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Abstract: For a smooth and geometrically irreducible variety X over a field k, thequotient of the absolute Galois group G {kX} by the commutator subgroup ofG {\bar kX} projects onto G k. We investigate the sections of thisprojection. We show that such sections correspond to -infinite divisions- ofthe elementary obstruction of Colliot-Th\-el\`ene and Sansuc. If k is a numberfield and the Tate-Shafarevich group of the Picard variety of X is finite, thensuch sections exist if and only if the elementary obstruction vanishes. Forcurves this condition also amounts to the existence of divisors of degree 1.Finally we show that the vanishing of the elementary obstruction is notpreserved by extensions of scalars.

Author: Hélène Esnault, Olivier Wittenberg

Source: https://arxiv.org/

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