On the continuous part of codimension two algebraic cycles on threefolds over a field

Abstract

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.