Vector bundles of rank 2 and linear systems on algebraic surfaces (following the paper of I. Reider)

We will prove the technical theorem about linear systems of the form $|K+L|$ (where $K$ is the canonical divisor and $L$ is a numerical effective divisor) on smooth algebraic surfaces. The proof will be based on the notion of Bogomolov's instability of rank 2 vector bundles. As a consequence of the theorem, we will get easy proofs for some quite deep results on the geometry of surfaces.