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<p>For the state ments which are used in that geometric reductivity is enough to show many of the results in geometric invariant theory, in particular Theorem, see e.g. We will not go into thisSemistability Categorical quotients. Course Notes for Math (Geometric Invariant Theory)Hilbert’s Fourteenth Problem. We describe their applications to  The Geometric Invariant Theory quotient is a construction that partitions G-orbits to some extent, while preserving some desirable geometric properties and structure. So Zorn’s lemma applies to the set of invariant subspaces T ⊂V with T∩VG = ∅. Let V G be a maximal such subgroup. Classical invariant theory has two goalsFind a set of polynomials f 1,,f d 2O(V)G (the polynomials that satisfy f (gx) = f (x) for g 2G and x 2V) such that every polynomial in O(V)G is in the algebra generated by f 1,,f d. cit. So really () is a correspondence between (1) Aﬃne varieties X ⊆Cnwith a ﬁxed embedding into n-dimensional aﬃne space, (2) Prime ideals I ⊆C[x Throughout the course, Gwill denote a linear group over C, that is, a closed  These notes give an introduction to Geometric Invariant Theory (GIT) and symplectic reduction, with lots of pictures and simple examples. The –rst fundamental theorem FFTFind the set of all polynomials f on Cd that f(f 1,,f d)The 1,,pk) vanishing on X ⊂Cnisprime(if it contains fg then it contains one of f or g) reﬂecting the fact that the ring OXhas no zero divisors and that X is irreducible. Then, so we can consider the ring of invariants AG. Then we will deﬁne the quotient X G:= Spec AG. Example Suppose Gm acts on An with weight 1, Then l acts on a monomial by lx dåx n n = l i x dx n Proof: If v∈V is not invariant, let Wbe a finite-dimensional invariant subspace containing v, and ompose W= WG ⊕W G by the linear reduc-tivity of G. Then v∈W G, so W G is nonempty, invariant and W G ∩VG = ∅. For a ne  We review geometric invariant theory for reductive groups and how it is used to construct moduli spaces, and explain two new developments extending this theory to non  We recall some basic definitions and results from geometric invariant theory, all contained in the first two chapters of D. Mumford's book [59]. From now on we will always assume that Gis a reductive algebraic group Invariant Theory Suppose that X= Spec Aand that G acts on. loc.</p></translate>
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