On the rationality problem for quadric bundles

Abstract

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 ${\mathbb{P}}_{\mathbb{C}}^n$ are not stably rational if $r\in [2^{n-1}-1,2^n-2]$. In our proofs we introduce a generalization of the specialization method of Voisin and of Colliot-Thélène and Pirutka which avoids universally ${\mathrm{CH}}_0$-trivial resolutions of singularities.