Does there exist a smooth non-rational projective variety whose bounded derived category of coherent sheaves admits a full exceptional collection? I could not find any examples in the literature (for instance projective spaces and intersections of quadrics, which admit full exceptional collections, are rational). 



A non-rational variety with a full exceptional collection?
Rationality of a variety with a full exceptional collection is a well-know folklore conjecture. In some form a similar open question is mentioned in the paper of Brown and Shipman "The McKay Correspondence, Tilting, and Rationality".


