Introduction to GIT We start with some facts about algebraic groups we will need later on. 1 Algebraic groups • Let X be a variety over K = K with char(K) = 0. • Let G be an ane algebraic group acting on X: σ : G × X ! X with g:x := σ(g; x) • Denote by XG := fx 2 X j g:x = x 8g 2 Gg the G-invariant points and by Gx := fg 2 G j g:x = xg the stabiliser of x. • The set G:x := fg:x j 8g 2 Gg is the orbit of x. • Let Ψ: G × X ! X × X be dened as Ψ(g; x) = (σ(g; x); x). • A map f : X ! Y is called G-invariant if f(g:x) = f(x) 8g 2 G; x 2 X. Denition 1.1. An ane algebraic group G over K is an ane algebraic variety over K with a group structure such that the inversion map β : G ! G and the group operation γ : G × G ! G are regular maps. Example. A torus T = (K∗)m is an ane algebraic group. The general linear group GL(n; K) is an ane algebraic group over K with matrix multiplication. Remark. Ane algebraic groups can be embedded into GL(n; K) and are therefore linear algebraic groups. Denition 1.2. A linear algebraic group G is called linearly reductive if for any rational representation ρ : G ! GL(V ) and any nonzero G-invariant vector v 2 V there exists a linear G-invariant function f on V such that f(v) 6= 0. Denition 1.3. A linear algebraic group G is called geometrically reductive if for any rational representation ρ : G ! GL(V ) and any nonzero G-invariant vector v 2 V there exists a homogeneous G-invariant polynomial f on V such that f(v) 6= 0. Remark. Linearly reductive implies geometrically reductive. 1 Examples of geometrically reductive groups are (C∗)k and GL(n) for all k and n. Products of geometrically reductive groups are geometrically reduc- tive. For further details see [Hum75, ch. VII] and [MFK65, p. 191-193]. The reductiveness of a group gives a useful property for a group action which we will need later. Lemma 1.1. Let G be a geometrically reductive group acting on some ane variety X. Let Z1 and Z2 be two closed G-invariant (g:Zi ⊆ Zi for all g 2 G) subsets of X with Z1 \ Z2 = ;. Then there exists a G-invariant G funcion Ψ 2 O(X) such that Ψ(Z1) = 1 and Ψ(Z2) = 0. Proof. Z1 \ Z2 = ;) (1) = I(Z1 \ Z2) = I(Z1) + I(Z2) ) 9f1 2 I(Z1); f2 2 I(Z2): f1 + f2 = 1 Let W ⊆ O(X) be the linear subspace spanned by fg:f2jg 2 Gg which is nite dimensional with some basis f'1; :::; 'ng. Set n . Since the are -invariant, we have f : X ! A ; x 7! ('1(x); :::; 'n(x)) Zi G (g:f2)(Zi) = −1 f2(g :Zi) = f2(Zi). ) 'j(Z1) = f2(Z1) = 1 and 'j(Z2) = f2(Z2) = 0 ) f(Z1) = (1; :::; 1) and f(Z2) = (0; :::; 0) is geometrically reductive homogeneous such that G ) 9F 2 K[x1; :::; xn] F (1; ::; 1) = 1 ) Ψ := f ∗(F ) has the desired properties. Just identifying points of the same orbit gives only a set-theoretical quotient. If we want to construct the quotient of a variety by the group action as a variety and the map to this quotient such that it be a morphism, we need to make the notion of quotients more precise. For this we follow [Dol03]. 2 Quotients Denition 2.1. A G-invariant map p : X ! Y is called a categorical quotient (of the G action), if for every G-invariant f : X ! Z there exists a unique f : Y ! Z, such that f ◦ p = f. Denition 2.2. • A G-equivariant map p : X ! Y is called a good categorical quotient (of the G-action), if p satises: 2 i) For all U ⊆ Y open, p∗ : O(U) !O(p−1(U)) is an isomorphism onto the subring O(p−1(U))G of G-invariant functions. ii) If W ⊆ X is closed and G-invariant, then p(W ) ⊆ Y is closed. iii) If V1;V2 ⊆ X are closed, G-invariant, and V1 \ V2 = ;, then p(V1) \ p(V2) = ;. • A (good) categorical quotient is called a (good) geometric quotient, if Im(Ψ) = X ×Y X. Remark. Satisfying the property for being a geometric quotient means that the preimage under p of one point in Y is exactly one G-orbit in X. Example. Let ∗ act on 2 by . Then G = C X = C n f0g g:(x1; x2) = (gx1; gx2) the map p : X ! 