A note on some moduli spaces of Ulrich bundles

We prove that the modular component M(r), constructed in the Main Theorem in Fania and Flamini (Adv Math 436:109409, 2024. https://doi.org/10.1016/j.aim.2023.109409), of Ulrich vector bundles of rank r and given Chern classes, on suitable threefold scrolls Xe over Hirzebruch surfaces F_e≥0, which arise as tautological embeddings of projectivization of very- ample vector bundles on F_e, is generically smooth, irreducible and unirational. A stronger result holds for the suitable associated moduli space M_Fe (r ) of vector bundles of rank r and given Chern classes on F_e, Ulrich w.r.t. the very ample polarization c_1(E_e) = O_Fe(3,b_e), which turns out to be generically smooth, irreducible and unirational.