38 VICTORIA HOSKINS 5. Projective GIT quotients In this section we extend the theory of affine GIT developed in the previous section to construct GIT quotients for reductive group actions on projective schemes. The idea is that we would like construct our GIT quotient by gluing affine GIT quotients. In order to do this we would like to cover our schemeX by affine open subsets which are invariant under the group action and glue the affine GIT quotients of these affine open subsets ofX. However, it may not be possible to cover all ofX by such compatible open invariant affine subsets. For a projective schemeX with an action of a reductive groupG, there is not a canonical way to produce an open subset ofX which is covered by open invariant affine subsets. Instead, this will depend on a choice of an equivariant projective embeddingX� P n, whereG acts onP n by → a linear representationG GL n+1. We recall that a projective embedding ofX corresponds to a choice of a (very) ample→ line bundle onX. We will shortly see that equivariant projective embeddings are given by an ample linearisationL of theG-action onX, which is a lift of the G-action to a ample line bundle onX such that the projection toX is equivariant and the action on thefibres is linear. In this section, we will show for a reductive groupG acting on a projective schemeX and a choice of ample linearisation of the action, there is a good quotient of an open subset of semistable points inX. Furthermore, this quotient is itself projective and restricts to a geometric quotient on an open subset of stable points. The main reference for the construction of the projective GIT quotient is Mumford’s book [25] and other excellent references are [4, 24, 31, 32, 42]. 5.1. Construction of the projective GIT quotient. Definition 5.1. LetX be a projective scheme with an action of an affine algebraic groupG. A linearG-equivariant projective embedding ofX is a group homomorphismG GL and → n+1 aG-equivariant projective embeddingX� P n. We will often simply say that theG-action on → X� P n is linear to mean that we have a linearG-equivariant projective embedding ofX as above.→ Suppose we have a linear action of a reductive groupG on a projective schemeX P n. ⊂ Then the action ofG onP n lifts to an action ofG on the affine coneA n+1 overP n. Since the projective embeddingX P n isG-equivariant, there is an induced action ofG on the affine ⊂ cone X˜ A n+1 overX P n. More precisely, we have ⊂ ⊂ n+1 0 n (A ) =k[x 0, . , xn] = k[x0, . , xn]r = H (P , X (r)) O O r 0 r 0 �≥ �≥ n and ifX P is the closed subscheme associated to a homogeneous idealI(X) k[x 0, . , xn], ⊂ ⊂ then X˜ = SpecR(X) whereR(X)=k[x 0, . , xn]/I(X). Thek-algebras (A n+1) andR(X) are graded by homogeneous degree and, as theG-action O onA n+1 is linear it preserves the graded pieces, so that the invariant subalgebra n+1 G G (A ) = k[x0, . , xn] O r r 0 �≥ G G G is a graded algebra and, similarly,R(X) = r 0R(X)r . By Nagata’s theorem,R(X) is finitely generated, asG is reductive. The inclusion⊕ ≥ offinitely generated gradedk-algebras R(X)G � R(X) determines a rational morphism of projective schemes → G X ��� ProjR(X) whose indeterminacy locus is the closed subscheme ofX defined by the homogeneous ideal R(X)G := R(X)G. + ⊕ r>0 r Definition 5.2. For a linear action of a reductive groupG on a projective schemeX P n, we define the nullconeN to be the closed subscheme ofX defined by the homogeneous⊂ ideal G ˜ R(X)+ inR(X) (strictly speaking the nullcone is really the affine cone N overN). We define the semistable setX ss =X N to be the open subset ofX given by the complement to the nullcone. More precisely,x −X is semistable if there exists aG-invariant homogeneous function ∈ MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 39 G f R(X) r forr> 0 such thatf(x)= 0. By construction, the semistable set is the open subset which∈ is the domain of definition of� the rational map G X ��� ProjR(X) . We call the morphismsX ss X//G := ProjR(X) G the GIT quotient of this action. → Theorem 5.3. For a linear action of a reductive groupG on a projective schemeX P n, the GIT quotientϕ:X ss X//G is a good quotient of theG-action on the open subset⊂ X ss of semistable points inX.