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 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 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. s → By definition,X is the open subset ofX c consisting of points with zero dimensional stabilisers s s s and we letY :=ϕ(X ) Y c. It remains to prove thatY is open. Asϕ:X c Y c is a geometric s ⊂ 1 s s s→ s quotient andX is aG-invariant subset ofX,ϕ − (Y ) =X and alsoY c Y =ϕ(X c X ). As s − s −s Xc X is closed inX c, property iv) of good quotient gives thatϕ(X c X ) =Y c Y is closed − s − s − inY c and soY is open inY c. SinceY c is open inY , we can conclude thatY Y is open. Finally, the geometric quotientϕ:X Y restricts to a geometric quotientϕ:X⊂ s Y s by c → c → Corollary 3.33. � Remark 5.7. We see from the proof of this theorem that to get a geometric quotient we do not have to impose the condition dimG x = 0 and in fact in Mumford’s original definition of stability this condition was omitted. However, the modern definition of stability, which asks for zero dimensional stabilisers, is now widely accepted. One advantage of the modern definition is that if the semistable set is nonempty, then the dimension of the geometric quotient equals its expected dimension. A second advantage of the modern definition of stability is that it is better suited to moduli problems. n Example 5.8. Consider the linear action ofG=G m onX=P by 1 t [x :x : :x ] = [t− x : tx : : tx ]. · 0 1 ··· n 0 1 ··· n In this caseR(X) =k[x 0, . . . , xn] which is graded into homogeneous pieces by degree. It is easy to see that the functionsx 0x1, . . . , x0xn are allG-invariant. In fact, we claim that these functions generate the ring of invariants. To prove the claim, suppose we havef R(X); then ∈ m0 m1 mn f= a(m)x0 x1 . . . xn m=(m�0,...,mn) and, fort G m, ∈ m0 m1 mn m0 m1 mn t f= a(m)t − −···− x x . . . x . · 0 1 n m=(m�0,...,mn) n Thenf isG-invariant if and only ifa(m ) = 0 for allm = (m0, . . . , mn) such thatm 0 = i=1 mi. n � Ifm satisfiesm 0 = mi, then i=1 � m0 m1 mn m1 mn � x0 x1 . . . xn = (x0x1) ...(x 0xn) ; that is, iff isG-invariant, thenf k[x x , . . . x x ]. Hence ∈ 0 1 0 n G R(X) =k[x 0x1, . . . , x0xn] = k[y0, . . . , yn 1] ∼ − n 1 and after taking the projective spectrum we obtain the X//G=P − . The explicit choice of generators forR(X) G allows us to write down the rational morphism n n 1 ϕ:X=P ��� X//G=P − [x :x : :x ] [x x : :x x ] 0 1 ··· n �→ 0 1 ··· 0 n MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 41

and its clear from this description that the nullcone n N= [x 0 : :x n] P :x 0 = 0 or (x1, ,x n) = 0 { ··· ∈ ··· } is the projective variety defined by the homogeneous idealI=(x x , ,x x ). In particular, 0 1 ··· 0 n n ss n n X = Xx0xi = [x 0 : :x n] P :x 0 = 0 and (x1, . . . , xn)= 0 = A 0 { ··· ∈ � � } ∼ −{ } i=1 � n n where we are identifying the affine chart on whichx 0 = 0 inP withA . Therefore � ss n n 1 ϕ:X =A 0 X//G=P − −{ } ��� is a good quotient of the action onX ss. As the preimage of each point in X//G is a single orbit, this is also a geometric quotient. Moreover, every semistable point is stable as all orbits are closed inA n 0 and have zero dimensional stabilisers. −{ } In general it can be difficult to determine which points are semistable and stable as it is necessary to have a description of the gradedk-algebra of invariant functions. The ideal situation is as above where we have an explicit set of generators for the invariant algebra which allows us to write down the quotient map. However,finding generators and relations for the invariant algebra in general can be hard. We will soon see that there are other criteria that we can use to determine (semi)stability of points. Lemma 5.9. LetG be a reductive group acting linearly onX P n.Ak-pointx X(k) is ⊂ ss ∈ stable if and only ifx is semistable and its orbitG x is closed inX and its stabiliserG x is zero dimensional. · ss 1 Proof. Supposex is stable andx � G x X ; thenϕ(x �) =ϕ(x) and sox � ϕ − (ϕ(x)) 1 s s s ∈ · ∩ ∈ ⊂ ϕ− (Y ) =X . AsG acts onX with zero-dimensional stabiliser, this action must be closed as the boundary of an orbit is a union of orbits of strictly lower