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This is a survey of recent examples of varieties that are not stablyrational. We review the specialization method based on properties of the Chowgroup of zero-cycles used in these examples and explain the point of view ofunramified cohomology for the construction of nontrivial stable invariants ofthe special fiber. In particular, we find an explicit formula for the Brauergroup of fourfolds fibered in quadrics of dimension two over a rationalsurface.

Varieties that are not stably rational, zero-cycles and unramified cohomology