Consequences of the existence of exceptional collections in arithmetic and rationality
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| Main Authors: | , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866916411171930112 |
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| author | Ballard, Matthew R. Duncan, Alexander Lamarche, Alicia McFaddin, Patrick K. |
| author_facet | Ballard, Matthew R. Duncan, Alexander Lamarche, Alicia McFaddin, Patrick K. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2009_10175 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Consequences of the existence of exceptional collections in arithmetic and rationality Ballard, Matthew R. Duncan, Alexander Lamarche, Alicia McFaddin, Patrick K. Algebraic Geometry 14E08 (Primary) 14F08, 14L99 (Secondary) 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. |
| title | Consequences of the existence of exceptional collections in arithmetic and rationality |
| topic | Algebraic Geometry 14E08 (Primary) 14F08, 14L99 (Secondary) |
| url | https://arxiv.org/abs/2009.10175 |