Moduli Theory and Geometric Invariant Theory

Transcription

1 Moduli Theory and Geometric Invariant Theory November 29, 2017

2 Chapter 1 Moduli theory 1.1 Category theory A category consists of a collection of objects, and for each pair of objects, a set of morphisms between them. The collection of objects of a category C is often denoted ob(c) and the collection of morphisms between A, B C is denoted Hom C (A, B). Let C and D be two categories. A morphism of categories C and D is given by a (covariant) functor F : C D which associates to every object C ob(c) an object F (C) ob(d) and to each morphism f Hom C (C, C ) a morphism F (f) : F (C) F (C ) in Hom D (F (C), F (C )) such that F preserves identity morphisms and composition. A contravariant functor F : C D reverses arrows of the morphism. The notion of a morphism of (covariant) functors F, G : C D is given by a natural transformation η : F G which associates to every object C ob(c) a morphism η C morphisms f : C C, i.e. we have a commutative square F (C) F (C ) G(C) G(C ) The contravariant functors from the category to sets are called presheaves. : F (C) G(C) which is compatible with (1.1.1) Example Let C be a category and (ets) be the category of sets. For any A ob(c), we have a contravariant functor h A : C (ets) defined by h A (B) = Hom(B, A). This functor is called functor of maps to A, and it is also known as the functor of points. e.g. Let ch be the category of schemes. Then the functor of points is a sheaf. 1

3 Theorem 1. (Yoneda s lemma) There is a bijection Ψ : {natural transforms η : h C F } F (C) for any presheaf F on C. In particular, if h C and h C are the identified functors, then C and C are canonically isomorphic. Proof. The map is given by Ψ(η) = η C (id C ). Conversely, if s F (C), we can define a natural transformation η as follows. For A C, let η A : h C (A) F (A) be the morphism η A (f) = F (f)(s), where F (f) denotes the morphism F (C) F (A). Definition A prefsheaf F is called representable if there exists an object C ob(c) and natural isomorphism F = h C. Example The global section functor Γ : ch (ets) X Γ(X, O X ) is representable by A 1. The natural transform between Γ and h A 1 is as below: the morphism ϕ : X A 1 corresponds to maps ϕ : k[t] Γ(X, O X ) and ϕ (t) is the function on X. imilarly, one can show that the functor j Γ(, O a j ) is representable by j Aa j = A a j. Is every presheaf F of ch representable by a scheme? We will show that this has a negative answer. 1.2 Moduli problem A moduli problem is essentially a classification problem: we have a collection of objects and we want to classify these objects up to equivalence. Definition A (naive) moduli problem (in algebraic geometry) is a collection M of objects (in algebraic geometry) and an equivalence relation on M. Example (Grassmannian) Let M be the set of k-dimensional linear subspaces of an n- dimensional vector space and be equality. 2. Let M be the set of n ordered distinct points on P 1 and be the equivalence relation given by the natural action of the automorphism group P GL 2 of P 1. 2

4 3. Let M be the set of hypersurfaces of degree d in P n and can be chosen to be either equality or the relation given by projective change of coordinates (i.e. corresponding to the natural P GL n+1 -action). 4. (Moduli of vector bundles) Let M be the collection of vector bundles on a fixed scheme X and be the relation given by isomorphisms of vector bundles. Let us work out some examples. Example Let M consist of 4 ordered distinct points (p 1, p 2, p 3, p 4 ) on P 1. We want to classify these quartuples up to the automorphisms of P 1. We recall that the automorphism group of P 1 is the projective linear group P GL 2, which acts as Mobius transformations. We define our equivalence relation by (p 1, p 2, p 3, p 4 ) (q 1, q 2, q 3, q 4 ) if there exists an automorphisms f : P 1 P 1 such that f(p i ) = q i for i = 1,..., 4. Note that for any 3 distinct points (p 1, p 2, p 3 ) on P 1, there exists a unique Möbius transformation f P GL2 which sends (p 1, p 2, p 3 ) to (0, 1, ) and the cross-ratio of 4 distinct points (p 1, p 2, p 3, p 4 ) on P 1 is given by f(p 4 ) P 1 {0, 1, }, where f is the unique Möbius transformation that sends (p 1, p 2, p 3 ) to (0, 1, ). Therefore, we see that the set M/ is in bijection with the set of k-points in the quasi-projective variety P 1 {0, 1, }. We want more than this, we want a moduli space which encodes how these objects vary continuously in families; this information is encoded in a moduli functor. It is of the form {Families over } Definition Let (M, ) be a naive moduli problem. Then an extended moduli problem is given by 1. sets M() of families over and an equivalence relation on M(), for all schemes, 2. pullback maps f : M() M(T ), for every morphism of schemes T, satisfying the following properties: (M(pt), pt ) = (M, ); for the identity Id : and any family U over, we have Id U = U; for a morphism f : T and equivalent families U U over, we have f U T f U ; for morphisms f : T and g : R, and a family W over R, we have an equivalence (g f) W T f g W. 1.3 Fine moduli space Definition Let M : C (ets) be a moduli functor; then M ob(c) is a fine moduli space for M if it represents M. Definition Let M be a fine moduli space for M. Then the family U M(M) corresponding to the identity morphism on M is called the universal family. 3

