{
  "id": "2002.12366",
  "version": "v1",
  "published": "2020-02-27T16:24:41.000Z",
  "updated": "2020-02-27T16:24:41.000Z",
  "title": "On the Hilbert scheme of smooth curves in $\\mathbb{P}^4$ of degree $d = g+1$ and genus $g$ with negative Brill-Noether number",
  "authors": [
    "Changho Keem",
    "Yun-Hwan Kim"
  ],
  "comment": "17 pages. arXiv admin note: text overlap with arXiv:1903.02307, arXiv:1904.07716",
  "categories": [
    "math.AG"
  ],
  "abstract": "We denote by $\\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\\PP^r$. In this article, we show that for low genus $g$ outside the Brill-Noether range, the Hilbert scheme $\\mathcal{H}_{g+1,g,4}$ is non-empty whenever $g\\ge 9$ and irreducible whose only component generically consists of linearly normal curves unless $g=9$ or $g=12$. This complements the validity of the original assertion of Severi regarding the irreducibility of $\\mathcal{H}_{d,g,r}$ outside the Brill-Nother range for $d=g+1$ and $r=4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-27T16:24:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C05",
      "14H10"
    ],
    "keywords": [
      "hilbert scheme",
      "negative brill-noether number",
      "smooth curves",
      "general point corresponds",
      "linearly normal curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}