{
  "id": "2205.09720",
  "version": "v1",
  "published": "2022-05-19T17:32:57.000Z",
  "updated": "2022-05-19T17:32:57.000Z",
  "title": "Smooth hypersurfaces in abelian varieties over arithmetic rings",
  "authors": [
    "Ariyan Javanpeykar",
    "Siddharth Mathur"
  ],
  "comment": "15 pages. Comments welcome",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $A$ be an abelian scheme of dimension at least four over a $\\mathbb{Z}$-finitely generated integral domain $R$ of characteristic zero, and let $L$ be an ample line bundle on $A$. We prove that the set of smooth hypersurfaces $D$ in $A$ representing $L$ is finite by showing that the moduli stack of such hypersurfaces has only finitely many $R$-points. We accomplish this by using level structures to interpolate finiteness results between this moduli stack and the stack of canonically polarized varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-19T17:32:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smooth hypersurfaces",
      "abelian varieties",
      "arithmetic rings",
      "moduli stack",
      "ample line bundle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}