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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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