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

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.