Universal triviality of zero-cycles

We will discuss a certain explicit cohomological conditions on a smooth projective threefold that is enough for universal triviality of zero-cycles on the threefold. We will consider two examples when these conditions are satisfied: a threefold with a full exceptional collection and a double covering of the projective space ramified over a quartic with 10 singular points. In particular, for the second example, this means that it is impossible to use Voisin's method in order to show non-rationality of the threefold.