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

Let \({\mathscr {E}}\) $\mathcal{E}$ be a vector bundle on a smooth projective variety \(X\subseteq {\mathbb {P}}^N\) $X\subseteq {\mathbb{P}}^N$ that is Ulrich with respect to the hyperplane section H. In this article, we study the Koszul property of \({\mathscr {E}}\) $\mathcal{E}$ , the slope-semistability of the k-th iterated syzygy bundle \({\mathscr {S}}_k({\mathscr {E}})\) ${\mathcal{S}}_k(\mathcal{E})$ for all \(k\ge 0\) $k\ge 0$ 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 \(d\ge 4\) $d\ge 4$ , then any Ulrich bundle \({\mathscr {E}}\) $\mathcal{E}$ satisfies the Koszul property and is slope-semistable. We also show that, for infinitely many Chern characters \(\textbf{v}=(r,c_1, c_2)\) $\mathbf{v}=(r,c_1,c_2)$ , the corresponding moduli spaces of slope-stable bundles \({\mathfrak {M}}_H(\textbf{v})\) ${\mathfrak{M}}_H(\mathbf{v})$ when non-empty, are rational, and thereby produce new evidences for a conjecture of Costa and Miró-Roig. As a consequence, we show that the iterated syzygy bundles of Ulrich bundles are dense in these moduli spaces.