dimension. Therefore,x � G x and so the orbitG x is closed inX ss. ∈ · Conversely, we· supposex is semistable with closed orbit inX ss and zero dimensional sta- G biliser. Asx is semistable, there is a homogeneousf R(X) + such thatx X f . AsG x ss ∈ ss ∈ · is closed inX , it is also closed in the open affine setX f X . By Proposition 3.21, the G-invariant set ⊂ Z := z X : dimG >0 { ∈ f z } is closed inX . SinceZ is disjoint fromG x and both sets are closed in the affine schemeX , f · f by Lemma 4.29, there existsh (X )G such that ∈O f h(Z) = 0 andh(G x) = 1. · We claim that from the functionh, we can produce aG-invariant homogeneous polynomial h R(X) G such thatx X andX is disjoint fromZ, as then all orbits inX have � ∈ + ∈ fh� fh� fh� zero dimensional stabilisers and so must be closed inX fh� (as the closure of an orbit is a union of lower dimensional orbits), in which case we can conclude thatx is stable. The proof of the above claim follows from Lemma 5.10 below and uses the fact thatG is geometrically reductive. More precisely, we have that (X ) = ( X˜ ) is a quotient ofA := (k[x , . . . , x ] ) and we O f O f 0 0 n f 0 takeI to be the kernel. Thenh r =h /f s A G/(I A G) for some homogeneous polynomialh � ∈ ∩ � and positive integersr ands. � Lemma 5.10. LetG be a geometrically reductive group acting rationally on afinitely generated k-algebraA. For aG-invariant idealI ofA anda (A/I) G, there is a positive integerr such thata r A G/(I A G). ∈ ∈ ∩ Proof. Letb A be an element whose image in A/I isa and we can assumea= 0. As the action is rational,∈ b is contained in afinite dimensionalG-invariant linear subspace� V A spanned by the translatesg b. Thenb/ V I asa= 0; however,g b b V I for allg G ⊂ asa isG-invariant. Therefore· dimV = dim(∈ ∩V I) +� 1 and every element· − ∈ ∩ inV can be∈ uniquely written asλb+b forλ k andb V I. Consider∩ the linear projectionl:V k onto the line � ∈ � ∈ ∩ → spanned byb, which isG-equivariant. In terms of the dual representationV ∨, the projectionl 42 VICTORIA HOSKINS corresponds to a non-zerofixed pointl ∗ and so, asG is geometrically reductive, there exists a G-invariant homogeneous functionF (V ) of positive degreer which is not vanishing atl . ∈O ∨ ∗ We can take a basis ofV (and dual basis ofV ∨) where thefirst basis vector corresponds tob. r Then the coefficientλ ofx 1 inF is non-zero. Consider the algebra homomorphism (V ∨) = Sym∗ V A O → G G r and letb 0 A be the image ofF (V ∨) . Thenb 0 λb I, as this belongs to the ideal generated by∈ a choice of basis vectors∈O forV I. Hencea −r A G∈/(I A G) as required. ∩ ∈ ∩ � Remark 5.11. IfG is linearly reductive, then takingG-invariants is exact, and so we can take r = 1 in the above lemma. 5.2. A description of thek-points of the GIT quotient. Definition 5.12. LetG be a reductive group acting linearly onX P n.Ak-pointx X(k) is said to be polystable if it is semistable and its orbit is closed in⊂X ss. We say two semistable∈ k-points are S-equivalent if their orbit closures meet inX ss. We write this equivalence relation ss ss onX (k) as S-equiv. and letX (k)/ S-equiv. denote theS-equivalence classes of semistable k-points. ∼ ∼ By Lemma 5.9 above, every stablek-point is polystable. Lemma 5.13. LetG be a reductive group acting linearly onX P n and letx X(k) be a ⊂ ∈ semistablek-point; then its orbit closure G x contains a unique polystable orbit. Moreover, if x is semistable but not stable, then this unique· polystable orbit is also not stable. Proof. Thefirst statement follows from Corollary 3.32:ϕ is constant on orbit closures and the preimage of ak-point underϕ contains a orbit which is closed inX ss; this is the polystable orbit. For the second statement we note that if a semistable orbitG x is not closed, then the unique closed orbit in G x has dimension strictly less thanG x by Proposition· 3.15 and so · · cannot be stable. � Corollary 5.14. LetG be a reductive group acting linearly onX P n. For two semistable ss ⊂ points x, x� X , we haveϕ(x) =ϕ(x �) if and only ifx andx � are S-equivalent. Moreover, there is a bijection∈ of sets X//G(k) = Xps(k)/G(k) = Xss(k)/ ∼ ∼ ∼ S-equiv. whereX ps(k) is the set of polystablek-points.