Exceptional collections, and the Neron-Severi lattice for surfaces

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.

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