Editing Linear algebraic group (section)

===Group actions and geometric invariant theory===
An '''[[group-scheme action|action]]''' of a linear algebraic group ''G'' on a variety (or scheme) ''X'' over a field ''k'' is a morphism
:<math>G \times_k X \to X</math>
that satisfies the axioms of a [[Group action (mathematics)|group action]]. As in other types of group theory, it is important to study group actions, since groups arise naturally as symmetries of geometric objects.

Part of the theory of group actions is [[geometric invariant theory]], which aims to construct a quotient variety ''X''/''G'', describing the set of [[orbit (group theory)|orbit]]s of a linear algebraic group ''G'' on ''X'' as an algebraic variety. Various complications arise. For example, if ''X'' is an affine variety, then one can try to construct ''X''/''G'' as [[spectrum of a ring|Spec]] of the [[ring of invariants]] ''O''(''X'')<sup>''G''</sup>. However, [[Masayoshi Nagata]] showed that the ring of invariants need not be finitely generated as a ''k''-algebra (and so Spec of the ring is a scheme but not a variety), a negative answer to [[Hilbert's 14th problem]]. In the positive direction, the ring of invariants is finitely generated if ''G'' is reductive, by [[Haboush's theorem]], proved in characteristic zero by [[David Hilbert|Hilbert]] and Nagata.

Geometric invariant theory involves further subtleties when a reductive group ''G'' acts on a projective variety ''X''. In particular, the theory defines open subsets of "stable" and "semistable" points in ''X'', with the quotient morphism only defined on the set of semistable points.