{
  "id": "2409.03960",
  "version": "v1",
  "published": "2024-09-06T00:48:30.000Z",
  "updated": "2024-09-06T00:48:30.000Z",
  "title": "Extendability of projective varieties via degeneration to ribbons with applications to Calabi-Yau threefolds",
  "authors": [
    "Purnaprajna Bangere",
    "Jayan Mukherjee"
  ],
  "comment": "30 pages, 3 figures, Comments are welcome !",
  "categories": [
    "math.AG"
  ],
  "abstract": "In this article we study the extendability of a smooth projective variety by degenerating it to a ribbon. We apply the techniques to study extendability of Calabi-Yau threefolds $X_t$ that are general deformations of Calabi-Yau double covers of Fano threefolds of Picard rank $1$. The Calabi-Yau threefolds $X_t \\hookrightarrow \\mathbb{P}^{N_l}$, embedded by the complete linear series $|lA_t|$, where $A_t$ is the generator of Pic$(X_t)$, $l \\geq j$ and $j$ is the index of $Y$, are general elements of a unique irreducible component $\\mathscr{H}_l^Y$ of the Hilbert scheme which contains embedded Calabi-Yau ribbons on $Y$ as a special locus. For $l = j$, using the classification of Mukai varieties, we show that the general Calabi-Yau threefold parameterized by $\\mathscr{H}_j^Y$ is as many times smoothly extendable as $Y$ itself. On the other hand, we find for each deformation type $Y$, an effective integer $l_Y$ such that for $l \\geq l_Y$, the general Calabi-Yau threefold parameterized by $\\mathscr{H}_l^Y$ is not extendable. These results provide a contrast and a parallel with the lower dimensional analogues; namely, $K3$ surfaces and canonical curves, which stems from the following result we prove: for $l \\geq l_Y$, the general hyperplane sections of elements of $\\mathscr{H}_l^Y$ fill out an entire irreducible component $\\mathscr{S}_l^Y$ of the Hilbert scheme of canonical surfaces which are precisely $1-$ extendable with $\\mathscr{H}^Y_l$ being the unique component dominating $\\mathscr{S}_l^Y$. The contrast lies in the fact that for polarized $K3$ surfaces of large degree, the canonical curve sections do not fill out an entire component while the parallel is in the fact that the canonical curve sections are exactly one-extendable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-06T00:48:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N05",
      "14J29",
      "14J30",
      "14J45"
    ],
    "keywords": [
      "projective variety",
      "general calabi-yau threefold",
      "extendability",
      "canonical curve sections",
      "hilbert scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}