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I found a neat result in Beauville's paper "VARIÉTÉS DE PRYM ET JACOBIENNES INTERMÉDIAIRES" : if $U \subset \mathbb{P}^n$ is an open and $V \to U$ is a conic bundle whose fibres are all smooth conics, then $A^2(V) = 0$, where $A^n$ are the $n$-cycles modulo algebraic equivalence. An important input is that $A^n(U) = 0$ for all $n\geq 1$.

In my situation, $W \to U$ is an etale cover, so crucially the fibres are not connected. This causes problems for the picard group, so in particular $A^1(W)$ need not be $0$. How about $A^n(W)$ for $n \geq 2$? Can we prove $A^2(W) = 0$ or can we find a counter-example?

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    $\begingroup$ Any smooth variety locally admits an etale map to (an open of) $\mathbf{P}^n$. $\endgroup$
    – Piotr Achinger
    Commented Apr 2, 2023 at 14:57
  • $\begingroup$ @PiotrAchinger ok I see. Well, is there anything we can say about the higher cycles on an etale cover? For the picard group, there are some known results (Riemann-Hurwitz etc...). $\endgroup$
    – TCiur
    Commented Apr 3, 2023 at 11:09

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