Characterization of Ulrich bundles on Hirzebruch surfaces

TL;DR

This paper characterizes Ulrich bundles on polarized rational ruled surfaces, showing their resolutions, existence conditions, stability, and providing explicit examples for various ranks and polarizations.

Contribution

It provides a comprehensive characterization of Ulrich bundles on Hirzebruch surfaces, including existence, stability, and explicit constructions for different ranks and polarizations.

Findings

Ulrich bundles admit resolutions in terms of line bundles.

Existence of Ulrich bundles for any admissible rank and first Chern class.

Construction of indecomposable Ulrich bundles for various polarizations and ranks.

Abstract

In this work we characterize Ulrich bundles of any rank on polarized rational ruled surfaces over $\mathbb{P}^1$. We show that every Ulrich bundle admits a resolution in terms of line bundles. Conversely, given an injective map between suitable totally decomposed vector bundles, we show that its cokernel is Ulrich if it satisfies a vanishing in cohomology. As a consequence we obtain, once we fix a polarization, the existence of Ulrich bundles for any admissible rank and first Chern class. Moreover we show the existence of stable Ulrich bundles for certain pairs $(\textrm{rk}(E),c_1(E))$ and with respect to a family of polarizations. Finally we construct examples of indecomposable Ulrich bundles for several different polarizations and ranks.