Asymptotic primes and the Chow group

TL;DR

This paper uncovers a surprising link between asymptotic prime divisors and Chow groups, demonstrating that certain Chow groups are torsion in specific graded rings, with implications for higher-dimensional cases.

Contribution

It establishes that the Chow group $A_1(R)$ is torsion for a class of graded rings constructed from specific polynomial relations, extending to higher dimensions.

Findings

Chow group $A_1(R)$ is torsion for certain graded rings

Connection between asymptotic primes and Chow groups established

Results extended to higher-dimensional analogues

Abstract

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.