{
  "abstract": "Let $\\mathcal{A}\\to S$ be an abelian scheme over a normal quasi-projective variety over a number field, and let $\\mathcal{X}\\subseteq\\mathcal{A}$ be a positive-dimensional subvariety. For every integer $t>0$, we prove that, outside the $t$-degeneracy locus of $\\mathcal{X}$, at most finitely many points lie in flat group subschemes of relative dimension less than $t$. In particular, this intersection is not Zariski dense whenever $\\mathcal{X}$ is $t$-nondegenerate. The proof combines a bounded height theorem with uniform large Galois orbit estimates, o-minimal point counting, and mixed Ax--Schanuel, following the Pila--Zannier strategy. The arithmetic input is an explicit bound for the complexity of endomorphism relations, obtained from geometry of numbers and height estimates for abelian varieties. As applications, we formulate a relative Mordell--Lang conjecture, prove it for section images in powers of certain abelian schemes, and establish injectivity of specialization of the group of sections outside a proper closed subset under variation and dimension hypotheses.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06518",
  "author_total": 4,
  "authors": [
    {
      "name": "Fabrizio Barroero"
    },
    {
      "name": "Laura Capuano"
    },
    {
      "name": "Tangli Ge"
    },
    {
      "name": "Francesco Tropeano"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8g32d1hqnscx5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06518",
  "fulltext_url": null,
  "id": "8g32d1hqnscx5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06518",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220729636",
  "pdf_url": "https://arxiv.org/pdf/2610.06518",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8g32d1hqnscx5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06518",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06518"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Degeneracy loci and unlikely intersections in abelian schemes",
  "title_text": "Degeneracy loci and unlikely intersections in abelian schemes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/8g32d1hqnscx5",
  "venue": "arXiv",
  "versions": []
}