Ulrich bundles on Brauer--Severi varieties

TL;DR

This paper proves the existence of Ulrich bundles on all Brauer--Severi varieties, establishes their minimal rank in relation to the variety's period, and provides a formula linking the rank to the period for certain dimensions.

Contribution

It demonstrates the existence of Ulrich bundles on all Brauer--Severi varieties and derives a formula connecting their rank to the variety's period, advancing understanding of their structure.

Findings

Ulrich bundles exist on any Brauer--Severi variety.

Minimal rank of Ulrich bundles equals the period in some cases.

Rank of Ulrich bundles is a multiple of the period for certain dimensions.

Abstract

We prove the existence of Ulrich bundles on any Brauer--Severi variety. In some cases, the minimal possible rank of the obtained Ulrich bundles equals the period of the Brauer--Severi variety. Moreover, we find a formula for the rank of an Ulrich bundle involving the period of the considered Brauer--Severi variety $X$, at least if $\mathrm{dim}(X)=p-1$ for an odd prime $p$. This formula implies that the rank of any Ulrich bundle on such a Brauer--Severi variety $X$ must be a multiple of the period.