On the continuous part of codimension 2 algebraic cycles on three-dimensional varieties

Let $X$ be a nonsingular projective threefold over an algebraically closed field and let $A^2(X)$ be the group of algebraically trivial codimension 2 algebraic cycles on $X$ modulo rational equivalence with coefficients in $\mathbb Q$. Assume that $X$ is birationally equivalent to a threefold $X'$ fibered over an integral curve $C$ with generic fiber $X_{\bar \eta }$ satisfying the following three conditions: the motive $M(X'_{\bar \eta })$ is finite-dimensional; $H^1_{\mathrm{et}}(X_{\bar\eta},{\mathbb Q}_l)=\nobreak0$; $H^2_{\mathrm{et}}(X_{\bar \eta },{\mathbb Q} _l(1))$ is spanned by divisors on $X_{\bar \eta }$. We prove that under these three assumptions the group $A^2(X)$ is weakly representable: there exist a curve $Y$ and a correspondence $z$ on $Y\times X$ such that $z$ induces an epimorphism $A^1(Y)\to A^2(X)$, where $A^1(Y)$ is isomorphic to ${\mathrm{Pic}}^0(Y)$ tensored with $\mathbb Q$. In particular, this result holds for threefolds birationally equivalent to three-dimensional del Pezzo fibrations over a curve.
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