1 , p(x ; x ) = (x : x ) is a good geometric quotient, PC 1 2 1 2 because the smallest closed and G-invariant subsets are exactly the orbits and two dierent orbits are mapped to two dierent points. Theorem 2.1. Let X be ane, G geometrically reductive and acting on X. Then O(X)G ⊆ O(X) is a nitely generated K-algebra and the canonical map p : X ! Y := Spec(O(X)G) is a good categorical quotient. Proof. Let R := O(X). Then RG is nitely generated because of Nagata's theorem [Dol03, p. 41-45]. i) We show it for basic open sets D(f) ⊆ Y for f 2 RG. g 2 (RG) , g 2 RG f n f g , G-invariant f n g , 2 (R )G f n f G G ) (R )f = (Rf ) ii) Let W ⊆ X be closed and G-invariant and assume that p(W ) is not closed. 9y 2 p(W ) n p(W ) with Z := p−1(y) ⊆ X closed, G-invariant, and Z \ W = ;. lemma 1.1 ) 9Φ 2 O(X)G : Φ(W ) = 1 and Φ(Z) = 0 p∗ isomorphism ) Φ = p∗(φ) for some φ 2 O(Y ) ) φ(y) = 0 and φ(p(W )) = 1, which is a contradiction. iii) Let V1;V2 2 X closed, G-invariant, and V1 \ V2 = ;. lemma 1.1 ) 9φ 2 O(Y ): φ(p(V1)) = 1 and φ(p(V2)) = 0 ) p(V1) \ p(V2) = ; 3 Remark. If G is not reductive but O(X)G is nitely generated, then theorem 2.1 still gives a categorical quotient. Example. Let the additive group act on 2 by C C g:(c1; c2) = (c1; c2 + gc1) with and 2. This corresponds to an action of on g 2 C (c1; c2) 2 C C C[x1; x2] by . Thus the invariant functions are C g:p(x1; x2) = p(x1; x2 +gx1) C[x1; x2] = and therefore the projection onto the rst coordinate is a categorical C[x1] quotient, but not a good one, because (0; 0) and (0; 1) are two closed C- invariant points but are projected onto the same point. We are now able to construct good categorical quotients of ane varieties, so we will approach projective varieties. For projective X we will embed X,! Pn. This is done by choosing a (very ample) line bundle L over X. The idea is then to cover the image of X with open, ane, G-invariant subsets. To do so, we need an extension of the G-action on X to a G-action on Pn. 3 Linearisation and stability Denition 3.1. Let L be a line bundle on X with projection π : L ! X.A G-linearisation of L is an extension of the action σ on X to an action σ on L, such that σ G × L / L i) id×π π commutes. σ G × X / X ii) The zero-section is G-invariant. Remark. Every line bundle L on X is associated to an invertible sheaf F on X (see [Har77, exercise 5.18]). So in terms of invertible sheafs the G- linearisation σ on L is the same as an isomorphism ∗ ∗ Φ: σ F!~ pr2F satisfying the co-cycle condition (see [MFK65, denition 1.6]). Example 3.1. Let L := X×C be the trivial line bundle. Then a linearisation can be given by choosing a character χ 2 Hom(G; C∗). An action σ is then given by σ(g; (x; z)) := (σ(g; x); χ(g)z) for g 2 G and (x; z) 2 X × C. Therefore a G-linearisation is not unique. 4 Remark. Let V := Γ(X; L) be the global sections of L. Then the linearisation induces an action on V: (g:s)(x) = σ(g; s(σ(g−1; x))) = g:(s(g−1:x)) (1) for s 2 V; g 2 G, and x 2 X. The next problem which occurs is that a cover of open, ane, and G-invariant subsets of X,! Pn where we can nd good categorical quotients need not exist, or we do not get a good categorical quotient by gluing them together. Example 3.2. Let C∗ act on P1 by with ∗ and 1 t:(z1 : z2) = (z1 : tz2) t 2 C (z1 : z2) 2 P : This action has three orbits f(1 : t); (1 : 0); (0 : 1)g. To be a good categorical quotient a morphism from P1 to some Y has to take the orbit (1 : t) to one point. As the other two points are in the closure of this orbit, they have to go to the same point. Therefore the only possible quotient is p : P1 7! pt, which does not satisfy property iii) of Def 2.2.