The geometry of Ulrich bundles on del Pezzo surfaces

Given a smooth del Pezzo surface of degree d, we isolate the essential geometric obstruction to a vector bundle on Xd being an Ulrich bundle by showing that an irreducible curve D of degree dr on Xd represents the first Chern class of a rank-r Ulrich bundle on Xd if and only if the kernel bundle of the general smooth element of |D| 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, Mustaţǎ and Popa to relate the existence of Ulrich bundles on Xd to the Minimal Resolution Conjecture for curves lying on Xd. In particular, we show that a smooth irreducible curve D of degree 3r lying on a smooth cubic surface X3 represents the first Chern class of an Ulrich bundle on X3 if and only if the Minimal Resolution Conjecture holds for the general smooth element of |D|.