Ulrich bundles on cyclic coverings of projective spaces

TL;DR

This paper demonstrates the existence of Ulrich bundles on cyclic coverings of projective spaces of any degree, providing constructions and rank estimations, advancing the understanding of vector bundles in algebraic geometry.

Contribution

It proves the existence of Ulrich bundles on cyclic coverings of projective spaces and constructs such bundles from subvarieties, offering new insights into their properties.

Findings

Existence of Ulrich bundles on cyclic coverings of any degree.

Construction method from subvarieties to ambient varieties.

Rank estimations for Ulrich bundles depending on divisibility conditions.

Abstract

We prove the existence of Ulrich bundles on cyclic coverings of $\mathbb{P}^n$ of arbitrary degree $d$. Given a relatively Ulrich bundle on a complete intersection subvariety, we construct a relatively Ulrich bundle on the ambient variety. As an application, we prove that there exists a rank $d$ Ulrich bundle on generic cyclic coverings of $\mathbb{P}^2$ of degree $d$ such that the degree of the branch divisor $d \cdot k$ is even. When $d \cdot k$ is odd, we also provide an estimation of the rank of the Ulrich bundle on generic cyclic coverings of $\mathbb{P}^2$.