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Zero-cycles on quadric fibrations: Finiteness theorems and the cycle map. (English) Zbl 0865.14002
Invent. Math. 122, No. 1, 83-117 (1995); erratum 123, No. 3, 611 (1996).
From the very nice introduction: “Let $$C$$ be a smooth projective integral curve over a field $$k$$ of characteristic not 2, let $$\pi:X\to C$$ be a quadric fibration of relative dimension $$d\geq 1$$. $$\pi$$ is called admissible if for every closed point $$P\in C$$ the scheme $$X\times_C\text{Spec} {\mathcal O}_{C,P}$$ is $${\mathcal O}_{C,P}$$-isomorphic to the scheme of zeros of a diagonal form $$\langle a_1, \dots, a_n \rangle$$ with $$0\leq v_P(a_i)\leq 1$$ and $$v_P(a_i)=0$$ for $$1\leq i\leq {n+1 \over 2}$$ $$(v_P$$ is the valuation at $$P)$$. (For any quadric fibration $$\pi:X \to C$$ there exists an admissible quadric fibration with isomorphic generic fiber.) Throughout $$\pi:X\to C$$ is assumed to be a quadric fibration of relative dimension $$d$$ and $$q$$ is the quadratic space defining its generic fiber, $$d(q)$$ is the discriminant of $$q$$. If $$d\leq 2$$ and $$k$$ is a number field (except if $$d=2$$ and $$d(q)$$ is not a square in $$k(C)$$ but $$d(q)$$ is a square in $$\overline k(C))$$ or $$k$$ is a local field, J.-L. Colliot-Thélène and A. N. Skorobogatov proved [$$K$$-Theory 7, No. 5, 477-500 (1993; Zbl 0837.14002)] that the relative Chow group $$\text{CH}_0(X/C)$$ is finite and that $$\text{CH}_0 (X/C)=0$$ if $$k$$ is of cohomological dimension 1. The paper of Colliot-Thélène and Skorobogatov is the starting point of the paper discussed here. Colliot-Thélène and Skorobogatov raised the following questions:
(i) If $$k$$ is a finitely generated field over $$\mathbb{Q}$$ is $$\text{CH}_0 (X/C)$$ finite?
(ii) If $$d\geq 3$$ and $$k$$ a field of cohomological dimension less than or equal to 2, is the group $$\text{CH}_0 (X/C)$$ zero or is it at least finite?
(iii) If $$\pi:X\to C$$ is a smooth conic fibration over a $$p$$-adic field is $$\text{CH}_0 (X/C)= 0$$?
(iv) If $$d=2$$, and $$k$$ is a $$p$$-adic field and the fibers of $$\pi$$ are all geometrically integral, is the group $$\text{CH}_0 (X/X)=0$$? (Is it true at least if $$C=\mathbb{P}^1?)$$
In the paper answers to these questions (under some constraints on $$k$$ and on the generic fiber) are given. A survey of the main results:
If $$C$$ is a smooth projective conic over $$k$$ and if the generic fiber is defined by a Pfister neighbor then the group $$A_0(X)=0$$ $$(A_0(X)$$ is the group of zero-cycles of degree zero modulo rational equivalence.) If $$C=\mathbb{P}^1$$ then the condition on the generic fiber can be dropped. – An example (3.6) with $$k=\mathbb{Q}(t)$$ or $$\mathbb{Q}_5 (t)$$ and $$X=Q\times C$$, $$Q$$ a quadric of dimension 2 over $$k$$ such that $$A_0(Q\times C) \neq 0$$ is given.
For $$k$$ a number field or a field of 2-cohomological dimension $$\leq 2$$, it is shown that $$A_0(X)=0$$ for any smooth quadric fibration over a conic (3.5).
Counterexamples to questions (iii) and (iv) are given (6.1 and 6.2). In case $$k$$ is a local field and $$\pi:X\to C$$ is an admissible quadric fibration of relative dimension $$\geq 1$$ over a smooth projective curve $$C$$ over $$k$$ it is shown that $$\text{CH}_0 (X/C)$$ is finite (4.8). This leads to finiteness of the torsion of $$\text{CH}_0 (X)$$ for such fibration (4.9). It also leads to finiteness of $$A_0(X)$$ for varieties $$X$$ that are smooth complete intersections of two quadrics in $$\mathbb{P}^1$$ and such that $$X(k)\neq \emptyset$$ (4.10).
The results of Colliot-Thélène and Skorobogatov mentioned above, are completed. It is proven that they also hold if $$d(q)$$ is not a square in $$k(C)$$ but is a square in $$\overline k(C)$$ (5.1 and 5.2). In (5.3) it is proven that for an admissible fibration over any smooth projective conic over a number field $$k$$, $$\text{CH}_0 (X_{k_v}/C_{k_v})=0$$ for all but finitely many places $$v$$ of $$k$$. The (counter)examples in section 6 (mentioned before) also lead to negative answers to other open questions. In (7.6) a smooth conic fibration $$\pi:X\to C$$ over a smooth hyperelliptic curve $$C$$ defined over $$\mathbb{Q}_3$$ is constructed such that $$X (\mathbb{Q}_3) \neq\emptyset$$ but for which the map $$\kappa_X$$ induced by the Brauer pairing is not injective. In (8.5) an example of a smooth projective surface with rational points over $$p$$-adic fields for which the cycle map $$\text{CH}^2 (X)/2\to H^4_{\text{et}} (X,\mathbb{Z}/2)$$ is not injective is given.”
The erratum that appeared in Invent. Math. 123, No. 3, 611 (1996), mentions that at several pages the sign $$\leq$$ was changed into the number 5. Although this is an unfortunate mistake, it is not hard for the reader the correct it.

##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14C25 Algebraic cycles 11G35 Varieties over global fields 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry
##### Keywords:
relative Chow group; zero-cycles
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