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Schreieder, Stefan (2019): ON THE RATIONALITY PROBLEM FOR QUADRIC BUNDLES. In: Duke Mathematical Journal, Vol. 168, No. 2: pp. 187-223
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We classify all positive integers n and r such that (stably) nonrational complex r - fold quadric bundles over rational n-folds exist. We show in particular that, for any n and r, a wide class of smooth r -fold quadric bundles over P-C(n) are not stably rational if r is an element of [2(n-1) - 1, 2(n) - 2]. In our proofs we introduce a generalization of the specialization method of Voisin and of Colliot-Thelene and Pirutka which avoids universally CH0-trivial resolutions of singularities.