Exceptional collections, and the Neron-Severi lattice for surfaces

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.

27 pages; fills a gap in the proof of the main theorem. The relevant linear algebra has been moved to an appendix

References in corpus (3)

• Semiorthogonal decompositions in algebraic geometry
• A simple counterexample to the Jordan-Hölder property for derived categories
• Exceptional collections on Dolgachev surfaces associated with degenerations

Cited by in corpus (11)

• Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields
• Cycles, derived categories, and rationality
• A Phantom on a Rational Surface
• Non-existence of exceptional collections on twisted flags and categorical representability via noncommutative motives
• Exceptional collections on Dolgachev surfaces associated with degenerations
• On derived categories of arithmetic toric varieties
• Exceptional collections in surface-like categories
• Rational points on symmetric powers and categorical representability
• Deformation of exceptional collections
• Consequences of the existence of exceptional collections in arithmetic and rationality
• Mutations of Numerically Exceptional Collections on Surfaces