A note on some moduli spaces of Ulrich Bundles

We prove that the modular component $\mathcal{M}(r)$, 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 $X_e$ 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 and unirational. A stronger result holds for the suitable associated moduli space ${\mathcal{M}}_{{\mathbb{F}}_e}(r)$ of vector bundles of rank $r$ and given Chern classes on ${\mathbb{F}}_e$, Ulrich w.r.t. the very ample polarization $c_1({\mathcal{E}}_e)={\mathcal{O}}_{{\mathbb{F}}_e}(3,b_e),$ which turns out to be generically smooth, irreducible and unirational.