We investigate on the existence of some "sporadic", rank-$r \geqslant 1$ Ulrich vector bundles on suitable $3$-fold scrolls $X$ over the Hirzebruch surface $\mathbb{ F}_0$, which arise as tautological embeddings of projectivization of very-ample vector bundles on $\mathbb{ F}_0$ that are uniform in the sense of Brosius and Aprodu--Brinzanescu. Such Ulrich bundles arise as deformations of ``iterative" extensions by means of "sporadic" Ulrich line bundles. We moreover explicitely describe irreducible components of the corresponding "sporadic" moduli spaces of rank $r \geqslant 1$ vector bundles which are Ulrich with respect to the tautological polarization on $X$. In some cases such irreducible components turn out to be a singleton, in some other cases such components are generically smooth, whose positive dimension has been computed and whose general point turns out to be a slope-stable vector bundle.
Categories: math.AG
