{
  "abstract": "We study the Brill--Noether theory of higher rank sheaves on surfaces $X$ with $H^1(X,\\mathcal{O}_X) = 0$ and its interactions with the classical Brill--Noether theory of line bundles on curves lying on $X$. Denoting by $\\mathrm{Spl}(v)$ the moduli space of simple sheaves on $X$ with Chern character $v = (r,c_1,\\mathrm{ch}_2)$, we introduce the Brill--Noether loci $BN^k(v)^\\circ \\subseteq \\mathrm{Spl}(v)$ of \\emph{generically $k$-generated} sheaves and show, under assumptions on $c_1$ and $K_X$, certain well-behavedness results. In particular, if $c_1$ is \\emph{indecomposable} and $X$ is a K3 surface, we generalize a Brill--Noether theorem for sheaves on K3s due to Yoshioka. This is achieved by employing He's deformation theory of the moduli space of coherent systems and a globalization of the Lazarsfeld--Mukai construction, providing a correspondence between the Brill--Noether theories of generically $r$-generated sheaves on $X$ and that of line bundles on curves $C \\in |c_1|$. The same techniques yield upper bounds on the dimension of $\\mathcal{W}^{r,\\mathrm{bpf}}_d(|c_1|) \\setminus \\mathcal{W}^{r+1}_d(|c_1|)$ consisting of basepoint-free line bundles in terms of the number of endomorphisms of associated Lazarsfeld--Mukai bundles, for arbitrary $c_1$. This generalizes a result of Aprodu and Farkas in the context of Green's conjecture beyond K3 surfaces. In the case $r = 1$ of pencils, we obtain smoothness and expected-dimension results under an effectivity condition on $-K_X|_C$ for $C \\in |c_1|$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06824",
  "author_total": 2,
  "authors": [
    {
      "name": "Samir Geiger"
    },
    {
      "name": "Andreas Kretschmer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bh3rdktyknw2b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06824",
  "fulltext_url": null,
  "id": "bh3rdktyknw2b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06824",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7212029868",
  "pdf_url": "https://arxiv.org/pdf/2609.06824",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bh3rdktyknw2b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06824",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06824"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Brill--Noether Theory of Sheaves on Regular Surfaces",
  "title_text": "On the Brill--Noether Theory of Sheaves on Regular Surfaces",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:29Z",
  "url": "https://research.draconitic.com/paper/bh3rdktyknw2b",
  "venue": "arXiv",
  "versions": []
}