Ulrich wildness of some decomposable threefold scrolls over $\mathbb F_a$

Abstract: The paper deals with Ulrich wildness of decomposable threefold scrolls $X$ over Hirzebruch surfaces $\mathbb{F}_a$, for any $a \geqslant 0$. Our Main Theorem enstablishes that for $a=0$, the moduli space of rank-$r$ Ulrich bundles, for any $r \geqslant 2$ and of given Chern classes, contains a generically smooth, unirational component $\mathcal{M}(r)$ of computed dimension whose general point corresponds to a slope-stable Ulrich bundle; in particular $X$ turns out to be {\em Ulrich wild}. When $a \geqslant 1$ and in presence of modular obstructions, $X$ is nevertheless shown to be Ulrich wild again.