  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.
