{
  "abstract": "In September 2026 four preprints and an AI-generated manuscript moved one numerical obstruction from projective geometry to local algebra. For a smooth complex projective surface $Y$ with an ample, globally generated divisor $H$ such that $H^2<σ(Y)$, the signature, one obtains in turn: no Ulrich bundles when $H$ is very ample (Anghel, arXiv:2609.02718; extended to every dimension by Lopez, arXiv:2609.06678), answering negatively the Eisenbud-Schreyer-Weyman question of 2003 for sheaves supported on the whole variety; no arithmetically Cohen-Macaulay bundles, i.e. section rings without graded maximal Cohen-Macaulay modules in characteristic zero (Anghel, arXiv:2609.15589); and complete local domains without small Cohen-Macaulay modules, counterexamples to Hochster's 1973 conjecture, if $K_Y\\equiv4H$ (Chen, arXiv:2609.24142, proposed) or $σ>2H^2$ (OpenAI, released 6 October). We recall the definitions and main statements, and add two things. First, through a finite projection to $\\mathbb P^2$ and an identity already present in the OpenAI manuscript, the three obstructions become one variance identity; for surfaces this explains the factor $2$ between the graded bound $σ\\le H^2$ and the local $σ\\le2H^2$. Second, we generalize the argument of the OpenAI manuscript to every dimension $m\\ge2$, with complete proofs: if $X$ is a smooth complex projective variety of dimension $m$ and $H$ an ample, globally generated divisor, a small Cohen-Macaulay module over the completed section ring of $(X,H)$ forces $(K_X^2-2c_2(X))\\cdot H^{m-2}\\le(m+4)H^m$. Consequently the completed section rings of all Hesse pairs, including the ball quotient of the first preprint, have no small Cohen-Macaulay module, and products of Hesse surfaces with projective spaces give complete local domains with an isolated singularity, of every dimension $\\ge3$, without small Cohen-Macaulay modules.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30027",
  "author_total": 1,
  "authors": [
    {
      "name": "Cristian Anghel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3hky2jadzkd4v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30027",
  "fulltext_url": null,
  "id": "3hky2jadzkd4v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30027",
  "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",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7223241606",
  "pdf_url": "https://arxiv.org/pdf/2610.30027",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3hky2jadzkd4v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.30027",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30027"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The recent counterexamples to the Eisenbud-Schreyer-Weyman Ulrich problem (2003) and to Hochster's small Cohen-Macaulay conjecture (1973): a survey, with a generalization to arbitrary dimension",
  "title_text": "The recent counterexamples to the Eisenbud-Schreyer-Weyman Ulrich problem (2003) and to Hochster's small Cohen-Macaulay conjecture (1973): a survey, with a generalization to arbitrary dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/3hky2jadzkd4v",
  "venue": "arXiv",
  "versions": []
}