Semiorthogonal decompositions for projective plane

A semiorthogonal decomposition is a way to decompose a derived category into smaller components. We know many examples, but we do not really understand the constraints on the structure of an arbitrary decomposition. In this talk I will show that all semiorthogonal decompositions of the derived category of coherent sheaves on P^2 arise from full exceptional collections, i.e., the known examples exhaust all possibilities. This implies that there are no phantom subcategories in P^2, making it the second nontrivial geometric example where we can prove this, the first being P^1.