We prove that the modular component , 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 \(X_e\) over Hirzebruch surfaces \({\mathbb {F}}_{e\ge 0}\) , which arise as tautological embeddings of projectivization of very-ample vector bundles on \({\mathbb {F}}_e\) , is generically smooth, irreducible and unirational. A stronger result holds for the suitable associated moduli space of vector bundles of rank r and given Chern classes on \({\mathbb {F}}_e\) , Ulrich w.r.t. the very ample polarization which turns out to be generically smooth, irreducible and unirational.
