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Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence

  • Sarah Frei , Brendan Hassett and Anthony Várilly-Alvarado ORCID logo EMAIL logo
Published/Copyright: September 30, 2022

Abstract

Given a smooth projective variety over a number field and an element of its Brauer group, we consider the specialization of the Brauer class at a place of good reduction for the variety and the class. We are interested in the case of K3 surfaces. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial after specialization at a set of places of positive natural density. We deduce that there exist cubic fourfolds over number fields that are conjecturally irrational, with rational reduction at a positive proportion of places. We also deduce that there are twisted derived equivalent K3 surfaces which become derived equivalent after reduction at a positive proportion of places.

Award Identifier / Grant number: DMS-1745670

Award Identifier / Grant number: DMS-1701659

Award Identifier / Grant number: DMS-1352291

Award Identifier / Grant number: DMS-1902274

Funding source: Simons Foundation

Award Identifier / Grant number: 546235

Funding statement: Sarah Frei was partially supported by NSF grant DMS-1745670. Brendan Hassett was partially supported by NSF grant DMS-1701659 and Simons Foundation Award 546235. Anthony Várilly-Alvarado was partially supported by NSF grants DMS-1352291 and DMS-1902274.

Acknowledgements

We thank Ravi Vakil for asking the third named author whether a statement like Theorem 1.5 could be true at the 2015 Arizona Winter School. We thank Nicolas Addington, Martin Bright, Jean-Louis Colliot-Thélène, Edgar Costa, Ofer Gabber, Daniel Huybrechts, Evis Ieronymou, Daniel Loughran, and Yuri Tschinkel for valuable mathematical comments and discussions, and Isabel Vogt for pointing out the reference [56]. We thank the anonymous referees for valuable comments and on earlier versions of this paper.

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Received: 2022-03-02
Revised: 2022-07-20
Published Online: 2022-09-30
Published in Print: 2022-11-01

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