{
  "id": "math/0610341",
  "version": "v2",
  "published": "2006-10-10T19:31:55.000Z",
  "updated": "2007-07-16T22:35:28.000Z",
  "title": "On the continuous part of codimension two algebraic cycles on threefolds over a field",
  "authors": [
    "Vladimir Guletskii"
  ],
  "comment": "16 pages",
  "categories": [
    "math.AG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-07-16T22:35:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C15",
      "14C25"
    ],
    "keywords": [
      "algebraic cycles",
      "continuous part",
      "three-dimensional del pezzo fibrations",
      "modulo rational equivalence",
      "threefolds birational"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10341G"
    }
  }
}