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

1. (i) the action of G preserves the fibres of ø,

2. (ii) every geometric fibre of ø is an orbit of a geometric point,

3. (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

4. (iv) the structure sheaf Ox is the G-invariant subsheaf of ø*(Oy).