T10:43:32.000Z",
  "title": "A footnote to a footnote to a paper of B. Segre",
  "authors": [
    "Luca Chiantini",
    "Giorgio Ottaviani"
  ],
  "comment": "Dedicated to Ciro Ciliberto, for his 70th birthday",
  "categories": [
    "math.AG"
  ],
  "abstract": "The paper is devoted to a detailed study of sextics in three variables having a decomposition as a sum of nine powers of linear forms. This is the unique case of a Veronese image of the plane which, in the terminology introduced by Ciliberto and the first author in [12], is weakly defective, and non-identifiable. The title originates from a paper of 1981, where Arbarello and Cornalba state and prove a result on plane curves with preassigned singularities, which is relevant to extend the studies of B. Segre on special linear series on curves. We explore the apolar ideal of a sextic $F$ and the associated catalecticant maps, in order to determine the minimal decompositions. A particular attention is played to the postulation of the decompositions. Starting with forms with a decomposition $A$ of length $9$, the postulation of $A$ determines several loci in the $9$-secant of the $6$-Veronese image of $\\mathbb P^2$, which include the lower secant varieties, and the ramification locus, where the decomposition is unique. We prove that equations of all these loci, including the $8$-th and the $7$-th secant varieties, are provided by minors of the catalecticant maps and by the invariant $H_{27}$ that we describe in Section 4.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-08T10:43:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N07"
    ],
    "keywords": [
      "decomposition",
      "veronese image",
      "th secant varieties",
      "lower secant varieties",
      "special linear series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}