Stable rationality of quadric and cubic surface bundle fourfolds

Research Article
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Abstract

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the specialization method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2, 3) in Open image in new window is not stably rational. Via projections onto the two factors, Open image in new window is a cubic surface bundle and Open image in new window is a conic bundle, and we analyze the stable rationality problem from both these points of view. Also, we introduce, for any \(n\geqslant 4\), new quadric surface bundle fourfolds Open image in new window with discriminant curve Open image in new window of degree 2n, such that \(X_n\) has nontrivial unramified Brauer group and admits a universally \(\mathrm {CH}_0\)-trivial resolution.

Keywords

Stable rationality Brauer group Quadric bundles Cubic surface bundles Fano fourfolds 

Mathematics Subject Classification

14C35 14D06 14E05 14E08 14F22 14J20 14J26 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Asher Auel
    • 1
  • Christian Böhning
    • 2
  • Alena Pirutka
    • 3
    • 4
  1. 1.Department of MathematicsYale UniversityNew HavenUSA
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  4. 4.National Research University Higher School of EconomicsMoscowRussian Federation

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