This question comes from my former question.
Let me formulate the question more precisely.
Let $C$ be a projective smooth curve over an algebraically closed field. Let $Y$ be any variety. Why for any closed point $y_1,y_2 \in Y$, $C \times y_1, C \times y_2$ are numerically equivalent in $C\times Y$ ? Here, numerically equivalence means they have same intersection number for any curve.
One can reduce the above case to the case where dimensional of $Y$ is $1$.