Koszul property of Ulrich bundles and rationality of moduli spaces of stable bundles on Del Pezzo surfaces

Let E be a vector bundle on a smooth projective variety X subset of P-N that is Ulrich with respect to the hyperplane section H. In this article, we study the Koszul property of E, the slope-semistability of the k-th iterated syzygy bundle S-k (E) for all k >= 0 and rationality of moduli spaces of slope-stable bundles on Del Pezzo surfaces. As a consequence of our study, we show that if X is a Del Pezzo surface of degree d >= 4, then any Ulrich bundle E satisfies the Koszul property and is slope-semistable. We also show that, for infinitely many Chern characters v = (r, c(1), c(2)), the corresponding moduli spaces of slope-stable bundles M-H(v) when non-empty, are rational, and thereby produce new evidences for a conjecture of Costa and Miro-Roig. As a consequence, we show that the iterated syzygy bundles of Ulrich bundles are dense in these moduli spaces.