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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.

##### MSC:
 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
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