We study the stable rationality problem for quadric and cubic surface bundlesover surfaces from the point of view of the degeneration method for the Chowgroup of 0-cycles. Our main result is that a very general hypersurface X ofbidegree (2,3) in P^2 x P^3 is not stably rational. Via projections onto thetwo 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 connectedfourfolds with rational and nonrational fibers. Finally, we introduce newquadric surface bundle fourfolds over P^2 with discriminant curve of any evendegree at least 8, having nontrivial unramified Brauer group and admitting auniversally CH_0-trivial resolution.
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机译：从零周期Chowgroup的退化方法的角度，研究了二次曲面和三次曲面束在曲面上的稳定合理性问题。我们的主要结果是，在P ^ 2 x P ^ 3中双度（2,3）的非常普遍的超曲面X不是稳定有理的。通过对这两个因素的投影，X是P ^ 2上的立方曲面束和P ^ 3上的圆锥形束，我们从这两个角度分析了稳定的合理性问题，这提供了一个光滑的有理连接四重族的例子与合理和非理性的纤维。最后，我们介绍了P ^ 2上四倍的新二次曲面束，具有至少8的均匀度的判别曲线，具有非平凡的不分叉的Brauer基团，并且通常允许CH_0的平凡分辨率。
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