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.