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Working paper

On exceptional collections of line bundles on weak del Pezzo surfaces

arxiv.org. math. Cornell University, 2017. No. 1710.03972.
Elagin Alexey, Xu J., Zhang S.
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.