Yusuf Mustopa

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 few questions about curves on surfaces.

In this note we address the following kind of question: let X be a smooth, irreducible, projective surface and

Pfaffian quartic surfaces and representations of Clifford algebras

D

Stability of syzygy bundles on an algebraic surface

D a divisor on X satisfying some sort of positivity hypothesis, then is there some multiple of D depending only on X which is effective or movable? We describe some examples, discuss some conjectures and prove some results that suggest that the answer should in general be negative, unless one puts some really strong hypotheses either on

The characteristic polynomial of an algebra and representations

D

Effective global generation on varieties with numerically trivial canonical class.

D or on