Derived categories of moduli of stable curves

I will report on a joint project with Ana-Maria Castravet on birational geometry and derived category of moduli spaces of pointed rational curves and related GIT quotients. The motivating question, due to Orlov, is whether these derived categories have a full exceptional collection permuted by automorphisms of the moduli space. A similar question for the K-group of a toric variety was raised by Merkurjev and Panin.