On the Continuous Part of Codimension Two Algebraic Cycles on Threefolds over a Field

Let X be a non-singular projective threefold over an algebraically closed field of any characteristic, and let A(X) be the group of algebraically trivial codimension 2 algebraic cycles on X modulo rational equivalence with coefficients in Q. Assume X is birationally equivalent to another threefold X ′ admitting a fibration over an integral curve C whose generic fiber X ′ η̄, where η̄ = Spec(k(C)), satisfies the following three conditions: (i) the motiveM(X ′ η̄) is finitedimensional, (ii) H et(Xη̄,Ql) = 0 and (iii) H 2 et(Xη̄,Ql(1)) is spanned by divisors on Xη̄. We prove that, provided these three assumptions, the group A(X) is representable in the weak sense: there exists a curve Y and a correspondence z on Y ×X , such that z induces an epimorphism A(Y ) → A(X), where A(Y ) is isomorphic to Pic(Y ) tensored with Q. In particular, the result holds for threefolds birational to three-dimensional Del Pezzo fibrations over a curve.