Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence
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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.
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Frei, Sarah, Hassett, Brendan and Várilly-Alvarado, Anthony. "Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence." Journal für die reine und angewandte Mathematik, 2022, no. 792 (2022) De Gruyter: 289-305. https://doi.org/10.1515/crelle-2022-0056.