It is well known that any smooth projective toric surface has a full exceptional collection. I will talk about a generalization of this fact for singular surfaces. First, if the class group of Weil divisors of the surface is torsion free (for instance, this holds for all weighted projective planes), I will construct a semiorthogonal decomposition of the derived category with components equivalent to derived categories of modules over certain local finite dimensional algebras. When the class group has torsion, a similar semiorthogonal decomposition will be constructed for an appropriately twisted derived category. Many of these results extend to non-necessarily toric rational surfaces. This is a joint work with Joseph Karmazyn and Evgeny Shinder.
Maxim Kontsevich
