Semiorthogonal decompositions for surfaces

The derived category of coherent sheaves on an algebraic variety is an important and complex invariant. Under certain birational transformations (blowups, sufficiently good flips in the three-dimensional case...), the derived category changes in a simple and natural way: it gets "glued" to another, smaller, category. This gluing is a special case of a semiorthogonal decomposition. The analogy between birational transformations and the study of semiorthogonal decompositions is attractive, but not completely one-to-one, even in the two-dimensional case. I will discuss what is known and unknown about the properties of semiorthogonal decompositions for smooth projective surfaces.