Ulrich bundles on smooth toric threefolds with Picard number $2$

TL;DR

This paper classifies and constructs Ulrich bundles on smooth toric threefolds with Picard number 2, providing explicit examples, resolutions, and showing the varieties are Ulrich wild.

Contribution

It introduces a complete classification and explicit constructions of Ulrich bundles on these threefolds, including pullback cases and wildness results.

Findings

Constructed resolutions and monads for Ulrich bundles of arbitrary rank

Provided explicit examples and classifications of Ulrich bundles

Proved the varieties are Ulrich wild

Abstract

In this paper, we study Ulrich bundles on smooth toric threefolds with Picard number$~2$, namely $\mathbb P(\mathcal O_{\mathbb P^{2}}(a_0) \oplus \mathcal O_{\mathbb P^{2}}(a_1))$. We construct resolutions and monads for Ulrich bundles of arbitrary rank, and provide explicit examples together with a complete classification of those arising as pullbacks from $\mathbb{P}^2$. As a consequence, we also show that these varieties are Ulrich wild.