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On the Hasse principle for zero-cycles on Severi-Brauer fibrations. (English) Zbl 1098.14005

The main object of the paper under review is a Severi-Brauer fibration \(\pi: X\to C\). This means that \(C\) is a smooth, projective, geometrically integral \(k\)-curve, \(X\) is a smooth, projective, geometrically integral \(k\)-variety, \(\pi\) is a proper, dominant \(k\)-morphism whose generic fibre \(X_{\eta}\) is a Severi-Brauer \(k(C)\)-variety. The index of such a fibration, defined as the least positive degree of a zero-cycle on \(X_{\eta}\), is assumed squarefree. The author focuses on the case where \(k\) is a number field and studies the Hasse principle for (relative) zero-cycles. More precisely, he defines \(\text{CH}_0(X/C)\) as the kernel of the induced map \(\pi_*: \text{CH}_0(X)\to \text{CH}_0(C)\). The deviation \(\text{Ш} \text{CH}_0(X/C)\) from the Hasse principle for this group is defined as the kernel of the localization map \(\text{CH}_0(X/C)\to \prod_v \text{CH}_0(X_v/C_v)\).
The main result of the paper (Theorem 1.1) relates this kernel with other arithmetical invariants. Namely, there exists an exact sequence
\[ 0\to \text{Ш} (\text{{Ker}}\, \Phi) \to \text{Ш} \text{CH}_0(X/C) \to \text{Ш}^1(T) \to 0, \] where \(T\) is the Néron-Severi torus of \(X\), \(\text{Ш}^1(T)\) is the Tate-Shafarevich group of \(T\), \(\Phi : \text{CH}_0(X/C) \to H^1(k,T)\) is the characteristic map, and \(\text{Ш} (\text{{Ker}}\, \Phi)\) is defined as the kernel of the corresponding localization map \(\text{{Ker}}\, \Phi \to \prod_v{\text{{Ker}}\, \Phi_v}\). Note that the left-hand term in the above exact sequence may be not zero as follows from a counter-example to a conjecture of Bloch constructed by V. Suresh [K-Theory 10, 597–610 (1996; Zbl 0874.14003)]. The author studies this kernel and obtains some sufficient conditions when it vanishes. The proofs heavily use earlier results by E. Frossard [J. Reine Angew. Math. 557, 81–101 (2003; Zbl 1095.14015)].

MSC:

14C25 Algebraic cycles
14G05 Rational points
11G35 Varieties over global fields
14G25 Global ground fields in algebraic geometry
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