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

TL;DR

This paper investigates the Koszul property and stability of Ulrich bundles on Del Pezzo surfaces, demonstrating their rational moduli spaces and the density of syzygy bundles, advancing understanding of vector bundle geometry.

Contribution

It proves Ulrich bundles on Del Pezzo surfaces are Koszul and slope-semistable, and shows many moduli spaces of stable bundles are rational, providing evidence for a conjecture and insights into syzygy bundle density.

Findings

Ulrich bundles on Del Pezzo surfaces are Koszul and slope-semistable.

Many moduli spaces of stable bundles are rational.

Syzygy bundles of Ulrich bundles are dense in moduli spaces.

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 $k\geq 0$ 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 $d\geq 4$, then any Ulrich bundle $\mathcal{E}$ satisfies the Koszul property and is slope-semistable. We also show that, for infinitely many Chern characters ${\bf v}=(r,c_1, c_2)$, the corresponding moduli spaces of slope-stable bundles $\mathfrak{M}_H({\bf v})$ 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.