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Chow groups of zero cycles on pencils of quadrics. (Groupe de Chow des zéro-cycles sur les fibrés en quadriques.) (French) Zbl 0837.14002
Let \(C\) be a smooth, projective, geometrically integral curve over a field \(k\), \(X\) an integral \(k\)-variety and \(p : X \to C\) a dominant \(k\)- morphism. The paper gives a formula for the relative Chow group \(\text{CH}_0 (X/C) = \ker (p_* : \text{CH}_0 (X) \to \text{CH}_0 (C))\) under the hypotheses that for each closed point \(P\) of \(C\), every zero-cycle of degree zero on the fibre \(X_P\) is rationally equivalent to 0 and the group \(p_* \text{CH}_0 (X_P)\) is generated by the image of a zero-cycle on \(X_P\) supported on the regular locus of \(X\).
For any variety \(Y\) over a field \(F\), let \(N_Y (F)\) be defined as the subgroup of \(F^*\) generated by all norms \(N_{L/F} (L^*)\) for all finite extensions \(L\) of \(F\) such that \(Y(L) \neq \emptyset\). Define \(k(C)^*_{dn}\) to be the group of nonzero rational functions on \(C\) which at any point of \(P \in C\) can be written as a product of a unit at \(P\) and an element of \(N_{X_\eta} (k(C))\). Then \[ \text{CH}_0 (X/C) \simeq k (C)^*_{dn}/k [C]^* \cdot N_{X_\eta} \bigl( k(C) \bigr). \] This theorem is then applied to the case of admissible quadric bundles. These are morphisms \(p:X\to C\) which are proper, surjective, with generic fibre \(X_\eta\) a smooth quadric hypersurface, whose localisation at any closed point \(P\) of \(C\) is isomorphic to a projective scheme over the local ring \(A_p\) at \(P\), given by a homogeneous equation \(\sum^n_{i = 1} a_i X^2_i = 0\) where the valuations \(v_p (a_i) \in \{0,1\}\) and \(v_p (a_i) = 0\) for \(i \leq (n + 1)/2\). If \(n \geq 3\) and \(k\) is a field of cohomological dimension one, then \(\text{CH}_0 (X/C) = 0\). If the relative dimension is two, the study of \(\text{CH}_0 (X/C)\) is related to the study of \(\text{CH}_0 (Y/ \widetilde C)\) for an associated conic bundle \(Y \to \widetilde C\) over a curve \(\widetilde C\) which is a double cover of \(C\) associated with the discriminant of the quadric bundle. When \(\widetilde C\) is geometrically integral, \(\text{CH}_0 (X/C)\) injects into \(\text{CH}_0 (Y/ \widetilde C)\). In the case where \(k\) is a number field or a local field, this implies that CH\(_0 (X/C)\) is finite.

14C05 Parametrization (Chow and Hilbert schemes)
14C25 Algebraic cycles
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
11E12 Quadratic forms over global rings and fields
Full Text: DOI
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