Koszul property of Ulrich bundles and rationality of moduli spaces of stable bundles on Del Pezzo surfaces (Q6564484)

Let \(X \subset {\mathbb P}^N\) be a smooth projective variety. A vector bundle \({\mathcal E}\) over \(X\) is said to be Ulrich if \(h^i(X, {\mathcal E}(-p))=0\) for \(p \geq 0\) and \(0 \leq i \leq n=\dim X\). Ulrich bundles are known to be globally generated and semistable with respect to the hyperplane section \(H\). The syzygy bundle \(M_{\mathcal E}\) is the kernel of the evaluation of global sections and it is a main object of study in the paper under review. As a first result, a sufficient conditions on \(H^n\) and \(H^{n-1}K_X\) to be \(M_{\mathcal E}\) slope-semistable with respect to \(H\) is presented (see Prop. 3.6). This condition is satisfied for anticanonical rational surfaces (see the Introduction of the paper under review and references therein). The twist by \(H\) of the syzygy bundle can be globally generated and leads to a new syzygy bundle, which inductively produces (see Def. 3.8) \({\mathcal S}_k({\mathcal E})\): \({\mathcal S}_0({\mathcal E})=M_{\mathcal E}\otimes H\), and if \({\mathcal S}_{k-1}({\mathcal E})\) exists and is globally generated then \({\mathcal S}_k({\mathcal E})=M_{{\mathcal S}_{k-1}({\mathcal E})} \otimes H\). The existence of \({\mathcal S}_p({\mathcal E})\), for \(p \leq k-1\) and the surjectivity of the product of sections \(H^0({\mathcal S}_p({\mathcal E})) \otimes H^0(H) \to H^0({\mathcal S}_p({\mathcal E})\otimes H)\) define the property Kos\(_k\) on \({\mathcal E}\). \N\NA main result in the paper under review is to show that for an anticanonically embedded Del Pezzo surface of degree \(d \geq 4\), any Ulrich bundle E satisfies Kos\(_\infty\) and that the iterated syzygy bundles are slope-semistable with respect to \(H\) (see Thm. 3.19 and Cor. 3.23). Moreover, the authors also prove (see Thm. 1.1) that, for such Del Pezzo surfaces, the association \({\mathcal F} \to {\mathcal S}_p({\mathcal F})\) defines a birational map between the moduli spaces of slope-stable bundles (with respect to \(H\)) of the corresponding Chern characters \(v({\mathcal E})\) (\(\mathcal E\) a stable Ulrich bundle of rank \(\ge 2\)) and \(v({\mathcal S}_p({\mathcal E}))\), whenever the latter space is non-empty. For \(d=3\) results in the case \(p=0\) are also obtained. This result, in combination with known results on the rank two case, shows the rationality of infinitely many moduli spaces of slope-stable bundles on Del Pezzo surfaces. Also, see Cor. 1.2, results on the Brill-Noether property of some moduli spaces are provided.