Unirational threefolds with no universal codimension $2$ cycle
Résumé
We prove that the general quartic double solid with $k\leq 7$ nodes does not admit a Chow theoretic decomposition of the diagonal, or equivalently has a nontrivial universal ${\rm CH}_0$ group. The same holds if we replace in this statement "Chow theoretic" by "cohomological". In particular, it is not stably rational. We also prove that the general quartic double solid with seven nodes does not admit a universal codimension $2$ cycle parameterized by its intermediate Jacobian, and even does not admit a parametrization with rationally connected fibres of its Jacobian by a family of $1$-cycles. This implies that its third unramified cohomology group is not universally trivial.