Ulrich subvarieties and the non-existence of low rank Ulrich bundles on complete intersections

TL;DR

This paper characterizes the existence of Ulrich bundles on varieties and proves that most high-dimensional complete intersections do not admit low-rank Ulrich bundles, except in specific cases involving quadrics.

Contribution

It provides a new characterization of Ulrich bundles via subvarieties and establishes non-existence results for low-rank Ulrich bundles on general complete intersections.

Findings

Most high-dimensional complete intersections lack low-rank Ulrich bundles.

Ulrich bundles exist only in specific cases, such as quadrics with rank 2.

The characterization links Ulrich bundles to particular subvarieties.

Abstract

We characterize the existence of an Ulrich vector bundle on a variety $X \subset P^N$ in terms of the existence of a subvariety satisfying some precise conditions. Then we use this fact to prove that a complete intersection of dimension $n \ge 4$, which if $n=4$ is very general and not of type $(2,2)$, does not carry any Ulrich bundles of rank $r \le 3$ unless $n=4, r=2$ and $X$ is a quadric.