Ulrich bundles on ruled surfaces

TL;DR

This paper investigates the existence of Ulrich bundles on ruled surfaces, establishing conditions for line bundles and constructing rank two bundles for various polarizations.

Contribution

It provides a complete characterization of when Ulrich line bundles exist and constructs rank two Ulrich bundles for different polarizations on ruled surfaces.

Findings

Ulrich line bundles exist if and only if the polarization coefficient equals one.

Rank two Ulrich bundles exist for polarizations with coefficients other than one.

The paper offers explicit conditions and constructions for Ulrich bundles on ruled surfaces.

Abstract

In this short note, we study the existence problem for Ulrich bundles on ruled surfaces, focusing our attention on the smallest possible rank. We show that existence of Ulrich line bundles occurs if and only if the coefficient $\alpha$ of the minimal section in the numerical class of the polarization equals one. For other polarizations, we prove the existence of rank two Ulrich bundles.