In the thesis we study codimension p algebraic cycles on a 2p-dimensional nonsingular projective variety X de�ned over an uncountable algebraically closed ground �eld k of characteristic 0. The main result (Theorem 4.7.1 in the thesis) says that, under some weak representability assumptions on the continuous parts of the Chow groups of the variety X and its nonsingular hyperplane sections Y , the kernel of the Gysin homomorphism from the codimension p Chow group of the very general Y to the codimension p + 1 Chow group of X is countable. As an application, we obtain the following concrete result. Let X be a nonsingular cubic hypersurface in P5 over k. Then, for a very general Y , there exists a countable set � of closed points on the Prym variety of the threefold Y , such that, if � and �0 are two linear combinations of lines of the same degree on Y , the one-cycle � is rationally equivalent to the one-cycle �0 on X if and only if the di�erence �