Stable rationality of quadric and cubic surface bundle fourfolds

Abstract

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2,3) in P^2 x P^3 is not stably rational. Via projections onto the two factors, X is a cubic surface bundle over P^2 and a conic bundle over P^3, and we analyze the stable rationality problem from both these points of view. This provides another example of a smooth family of rationally connected fourfolds with rational and nonrational fibers. Finally, we introduce new quadric surface bundle fourfolds over P^2 with discriminant curve of any even degree at least 8, having nontrivial unramified Brauer group and admitting a universally CH_0-trivial resolution.

27 pages, comments welcome!

References in corpus (5)

• Quadric surface bundles over surfaces and stable rationality
• Stable rationality of del Pezzo fibrations of low degree over projective spaces
• Intersections of three quadrics in ${\mathbb{P}}^7$
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Cited by in corpus (4)

• Quadric surface bundles over surfaces and stable rationality
• Torsion orders of Fano hypersurfaces
• On the rationality of quadric surface bundles
• Brauer groups of involution surface bundles