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

Guletskii, V (2009) On the continuous part of codimension two algebraic cycles on threefolds over a field. Sbornik: Mathematics, Volume (Number). pp. 17-30.

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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.

Item Type:	Article
Additional Information:	## TULIP Type: Articles/Papers (Journal) ##
Uncontrolled Keywords:	math.AG, math.AG, 14C15; 14C25
Depositing User:	Symplectic Admin
Date Deposited:	19 Dec 2018 09:43
Last Modified:	26 Jun 2023 13:34
DOI:	10.1070/SM2009v200n03ABEH003998
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3029506