Exceptional collections, and the Néron–Severi lattice for surfaces

Charles Vial

doi:10.1016/j.aim.2016.10.012

Abstract

We work out properties of smooth projective varieties X over a (not necessarily algebraically closed) field k that admit collections of objects in the bounded derived category of coherent sheaves Db(X) that are either full exceptional, or numerically exceptional of maximal length. Our main result gives a necessary and sufficient condition on the Néron–Severi lattice for a smooth projective surface S with χ(OS)=1 to admit a numerically exceptional collection of maximal length, consisting of line-bundles. As a consequence we determine exactly which complex surfaces with pg=q=0 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.

Highlights

A substantial amount of work [1, 7, 8, 16] was carried out in order to exhibit exceptional collections of line-bundles of maximal length on complex surfaces of general type with pg = q = 0, motivated by the will to exhibit geometric-phantom triangulated categories, i.e., categories with trivial or torsion Grothendieck group K0 ; see [17]
On the geometric side, we show that there are no numerical obstructions to the existence of exceptional collections of maximal length on complex surfaces of general type with pg = q = 0
We show that a numerically exceptional collection of maximal length, consisting of linebundles, on a surface S defined over an arbitrary field k remains of maximal length after any field extension (Theorem 3.3)

Summary

Introduction

We give a complete classification of smooth projective complex surfaces, with pg = q = 0, that admit numerically exceptional collections of maximal length : Theorem 3 (Theorem 3.10). We find arithmetic obstructions for geometrically rational surfaces to admit a numerically exceptional collection of maximal length : Theorem 5 (Theorem 3.7). Note that a surface may be rational but not admit a numerically exceptional collection of maximal length ; see Remark 3.9. A recent result of Perling [40] (see Theorem 2.2) implies that, for complex surfaces with pg = q = 0, the conditions of Theorem 3.1 are further equivalent to the existence of a numerically exceptional collection of maximal length (without any assumptions on the ranks of the objects of the collection).

A characterization of projective space

Full exceptional collections on surfaces and Lefschetz motives

Numerically exceptional collections of maximal length on surfaces