Abstract:
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.
