# Stable rationality of quadric and cubic surface bundle fourfolds

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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 2*n*, 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## References

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