The geometry of Ulrich bundles on del Pezzo surfaces

Given a smooth del Pezzo surface X-d subset of P-d of degree d, we isolate the essential geometric obstruction to a vector bundle on X-d being an Ulrich bundle by showing that an irreducible curve D of degree dr on X-d represents the first Chern class of a rank-r Ulrich bundle on X-d if and only if the kernel bundle of the general. smooth element of vertical bar D vertical bar admits a generalized theta-divisor. Moreover, we show that any smooth arithmetically Gorenstein surface whose Ulrich bundles admit such a characterization is necessarily del Pezzo. This result is applied to produce new examples of complete intersection curves with semistable kernel bundle, and also combined, with work of Farkas, Mustata and Popa to relate the existence of Ulrich bundles on X-d to the Minimal Resolution Conjecture for curves lying on X-d. In particular, we show that a smooth irreducible curve D of degree 3r lying on a smooth cubic surface X-3 represents the first Chern class of an Ulrich bundle on X-3 if and, only if the Minimal Resolution Conjecture holds for the general smooth element of vertical bar D vertical bar. (C) 2012 Elsevier Inc. All rights reserved.