Rationality of Motivic Chow series modulo A1-homotopy

Consider the formal power series ∑[Cp,α(X)]tα (called the Motivic Chow series), where Cp(X)=⨿Cp,α(X) is the Chow variety of X parametrizing the p-dimensional effective cycles on X with Cp,α(X) its connected components, and [Cp,α(X)] its class in K(ChM)A1, the K-ring of Chow motives modulo A1 homotopy. Using the Picard product formula and torus action, we will show that the Motivic Chow series is rational in many cases.