{
  "id": "2109.06303",
  "version": "v1",
  "published": "2021-09-13T20:29:24.000Z",
  "updated": "2021-09-13T20:29:24.000Z",
  "title": "On the degree of algebraic cycles on hypersurfaces",
  "authors": [
    "Matthias Paulsen"
  ],
  "comment": "12 pages; comments welcome",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X\\subset\\mathbb P^4$ be a very general hypersurface of degree $d\\ge6$. Griffiths and Harris conjectured in 1985 that the degree of every curve $C\\subset X$ is divisible by $d$. Despite substantial progress by Koll\\'ar in 1991, this conjecture is not known for a single value of $d$. Building on Koll\\'ar's method, we prove this conjecture for infinitely many $d$, the smallest one being $d=5005$. The set of these degrees $d$ has positive density. We also prove a higher-dimensional analogue of this result and construct smooth hypersurfaces defined over $\\mathbb Q$ that satisfy the conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-13T20:29:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C25",
      "14C30",
      "14D06",
      "14J70"
    ],
    "keywords": [
      "algebraic cycles",
      "despite substantial progress",
      "conjecture",
      "construct smooth hypersurfaces",
      "general hypersurface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}