{
  "abstract": "Let $X$ be an abelian surface of Picard rank 1 with ample generator $H$. We show that, for a Mukai vector $\\mathbf{v}=(r,mH,χ)$ with $r,m\\geq 1$ and $χ\\leq 0$, the first Brill-Noether loci $\\{E \\;|\\; H^0(E)\\neq 0\\}$ in $M_X(\\mathbf{v})$ is nonempty of the expected codimension $1-χ$. As an application, we study the rank 1 Segre invariant on these abelian surfaces. Specifically, for $\\mathbf{v}$ a Mukai vector of a $μ$-stable sheaf of $\\mathrm{rank}\\geq 2$, we compute the Segre invariant of the general sheaf in $M_X(\\mathbf{v})$, and we determine the numbers which arise as the Segre invariant of a sheaf in $M_X(\\mathbf{v})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24251",
  "author_total": 1,
  "authors": [
    {
      "name": "Jesse Chien"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/71aq6w33cdvba.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24251",
  "fulltext_url": null,
  "id": "71aq6w33cdvba",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24251",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223056050",
  "pdf_url": "https://arxiv.org/pdf/2610.24251",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/71aq6w33cdvba.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24251",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24251"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Codimension of Brill-Noether Loci on Abelian Surfaces and the First Segre Invariant",
  "title_text": "Codimension of Brill-Noether Loci on Abelian Surfaces and the First Segre Invariant",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/71aq6w33cdvba",
  "venue": "arXiv",
  "versions": []
}