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

Abstract

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

19 pages, v2: minor changes following referee's suggestions. Final version, accepted for publication in Manuscripta Mathematica