Ulrich bundles on some threefold scrolls over F_e

We investigate the existence of Ulrich vector bundles on suitable 3-fold scrolls Xe over Hirzebruch surfaces Fe, for any e⩾0, which arise as tautological embedding of projectivization of very-ample vector bundles on Fe which are uniform in the sense of Brosius and Aprodu--Brinzanescu (cf. [8] and [3], respectively, in Bibliography). We explicitely describe components of moduli spaces of rank r⩾1 Ulrich vector bundles whose general point is a slope-stable, indecomposable vector bundle. We moreover determine the dimension of such components as well as we prove that they are generically smooth. As a direct consequence of these facts, we also compute the Ulrich complexity of any such Xe and give an effective proof of the fact that such Xe's turn out to be geometrically Ulrich wild.