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 |
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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: | |
URI: | https://livrepository.liverpool.ac.uk/id/eprint/3029506 |