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

2022/02/28 by Bangere, Purnaprajna, Mukherjee, Jayan, Raychaudhury, Debaditya
#14J26 #14J60 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2202.13631

Abstract

Let E be a vector bundle on a smooth projective variety X⊆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 \bf v=(r,c₁, c₂), the corresponding moduli spaces of slope-stable bundles \mathfrakM_H(\bf 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.

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