Emre Coskun

THE GEOMETRY OF ULRICH BUNDLES ON DEL PEZZO SURFACES

Abstract Given a smooth del Pezzo surface X d ⊆ 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 | 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 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 | D | .

A Survey of Ulrich Bundles

The purpose of this article is to serve as an introduction to Ulrich bundles for interested readers. We discuss the origins of the study of Ulrich bundles, their elementary properties, and we give an exposition of the known results in their classification on various kinds of smooth projective varieties. As an illustration of a method recently used by Marta Casanellas and Robin Hartshorne for the classification of Ulrich bundles on a cubic surface, we classify the stable Ulrich bundles on a smooth quartic surface in ℙ3 of Picard number 2 by first Chern class.