Exceptional collections, and the Neron-Severi lattice for surfaces

TL;DR

This paper characterizes when smooth projective surfaces over various fields admit maximal-length exceptional collections in their derived categories, linking geometric properties to lattice conditions and classifying certain rational surfaces.

Contribution

It provides a necessary and sufficient condition on the Neron-Severi lattice for such collections to exist on surfaces with Euler characteristic 1, and classifies complex surfaces with specific invariants.

Findings

Characterization of surfaces admitting maximal-length collections

Identification of conditions on the Neron-Severi lattice

Classification of rational surfaces with these collections

Abstract

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.