{
  "abstract": "In their 2003 article in the Journal of the American Mathematical Society, Eisenbud, Schreyer and Weyman asked whether every embedded projective variety carries an Ulrich sheaf. In 2017 Beauville proposed a numerical route toward a surface with no Ulrich bundles: in Picard rank one, existence forces $H^2\\ge K_S^2-8χ(\\mathcal O_S)$, suggesting a search near the Bogomolov--Miyaoka--Yau boundary. We show that the Picard-rank-one hypothesis is not needed for the obstruction: a rank-independent Bogomolov--Hodge argument gives the same inequality on every smooth polarized surface. Using additional Neron--Severi directions on Hirzebruch--Kummer resolutions, the exponent-$3$ Hesse surface admits a very ample class $H=4A-E$ with $H^2=7\\cdot3^9<16\\cdot3^9=K_Y^2-8χ(\\mathcal O_Y)$, hence a smooth counterexample to the Eisenbud--Schreyer--Weyman problem in its standard formulation. Consequently its Chow form has no ESW-type linear determinantal representation arising from an Ulrich sheaf on the embedded surface. Moreover, for every $n\\ge3$ the Hesse pair $(Y_n,4A_n-E_n)$ is a counterexample, the surfaces are pairwise non-isomorphic, and $H_n^2/σ(Y_n)=7/(3n^2-11)\\to0$, while $K_{Y_n}^2/χ(\\mathcal O_{Y_n})\\to60/7\\approx8.5714$. A general arrangement-theoretic Rees-algebra/Segre mechanism yields further infinite families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02718",
  "author_total": 1,
  "authors": [
    {
      "name": "Cristian Anghel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9tf8kgjgkfhet.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02718",
  "fulltext_url": null,
  "id": "9tf8kgjgkfhet",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02718",
  "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.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7207535803",
  "pdf_url": "https://arxiv.org/pdf/2609.02718",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9tf8kgjgkfhet.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02718",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02718"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Smooth Counterexamples to the Eisenbud--Schreyer--Weyman Ulrich Existence Problem",
  "title_text": "Smooth Counterexamples to the Eisenbud--Schreyer--Weyman Ulrich Existence Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/9tf8kgjgkfhet",
  "venue": "arXiv",
  "versions": []
}