On Ulrich bundles on projective bundles (Q2170466)

Given a polarized surface \((X,\mathcal{O}_X(h))\), a coherent sheaf \(\mathcal{F}\) on \(X\) is called Ulrich if \(\mathcal{F}\) is arithmetically Cohen-Macaulay with the largest permitted number of global sections. A leading problem in the current theory of coherent sheaves on projective varieties is the construction of Ulrich bundles. In the paper under review, when \(X\) is a either a projective curve or a surface, the author gives some criteria for the existence of Ulrich bundles on projective bundles \(\pi:\mathbb{P}(\mathcal{E})\rightarrow X\) with respect to a fixed polarization of the type \(\pi^*A+H\) for \(A\) a divisor on \(X\) and \(H\) the relative hyperplane section. The result is based on the existance of a suitable semiortoghonal decomposition of the derived category \(D^b(\mathbb{P}(\mathcal{E}))\) [\textit{D. O. Orlov}, Russ. Acad. Sci., Izv., Math. 41, No. 1, 1 (1992; Zbl 0798.14007); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 4, 852--862 (1992)].