Existence of Ulrich bundles on polarized surfaces and weakly Ulrich bundles on polarized varieties in positive characteristic

Source abstract

In their seminal paper \cite{eisenbud2003resultants}, Eisenbud and Schreyer asked whether every smooth projective variety admits an Ulrich bundle with respect to each very ample polarization. This question has stimulated extensive research for more than two decades. Recently, Anghel \cite{anghel2026smooth} constructed counterexamples over $\mathbb C$ in dimension two, and Lopez \cite{lopez2026existence} extended this nonexistence result to higher dimensions. In this article, we study the existence problem for sufficiently large multiples of a fixed polarization and derive applications in positive characteristic. Over an algebraically closed field of positive characteristic, we prove that every polarized smooth surface admits an Ulrich bundle and every polarized smooth variety admits a weakly Ulrich bundle. Consequently, the Chow forms of surfaces (resp. varieties) admit linear determinantal (resp. determinantal) representations. Moreover, for every such polarized surface, the cone of cohomology tables of coherent sheaves coincides with that on $(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(1))$.