A note on some moduli spaces of Ulrich bundles

We prove that the modular component M(r), constructed in the Main Theorem of a former paper of us (published in Adv. Math on 2024), paramatrizing (isomorphism classes of) Ulrich vector bundles of rank r and given Chern classes, on suitable 3-fold scrolls Xe over Hirzebruch surfaces Fe≥0, which arise as tautological embeddings of projectivization of very-ample vector bundles on Fe, is generically smooth and unirational. A stronger result holds for the suitable associated moduli space MFe(r) of vector bundles of rank r and given Chern classes on Fe, Ulrich w.r.t. the very ample polarization c1(Ee)=OFe(3,be), which turns out to be generically smooth, irreducible and unirational.