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Consequences of the existence of exceptional collections in arithmetic and rationality

Matthew R. Ballard, Alexander Duncan, Alicia Lamarche, Patrick K. McFaddin

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 étale-exceptional collection but not a rational point. The counterexample comes from twisting a non-retract rational variety with a rational point and full étale-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.

Consequences of the existence of exceptional collections in arithmetic and rationality

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 étale-exceptional collection but not a rational point. The counterexample comes from twisting a non-retract rational variety with a rational point and full étale-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.

Paper Structure

This paper contains 15 sections, 25 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Derived categories, semi-orthogonal decompositions, and exceptional collections
  4. Castravet and Tevelev's Exceptional Collection
  5. Arithmetic toric varieties
  6. Torsors invisible to Brauer invariants
  7. Linearizing étale-exceptional objects
  8. Étale-exceptional objects on neutral toric varieties
  9. TCI-type collections on toric varieties
  10. Existence of a full TCI-type collection
  11. Proofs of the main results
  12. Orlov's Conjecture for toric varieties
  13. Non-rational toric varieties with full étale exceptional collections
  14. Proof of Proposition \ref{['proposition:isom_group']}
  15. Representability Results for GE

Key Result

Theorem 1

Let $X$ be a smooth projective toric variety over a field $k$ with $X(k) \neq \emptyset$. If $\operatorname{D^b}(X)$ has a full $k$-exceptional collection then $X$ is $k$-rational.

Theorems & Definitions (68)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.1: see BK
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6
  • ...and 58 more