On exceptional collections of line bundles on weak del Pezzo surfaces

TL;DR

This paper investigates the structure of exceptional line bundle collections on weak del Pezzo surfaces, establishing conditions for their augmentation and classifying surfaces with maximal-length cyclic collections.

Contribution

It proves that full strong exceptional collections are augmentations on degree ≥ 3 surfaces and classifies surfaces with maximal-length cyclic collections of line bundles.

Findings

Full strong exceptional collections are augmentations for degree ≥ 3.

Classification of surfaces with maximal-length cyclic collections.

Criteria for exceptionality and strong exceptionality of line bundle collections.

Abstract

We study full exceptional collections of line bundles on surfaces. We prove that any full strong exceptional collection of line bundles on a weak del Pezzo surface of degree $\ge 3$ is an augmentation in the sense of L.Hille and M.Perling, while for some weak del Pezzo surfaces of degree $2$ the above is not true. We classify smooth projective surfaces possessing a cyclic strong exceptional collection of line bundles of maximal length: we prove that they are weak del Pezzo surfaces and find all types of weak del Pezzo surfaces admitting such a collection. We find simple criteria of exceptionality/strong exceptionality for collections of line bundles on weak del Pezzo surfaces.