For every smooth projective variety X over an algebraically closed field and every very ample line bundle L on X, there exists a nonzero vector bundle E on X such that H^i bigl(X,E otimes L^-jbigr)=0 for all i≥ 0 and all integers j with 1≤ j≤ dim X. Such an E is an Ulrich bundle with respect to L.

Eisenbud–Schreyer Ulrich-bundle existence question

Eisenbud, Schreyer, and Weyman asked in 2003 whether every projective variety carries an Ulrich sheaf. The question remains the universal existence assertion, including arbitrary varieties and characteristics.

Known results

Curves, complete intersections, Grassmannians, del Pezzo surfaces, general $K3$ surfaces, abelian surfaces, and several other classes have positive results.
Rational surfaces under suitable hypotheses admit stable rank- $2$ Ulrich bundles (Anghel and collaborators, 2014), yielding Pfaffian Chow-form expressions.
In positive characteristic, weakly or almost Ulrich bundles exist in several surface and low-dimensional settings (2019).
Rank- $2$ existence has additional obstructions; these do not exclude Ulrich bundles of arbitrary rank.

Recent claimed counterexamples and positive-characteristic results

Two recent arXiv papers claim surfaces, and in one case varieties in every dimension $n\ge 3$ , with no Ulrich bundle for specified polarizations; one derives an obstruction $H^2\ge K_S^2-8\chi(\mathcal O_S)$ and applies it to Hesse surfaces. The retrieved sources provide no independent assessment, so these remain claims. Jagadish Pine proves positive-characteristic existence results for polarized surfaces and weakly Ulrich bundles in higher dimensions, which does not restore the universal assertion.

Current status (as of October 2026): Classical special cases and positive-characteristic weak-existence results are established, while the universal existence question remains open and two purported counterexamples are unverified.

Eisenbud–Schreyer Ulrich-bundle existence question

Progress summary

Solutions 0