Semi-orthogonal indecomposability for minimal surfaces

For a blow-up of a smooth point on an algebraic surface we can give a rather simple description of its derived category of coherent sheaves: take the derived category of the original surface, take a single object corresponding to the exceptional divisor, and glue them together in a natural way. This parallels the relation between Picard groups, and forms the basis of the analogy between semi-orthogonal decompositions and the minimal model program. This analogy is not an exact correspondence, but in lower dimensions there are specific results and conjectures. I will discuss some general observations and explain why a smooth projective surface with a nef and effective canonical class has indecomposable derived category, confirming the conjecture of Okawa.