Exceptional and rigid sheaves on surfaces with anticanonical class without base components

This paper consists of three parts. In the first different kinds stability are discussed, in particular, the concept of stability with respect to nef divisor is introduced. The structure of rigid and superrigid vector bundles on smooth projective surfaces with nef anticanonical class without base components and a positive square is studied. We prove that any superrigid bundle has an exceptional filtration. In the last part we give a constructible description of exceptional bundles on the indicated surfaces proceeding from the results of the first two parts.