Let K be an uncountable algebraically closed field of characteristic , and let be a smooth projective connected variety of dimension , embedded into over . Let be a hyperplane section of , and let and be the groups of algebraically trivial algebraic cycles of codimension and modulo rational equivalence on and , respectively. Assume that, whenever is smooth, the group is regularly parametrized by an abelian variety and coincides with the subgroup of degree classes in the Chow group. We prove that the kernel of the push-forward homomorphism from is the union of a countable collection of shifts of a certain abelian subvariety inside . For a very general hyperplane section whose tangent space is the group of vanishing cycles.