  Let $X$ be a non-singular projective threefold over an algebraically closed
field of any characteristic, 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 $X$ is birationally equivalent to another
threefold $X'$ admitting a fibration over an integral curve $C$ whose generic
fiber $X'_{\bar \eta}$, where $\bar \eta =Spec(\bar {k(C)})$, satisfies the
following three conditions: (i) the motive $M(X'_{\bar \eta})$ is
finite-dimensional, (ii) $H^1_{et}(X_{\bar \eta},\mathbb Q_l)=0$ and (iii)
$H^2_{et}(X_{\bar \eta},\mathbb Q_l(1))$ is spanned by divisors on $X_{\bar
\eta}$. We prove that, provided these three assumptions, the group $A^2(X)$ is
representable in the weak sense: there exists 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 $Pic^0(Y)$ tensored with $\mathbb Q$. In
particular, the result holds for threefolds birational to three-dimensional Del
Pezzo fibrations over a curve.
