Stable rationality of quadric and cubic surface bundle fourfolds

TL;DR

This paper investigates the stable rationality of quadric and cubic surface bundle fourfolds, showing that a general hypersurface of bidegree (2,3) in P^2 x P^3 is not stably rational, and introduces new examples with specific properties.

Contribution

It demonstrates the non-stable rationality of a general hypersurface in P^2 x P^3 and constructs new quadric surface bundle fourfolds with particular discriminant curves and nontrivial Brauer groups.

Findings

A very general hypersurface of bidegree (2,3) in P^2 x P^3 is not stably rational.

New quadric surface bundle fourfolds over P^2 with discriminant degree at least 8 are constructed.

The study provides examples of rationally connected fourfolds with both rational and nonrational fibers.

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.