{
  "id": "2408.03715",
  "version": "v1",
  "published": "2024-08-07T12:04:01.000Z",
  "updated": "2024-08-07T12:04:01.000Z",
  "title": "On the genus of projective curves not contained in hypersurfaces of given degree, II",
  "authors": [
    "Vincenzo Di Gennaro",
    "Giambattista Marini"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Fix integers $r\\geq 4$ and $i\\geq 2$. Let $C$ be a non-degenerate, reduced and irreducible complex projective curve in $\\mathbb P^r$, of degree $d$, not contained in a hypersurface of degree $\\leq i$. Let $p_a(C)$ be the arithmetic genus of $C$. Continuing previous research, under the assumption $d\\gg \\max\\{r,i\\}$, in the present paper we exhibit a Castelnuovo bound $G_0(r;d,i)$ for $p_a(C)$. In general, we do not know whether this bound is sharp. However, we are able to prove it is sharp when $i=2$, $r=6$ and $d\\equiv 0,3,6$ (mod $9$). Moreover, when $i=2$, $r\\geq 9$, $r$ is divisible by $3$, and $d\\equiv 0$ (mod $r(r+3)/6$), we prove that if $G_0(r;d,i)$ is not sharp, then for the maximal value of $p_a(C)$ there are only three possibilities. The case in which $i=2$ and $r$ is not divisible by $3$ has already been examined in the literature. We give some information on the extremal curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-07T12:04:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N15",
      "14N25",
      "14M05",
      "14J26",
      "14J70"
    ],
    "keywords": [
      "hypersurface",
      "irreducible complex projective curve",
      "castelnuovo bound",
      "maximal value",
      "fix integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}