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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
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