Ulrich Bundles on decomposable threefold scrolls over $\mathbb F_a$

Abstract

Ulrich bundles provide profound insights into the underlying geometry and derived categories of projective varieties supporting them, yet their existence and modular properties remain sometimes largely obscure. In this paper we study the geometry and the moduli spaces of Ulrich bundles on broad families of decomposable threefold scrolls $X$ over Hirzebruch surfaces $\mathbb F_a$, proving that their Ulrich complexity is 1. Exploiting the double-scroll structure enjoyed by $X$, we also introduce a (geometric) involution acting on the set of classified Ulrich line bundles which, together with the natural one, shapes the study of higher-rank extensions and significantly streamlines their modular study. We moreover prove that some higher-rank extensions yield indecomposable Ulrich bundles whose existence is intrinsically $3$-dimensional, i.e. going beyond natural pullbacks from the base surfaces. In rank two, for any choice of the parameters involved, we provide a comprehensive description of associated modular irreducible components, determining their dimensions, their generic smoothness and the description of their birational structure. Ultimately, for noteworthy parameter cases, we focus on the Ulrich representation type of $X$ proving that it is Ulrich wild by the existence of generically smooth modular components of slope-stable Ulrich bundles of arbitrary rank $r$ and of dimension growing quadratically with $r$, which reveals the unbounded complexity of Ulrich modules supported on these threefolds.