Eisenbud–Schreyer conjecture on the existence of Ulrich sheaves

Establish that for every n-dimensional closed subscheme X ⊂ P^N polarized by a very ample divisor class D, there exists a non-zero coherent sheaf E on X that is Ulrich with respect to D; equivalently, determine that the set of Ulrich ranks (X) is non-empty for every such polarized variety.

Background

The paper defines the set of Ulrich ranks (X) as the set of positive integers r for which there exists an Ulrich sheaf on X of rank r. An Ulrich sheaf E on an n-dimensional closed subscheme X ⊂ PN polarized by a very ample divisor class D is a non-zero coherent sheaf satisfying H*(E(−tD))=0 for 1 ≤ t ≤ n.

The cited conjecture of Eisenbud and Schreyer posits that every polarized projective variety admits an Ulrich sheaf, i.e., that (X) is always non-empty. This remains a central open problem in the area of Ulrich bundles and is independent of the specific focus of the present paper on Veronese varieties.

References

A fundamental conjecture of states that $(X)$ should be always non empty, while the main point of is to compute the lower bound of $(X)$, called the Ulrich complexity of $X$.

— Ulrich ranks of Veronese varieties and equivariant instantons  (2405.12574 - Faenzi et al., 2024) in Introduction