## Duke Mathematical Journal

### On the rationality problem for quadric bundles

Stefan Schreieder

#### 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 $\mathbb{P}^{n}_{\mathbb{C}}$ 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.

#### Article information

Source
Duke Math. J., Volume 168, Number 2 (2019), 187-223.

Dates
Revised: 12 July 2018
First available in Project Euclid: 8 November 2018

https://projecteuclid.org/euclid.dmj/1541646041

Digital Object Identifier
doi:10.1215/00127094-2018-0041

Mathematical Reviews number (MathSciNet)
MR3909896

Zentralblatt MATH identifier
07036862

#### Citation

Schreieder, Stefan. On the rationality problem for quadric bundles. Duke Math. J. 168 (2019), no. 2, 187--223. doi:10.1215/00127094-2018-0041. https://projecteuclid.org/euclid.dmj/1541646041

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