Asymptotic primes and the Chow group

In this paper we present an unexpected connection between the theory of asymptotic prime divisors and Chow groups. As an application we show that the Chow group $A_1(R)$ is a torsion group when $R$ is any graded ring such that we have an inclusion of graded rings $T \subseteq R \subseteq S$ where $S = k[X_1, X_2, Y]/(X_1^m + X_2^m + Y^n)$ where $k$ is algebraically field, $(m,n) = 1$, $(mn)^{-1} \in k$, $m, n \geq 2$. We consider $S$ graded with $deg \ X_1 = deg \ X_2 = n$ and $deg \ Y = m$. Also $T = k[X_1, X_2]$ where $deg \ X_1 = deg \ X_2 = n$. We also consider higher dimensional analogues of this result.