I am trying to understand on which kinds of varieties we can find strong full exceptional collections of sheaves. (Classical examples are of course projective spaces (Beilinson) and Grassmann varieties and quadrics (Kapranov).)
As far as I understand, the "strongness" of an exceptional collection implies that the derived category of coherent sheaves is equivalent to the derived category of modules over an associative algebra.
If one is content with working with DG algebras one can just focus on exceptional collections, while allowing higher Ext groups between the sheaves in the collection. However, I would like to know what currently is known about varieties with strong exceptional collections.