Ulrich bundles on some threefold scrolls over Fe

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 embeddings of projectivization of very-ample vector bundles on Fe that are uniform in the sense of Brosius and Aprodu–Brinzanescu, cf. [10] and [3] respectively. We explicitly describe components of moduli spaces of rank r⩾1 vector bundles which are Ulrich with respect to the tautological polarization on Xe and whose general point is a slope-stable, indecomposable vector bundle. We moreover determine the dimension of such components, proving also 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 these Xe's turn out to be geometrically Ulrich wild. At last, the machinery developed for 3–fold scrolls Xe allows us to deduce Ulrichness results on rank r⩾1 vector bundles on Fe, for any e⩾0, with respect to a naturally associated (very ample) polarization.