→ Furthermore, X//G is a projective scheme. Proof. We letϕ:X ss Y := X//G denote the projective GIT quotient. By construction X//G is the projective spectrum→ of thefinitely generated gradedk-algebraR(X) G. We claim G G G that ProjR(X) is projective over SpecR(X) 0 = Speck. IfR(X) isfinitely generated by G R(X)1 as ak-algebra, this result follows immediately from [14] II Corollary 5.16. If not, then as G R(X) is afinitely generatedk-algebra, we can pick generatorsf 1, . , fr in degreesd 1, . , dr. Letd :=d 1 ... d r; then · · G (d) G (R(X) ) = R(X)dl l 0 �≥ G (d) G (d) isfinitely generated by (R(X) )1 ask-algebra and so Proj (R(X) ) is projective over Speck. Since X//G := ProjR(X) G = Proj (R(X)G)(d) (see [14] II Exercise 5.13), we can ∼ � � conclude that X//G is projective. G � � Forf R +, the open affine subsetsY f Y form a basis of the open sets onY . Since ∈ G ⊂ f R(X) + R(X) +, we can also consider the open affine subsetX f X and, by construction ∈ ⊂ 1 ⊂ ofϕ, we have thatϕ − (Yf ) =X f . Let X˜f (respectively Y˜f ) denote the affine cone overX f (respectivelyY f ). Then (Y ) = (Y˜ ) = ((R(X)G) ) = ((R(X) ) )G = ( (X˜ ) )G = (X )G O f ∼ O f 0 ∼ f 0 ∼ f 0 ∼ O f 0 ∼ O f G and so the corresponding morphism of affine schemesϕ f :X f Y f ∼= Spec (X f ) is an affine GIT quotient, and so also a good quotient by Theorem 4.30.→ The morphismO ϕ:X ss Y is → ss obtained by gluing the good quotientsϕ f :X f Y f . SinceY f coverY (andX f coverX ) and being a good quotient is local on the target Remark→ 3.34, we can conclude thatϕ is also a good quotient. � ss ss We recall that asϕ:X X//G is a good quotient, for two semistable pointsx 1, x2 inX , we have → G x G x X ss = ϕ(x ) =ϕ(x ). · 1 ∩ · 2 ∩ � ∅ ⇐⇒ 1 2 Furthermore, the preimage of each point in X//G contains a unique closed orbit. The presence of non-closed orbits in the semistable locus will prevent the good quotientϕ:X ss X//G from being a geometric quotient. → We can now ask if there is an open subsetX s ofX ss on which this quotient becomes a geometric quotient. For this we want the action to be closed onX s. This motivates the definition of stability (see also Definition 4.35). Definition 5.4. Consider a linear action of a reductive groupG on a closed subschemeX P n. Then a pointx X is ⊂ ∈ (1) stable if dimG = 0 and there is aG-invariant homogeneous polynomialf R(X) G x ∈ + such thatx X f and the action ofG onX f is closed. (2) unstable if∈ it is not semistable. We denote the set of stable points byX s and the set of unstable points byX us :=X X ss =N. − We emphasise that, somewhat confusingly, unstable does not mean not stable, but this ter- minology has long been accepted by the mathematical community. Lemma 5.5. The stable and semistable setsX s andX ss are open inX. 40 VICTORIA HOSKINS Proof. By construction, the semistable set is open inX as it is the complement to the nullcone N, which is closed. To prove that the stable set is open, we consider the subsetX c := X f G ∪ where the union is taken overf R(X) + such that the action ofG onX f is closed; then ∈ s Xc is open inX and it remains to showX is open inX c. By Proposition 3.21, the function x dimG x is an upper semi-continuous function onX and so the set of points with zero dimensional �→ stabiliser is open. Hence, we have open inclusionsX s X X. ⊂ c ⊂ � Theorem 5.6. For a linear action of a reductive groupG on a closed subschemeX P n, let ϕ:X ss Y := X//G denote the GIT quotient. Then there is an open subschemeY ⊂s Y such thatϕ 1→(Y s) =X s and that the GIT quotient restricts to a geometric quotientϕ:X s⊂ Y s. − → Proof. LetY be the union ofY forf R(X) G such that theG-action onX is closed and c f ∈ + f letX be the union ofX over the same index set so thatX =ϕ 1(Y ). Thenϕ:X Y c f c − c c → c is constructed by gluingϕ :X Y forf R(X) G such that theG-action onX is closed. f f → f ∈ + f Eachϕ f is a good quotient and as the action onX f is closed,ϕ f is also a geometric quotient by Corollary 3.32. Henceϕ:X c Y c is a geometric quotient by Remark 3.34.