3 - Examples of torsors
from Part one - TORSORS
Published online by Cambridge University Press: 05 May 2010
Summary
The point of view of this chapter is that a torsor can be considered as the morphism of passing to the quotient by a freely acting algebraic group. To make this statement precise we evoke the basics of geometric invariant theory, and then treat in detail the example of a maximal torus of PGL(5) acting on the Grassmannian G(3,5). This leads to classification of Del Pezzo surfaces of degree 5. After a discussion of properties of torsors related to central extensions of algebraic groups, we describe explicit 2- and 4- descent on elliptic curves. Our intention in this chapter is to demonstrate in the examples the rôle played by the general concepts such as the type of a torsor, universal torsors, and so on.
Torsors in geometric invariant theory
Suppose that an algebraic k-group G acts on a k-variety Y. The following definition describes what could reasonably be called ‘the quotient variety Y/G’.
Definition 3.1.1 ([Mumford, GIT], Def. 0.6)The morphism ø : Y → X is called ageometric quotientof X by G if
(i) the action of G preserves the fibres of ø,
(ii) every geometric fibre of ø is an orbit of a geometric point,
(iii) ø is universally open (for any base change T/X a subset U ⊂ T is open if and only if U ×TYT is open in YT), and
(iv) the structure sheaf Ox is the G-invariant subsheaf of ø*(Oy).
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- Torsors and Rational Points , pp. 42 - 61Publisher: Cambridge University PressPrint publication year: 2001