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Zero-cycles on rational surfaces over number fields. (English) Zbl 0688.14008
The author answers, in the case of rational surfaces with a conic bundle structure \(X/{\mathbb{P}}^ 1_ k\), several statements conjectured by J.-L. Colliot-Thélène and J.-J. Sansuc [cf. Duke Math. J. 48, 421-447 (1981 Zbl 0479.14006)].
More precisely let X be a smooth projective, geometrically integral surface over a perfect \(field\quad k,\) and \(\bar F\) the function field of \(\bar X=\bar k\times_ kX\). A map from \(K_ 2(\bar F)\) to the sum of function fields of all irreducible divisors on X is constructed. Let M be the cokernel of this map. - In this paper it is shown that in the case X is a rational surface with conic bundle structure \(X/{\mathbb{P}}^ 1_ k\) the kernel of the natural map \(H^ 1(Gal(\bar k/k),M)\) to \(\prod_{v}H^ 1(Gal(\bar k_ v/k_ v),M_ v) \) is 0 (where \(k_ v\) is the completion at a place v of k). - As a corollary it is shown that if furthermore \(H^ 1(Gal(\bar k,k),Pic(\bar X))=0\) then there exists a 0-cycle of degree \( 1\) on X if and only if there are 0-cycles of degree \( 1\) on \(X_ v\) at all places v of k. It is also explained how to calculate the order of \(A_ 0(X)\).
Reviewer: A.Papantonopoulou

MSC:
14C25 Algebraic cycles
14M20 Rational and unirational varieties
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
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References:
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