Descent obstruction and fundamental exact sequence

Details Descent obstruction and fundamental exact sequence Descent obstruction and fundamental exact sequence

2010-05-07

arxiv.org/abs/1005.1302v2

Mathematics

Algebraic Geometry

20 pages - in this second version the applications (section 3) are reorganised as "abelian" applications (section 3) and "non-

Scientific paper

A torsor under a k-group scheme G on a variety X over a number field k imposes a descent obstruction against the existence of rational points on X. We discuss the finite descent obstruction, that is for all such torsors under finite k-groups G, in view of a local-global interpolation property for sections of the fundamental group short exact sequence of X/k. There are applications to the Brauer-Manin obstruction, to the descent obstruction by torsors under linear groups, and to the birational version of Grothendieck's section conjecture over number fields. In particular, we obtain examples of families of curves over number fields, such that the birational section conjecture is true in a non-trivial way.

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