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There is a variant of Standard Conjecture D for projective varieties over finite fields. It claims that rational and homological equivalences are equivalent on cycles after tensoring with $\mathbb{Q}$. This is often referred to as Beilinson's conjecture. Is it known whether this is true for product of curves?

Another related question: Is it known whether this conjecture is birationally invariant?

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    $\begingroup$ Surely birational equivalence of this conjecture is equivalent to the general case because you can embed any proojective variety in $\mathbb P^n$ and take the blow-up, and then view cycles on that variety as cycles on a variety birational to $\mathbb P^n$. $\endgroup$
    – Will Sawin
    Commented Feb 19, 2021 at 14:58

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