We classify all positive integers and such that (stably) nonrational complex -fold quadric bundles over rational -folds exist. We show in particular that, for any and , a wide class of smooth -fold quadric bundles over are not stably rational if . In our proofs we introduce a generalization of the specialization method of Voisin and of Colliot-Thélène and Pirutka which avoids universally -trivial resolutions of singularities.
"On the rationality problem for quadric bundles." Duke Math. J. 168 (2) 187 - 223, 1 February 2019. https://doi.org/10.1215/00127094-2018-0041