Ulrich trichotomy on del Pezzo surfaces (Q2684906)

An Ulrich bundle \(\mathcal{E}\) on a polarized variety \((X,H)\) is an arithmetically Cohen-Macaulay vector bundle whose associated module \(\bigoplus_{t} \mbox{H}^0 (X,\mathcal{E}(tH))\) has the maximum possible number of generators, which is \(\mathrm{deg}(X)\mathrm{rank}(\mathcal{E})\). Many mathematicians have studied the existence of Ulrich bundles, and since then, the existence problem has been solved for many varieties. Conversely, no variety that does not have an Ulrich bundle on it is known. Another way to consider the existence problem of Ulrich bundles on varieties is through the Ulrich representation type of varieties. An analogous problem is the representation type of quivers, which can be divided into finite type, tame, and wild according to the behavior of their representations. The classification of finite type quivers and giving a complete description of tame and wild quivers have already been extensively studied by \textit{A. Kirillov jun.} [Quiver representations and quiver varieties. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1355.16002)]. The authors use a correspondence between Ulrich bundles on a projective variety and quiver representations to prove that certain del Pezzo surfaces satisfy the Ulrich trichotomy, for any given polarization.