Geodesic
On abelian birational sections
Esnault, Hélène1 ; Wittenberg, Olivier2
1 Universität Duisburg–Essen, Mathematik, 45117 Essen, Germany
2 Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, 75320 Paris Cedex 05, France
Journal of the American Mathematical Society, Tome 23 (2010) no. 3, pp. 713-724

Voir la notice de l'article provenant de la source American Mathematical Society

For a smooth and geometrically irreducible variety $X$ over a field $k$, the quotient of the absolute Galois group $G_{k(X)}$ by the commutator subgroup of $G_{\bar k(X)}$ projects onto $G_k$. We investigate the sections of this projection. We show that such sections correspond to “infinite divisions” of the elementary obstruction of Colliot-Thélène and Sansuc. If $k$ is a number field and the Tate–Shafarevich group of the Picard variety of $X$ is finite, then such sections exist if and only if the elementary obstruction vanishes. For curves this condition also amounts to the existence of divisors of degree $1$. Finally we show that the vanishing of the elementary obstruction is not preserved by extensions of scalars.
DOI : 10.1090/S0894-0347-10-00660-0

Esnault, Hélène 1 ; Wittenberg, Olivier 2

1 Universität Duisburg–Essen, Mathematik, 45117 Essen, Germany
2 Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, 75320 Paris Cedex 05, France 
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Esnault, Hélène; Wittenberg, Olivier. On abelian birational sections. Journal of the American Mathematical Society, Tome 23 (2010) no. 3, pp. 713-724. doi: 10.1090/S0894-0347-10-00660-0

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