Ulrich Sheaves on the Hilbert Square of K3 and Abelian Surfaces

TL;DR

This paper demonstrates the existence of Ulrich sheaves on the Hilbert square of K3 and abelian surfaces, providing a new construction method and bounds for their complexity.

Contribution

It introduces a novel approach to constructing Ulrich sheaves on Hilbert squares via descent and lifting techniques, and estimates their complexity bounds.

Findings

Ulrich sheaves exist on the Hilbert square of K3 and abelian surfaces.

A new construction method using descent and lifting is proposed.

Bounds for Ulrich complexity of the Hilbert square are established.

Abstract

We prove the existence of Ulrich sheaves on the Hilbert scheme of two points on a polarized K3 surface or an abelian surface. The construction proceeds by descending Ulrich bundles on the surface to the symmetric square and lifting them to the Hilbert square via the crepant Hilbert--Chow resolution. Finally, we estimate a bound for Ulrich complexity of the Hilbert Square.