Let
be a vector bundle on a smooth projective variety
that is Ulrich with respect to the hyperplane section
H. In this article, we study the Koszul property of
, the slope--semistability of the
k--th iterated syzygy bundle
for all
and rationality of moduli spaces of slope--stable bundles on del--Pezzo
... [Show full abstract] surfaces. As a consequence of our study, we show that if X is a del--Pezzo surface of degree , then any Ulrich bundle satisfies the Koszul property and is slope--semistable. We also show that, for infinitely many Chern characters , the corresponding moduli spaces of slope--stable bundles when non--empty, are rational, and thereby produce new evidences for a conjecture of Costa and Mir\'o-Roig. As a consequence, we show that the iterated syzygy bundles of Ulrich bundles are dense in these moduli spaces.