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 F₄ 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 M ₅䂰 (r) of vector bundles of rank r and given Chern classes on Fₑ, Ulrich w. r. t. the very ample polarization c₁ (Eₑ) = O ₅䂰 (3, bₑ), which turns out to be generically smooth, irreducible and unirational.