Duke Mathematical Journal

On the rationality problem for quadric bundles

Stefan Schreieder

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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 PCn are not stably rational if r[2n11,2n2]. In our proofs we introduce a generalization of the specialization method of Voisin and of Colliot-Thélène and Pirutka which avoids universally CH0-trivial resolutions of singularities.

Article information

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

Dates
Received: 19 October 2017
Revised: 12 July 2018
First available in Project Euclid: 8 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1541646041

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

Mathematical Reviews number (MathSciNet)
MR3909896

Zentralblatt MATH identifier
07036862

Subjects
Primary: 14E08: Rationality questions [See also 14M20]
Secondary: 14D06: Fibrations, degenerations 14M20: Rational and unirational varieties [See also 14E08] 14J35: $4$-folds

Keywords
rationality problem stable rationality decomposition of the diagonal unramified cohomology Lüroth problem

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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