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

@misc{anghel2026recent,
  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}},
  author = {Cristian Anghel},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30027},
  eprint = {2610.30027},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30027},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - 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
AU  - Cristian Anghel
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.30027
UR  - https://arxiv.org/abs/2610.30027
AB  - 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≡4H (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 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≤ H^2 and the local σ≤2H^2. Second, we generalize the argument of the OpenAI manuscript to every dimension m≥2, 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))· H^m−2≤(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 ≥3, without small Cohen-Macaulay modules.
ER  -