  We study exceptional collections of line bundles on surfaces. We prove that
any full cyclic strong exceptional collection of line bundles on a rational
surface is an augmentation in the sense of L.Hille and M.Perling. We find
simple geometric criteria of exceptionality (strong exceptionality, cyclic
strong exceptionality) for collections of line bundles on weak del Pezzo
surfaces. As a result, we classify smooth projective surfaces admitting a full
cyclic strong exceptional collection of line bundles. Also, we provide an
example of a weak del Pezzo surface of degree 2 and a full strong exceptional
collection of line bundles on it which does not come from augmentations, thus
answering a question by Hille and Perling.
