By Alexei Skorobogatov
The classical descent on curves of genus you'll be interpreted as supplying stipulations at the set of rational issues of an algebraic sort X outlined over a host box, seen as a subset of its adelic issues. this is often the usual set-up of the Hasse precept and numerous approximation houses of rational issues. the main recognized between such stipulations is the Manin obstruction exploiting the Brauer-Grothendieck staff of X. It emerged lately non-abelian generalization of descent occasionally presents more desirable stipulations on rational issues. An all-encompassing 'obstruction' is said to the X-torsors (families of central homogenous areas with base X) less than algebraic teams. This booklet, first released in 2001, is an in depth exposition of the final conception of torsors with key examples, the relation of descent to the Manin obstruction, and functions of descent: to conic bundles, to bielliptic surfaces, and to homogenous areas of algebraic teams.
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