  We work out properties of smooth projective varieties over a (not necessarily
algebraically closed) field that admit collections of objects in the bounded
derived category of coherent sheaves that are either full exceptional, or
numerically exceptional of maximal length. Our main result gives a necessary
and sufficient condition on the Neron-Severi lattice for a smooth projective
surface with holomorphic Euler characteristic 1 to admit a numerically
exceptional collection of maximal length, consisting of line-bundles. As a
consequence we determine exactly which complex surfaces with vanishing
irregularity and geometric genus admit a numerically exceptional collection of
maximal length. Another consequence is that a minimal geometrically rational
surface with a numerically exceptional collection of maximal length is
rational.
