On the rationality problem for quadric bundles

Stefan Schreieder

Published 2017-06-05Version 1

We show that a wide class of smooth quadric bundles over rational bases are not stably rational. If the base has dimension n>1, then the possible fibre dimensions range from $2^{n-1}-1$ to $2^{n}-2$. For any r>2, this yields the first known smooth r-fold quadric bundles over rational bases which are not (stably) rational. In our proofs we introduce a generalization of the specialization method of Voisin and Colliot-Th\'el\`ene--Pirutka which avoids universally $CH_0$-trivial resolutions of singularities.