Consequences of the existence of exceptional collections in arithmetic and rationality

TL;DR

This paper investigates the link between exceptional collections and rationality of varieties, proving a conjecture for certain toric varieties and providing counterexamples for a generalized conjecture over number fields.

Contribution

It proves Orlov's conjecture for arithmetic toric varieties and constructs a counterexample to a generalized conjecture involving étale-exceptional collections.

Findings

Proved Orlov's conjecture for arithmetic toric varieties.

Constructed a counterexample over the rationals with an étale-exceptional collection but no rational point.

Developed tools for linearizing objects and controlling their linearizations.

Abstract

A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a slight generalization of this conjecture, where the endomorphism algebras of the exceptional objects are allowed to be separable field extensions of the base field. We show this generalization is false by exhibiting a geometrically rational, smooth, projective threefold over the the field of rational numbers that possesses a full \'etale-exceptional collection but not a rational point. The counterexample comes from twisting a non-retract rational variety with a rational point and full \'etale-exceptional collection by a torsor that is invisible to Brauer invariants. Along the way, we develop some tools for linearizing objects, including a group that controls linearizations.