"Exceptional collections on rational vs general type surfaces"

It is expected that a smooth projective surface carries a full
exceptional sequence if and only if it is rational.
However many non-rational surfaces admit (numerical) exceptional
sequences of maximal length, that is exceptional sequences generating
the Grothendieck group up to torsion, and it is then very difficult to
prove the "only if" part of the above conjecture.
In this talk, I will motivate the expectation, present examples of
phantom categories, and explain how the study of autoequivalences of
categories generated by exceptional objects can potentially be applied
to this open problem.