For every projective variety X⊆P^N over a field, equipped with its given embedding, does there exist a nonzero coherent sheaf E on X that is Ulrich with respect to O_X(1)? Equivalently, writing S=k[x_0,…,x_N] and M(E)=bigoplus_t∈ZH^0(X,E(t)), does there exist such an E for which M(E) has a linear minimal graded free resolution over S, namely 0→ F_c→…→ F_0→ M(E)→0 with c=N-dim X and each F_i a direct sum of copies of S(-i)?

Eisenbud–Schreyer–Weyman Ulrich Existence Problem

For every projective variety $X\subseteq\mathbb{P}^N$ over a field, equipped with its given embedding, does there exist a nonzero coherent sheaf $\mathcal{E}$ on $X$ that is Ulrich with respect to $\mathcal{O}_X(1)$ ? Equivalently, writing $S=k[x_0,\ldots,x_N]$ and $M(\mathcal{E})=\bigoplus_{t\in\mathbb{Z}}H^0(X,\mathcal{E}(t))$ , does there exist such an $\mathcal{E}$ for which $M(\mathcal{E})$ has a linear minimal graded free resolution over $S$ , namely $0\to F_c\to\cdots\to F_0\to M(\mathcal{E})\to0$ with $c=N-\dim X$ and each $F_i$ a direct sum of copies of $S(-i)$ ?

References

Primary source

arXiv

Additional references

Smooth Counterexamples to the Eisenbud--Schreyer--Weyman Ulrich Existence Problem — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A September 2026 preprint claims to disprove the general existence conjecture by giving smooth-surface counterexamples, but the result has not been independently confirmed.

The problem asks whether every projective variety admits an Ulrich sheaf, a conjecture associated with Eisenbud, Schreyer, and Weyman. The new work claims that the standard formulation is false.

Known results

The conjecture is known for curves, hypersurfaces, complete intersections, linear determinantal varieties, Grassmannians, Segre varieties, and generic K3 surfaces.
For $n \ge 4$ and $a \ge 2$ , the Veronese embedding $(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n}(a))$ has no Ulrich vector bundle of rank $r \le 3$ (2024).
Derived-categorical “Ulrich objects” generalize Ulrich bundles, but do not solve the standard existence problem (2025).

September 2026 smooth-surface counterexamples

The preprint Smooth Counterexamples to the Eisenbud--Schreyer--Weyman Ulrich Existence Problem removes the Picard-rank-one restriction from a numerical obstruction and exhibits Hesse-surface embeddings violating it. It therefore claims counterexamples, as well as further infinite families, and rules out the corresponding Ulrich-based determinantal representations for those embeddings.

Current status (as of September 2026): The general existence conjecture is claimed false by an unrefereed preprint, while its counterexamples and the resulting standard-formulation conclusion remain unverified.

Sources

No solutions have been posted yet.