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 $X^{\prime }$ admitting a fibration over an integral curve $C$ whose generic fiber $X_{\bar{\eta }}^{\prime }$, where $\bar{\eta }=Spec(\overline{k(C)})$, satisfies the following three conditions: (i) the motive $M(X_{\bar{\eta }}^{\prime })$ is finite-dimensional, (ii) $H_{et}^1(X_{\bar{\eta }},{\mathbb{Q}}_l)=0$ and (iii) $H_{et}^2(X_{\bar{\eta }},{\mathbb{Q}}_l(1))$ 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 $Pi{c}^0(Y)$ tensored with $\mathbb{Q}$. In particular, the result holds for threefolds birational to three-dimensional Del Pezzo fibrations over a curve.

16 pages