On Ulrich bundles on some decomposable threefold scrolls over $\mathbb F_a$

This paper investigates Ulrich bundles on decomposable threefold scrolls X over the Hirzebruch surface $\mathbb F_a$, for any integer $a \geq 0$, focusing on the study of their structure and classification. We prove existence of such Ulrich bundles, studying their properties, determining conditions for the Ulrich complexity of their support variety X and analyzing instances of Ulrich wildness for X. Our results delve also into the moduli spaces of such Ulrich bundles, characterizing generic smoothness (and sometimes even birational classification) of their modular components and computing their dimensions. Through a detailed analysis of Chern classes, we also provide understanding of the interplay between the geometric properties of the underlying variety X and the algebro-geometric features of Ulrich bundles on it, contributing to their construction as well as to their modular and enumerative theory.