Derived categories of Fano threefolds of index two with ordinary double points

A Kawamata decomposition of the derived category of a singular variety is a semiorthogonal decomposition into a perfect part and derived categories of finite-dimensional algebras. Previously known examples of singular threefolds admitting Kawamata decompositions are nodal V_6 and a nodal quadric. In this talk I will explain that a necessary and sufficient condition for a nodal Fano threefold of index two to admit a Kawamata decomposition is maximal nonfactoriality, which means that Weil divisors separate the singular points. This is joint work in progress with Nebojsa Pavic (Hannover).