5 Any family is equivalent to a family obtained by pulling back the universal family. Remark 1. If a fine moduli space for M exists, it is unique up to unique isomorphism: that is, if (M, η) and (M, η ) are two fine moduli spaces, then they are related by unique isomorphisms η M ((η M ) 1 (Id M )) : M M, and η M ((η M ) 1 (Id M )) : M M. Definition Let F and G be functors from ch to (ets). We say G is a subfunctor of F if G() F () for all schemes and G( ) G() is the restriction of F ( ) F () for any. We say G is an open subfunctor if ch and ξ F (), there exists an open subscheme Uξ G for any f Hom(, ), we have f ξ G( ) if and only if f factors through Uξ G. such that Definition Let {G i } be a collection of open subfunctors of F. We say that {G i } form an open covering of F if for all schemes and ξ F (T ), the set {U G i ξ } form an open cover of. Proposition 1. A functor F : ch (ets) is said to be a Zariski sheaf if it is compatible with gluing, i.e. the sequence F () F (U α ) F (Uα U β ) is exact for any open cover {U α } of. If F is a Zariski sheaf which admits an open covering by representable subfunctors G i then F is representable. Proof. By assumption, each G i = hxi is represented by a scheme X i. The functor G i F G j is an open subfunctor of G i and G j, which allows us to glue X i together to a scheme X. Now, as h X and F are Zariski sheaves, and they coincide on an open covering. By sheaf axiom, they must be isomorphic, and F = h X Grassmannian Consider the moduli problem G(r, V ) of r-dimensional linear subspaces in a fixed n-dimensional vector space V, where a family over is a rank r vector subbundle E of V and the equivalence relation is equality. In other words, G(r, V ) : ch (ets) { } subvector bundles of G(r, V )() = rank r of V Note that the trivial vector bundle corresponds to the free sheaf O n. Then subbundle E corresponds to a surjection O n Q, where Q is a finite locally free sheaf of rank n r. Thus we can regard G(r, V ) as the functor G(r, n)() := {O n Q} Theorem 2. The functor G(r, V ) is represented by a smooth projective scheme Grass(r, V ). 4

6 Proof. For each subset I = {j 1,..., j n r } {1, 2,..., n} of cardinality n r, we let G I be the subfunctor of G(r, n) corresponding to the surjection Q such that the induced map O I = j i each scheme the subset where s I : O n r O n G I () = {q : O n O n O e ji = O n r Q is an isomorphism. More precisely, G I associates Q q s I is surjective} G(r, n)() is the direct sum of the natural coprojection (j i -th position) O O n. However, the functor G I is isomorphic to the functor {q : O n O I } by composing s I. To determine q, we have to fix the images of q(e i ) in Γ(, O n r ). o G I is isomorphic to the functor Γ(, O n r ). j This functor is representable by A r(n r). As G(r, V ) is a Zariski sheaf and G I forms an open covering as I varies, then G(r, V ) is represented by a smooth scheme Grass(r, V ). To prove the projectivity of Grass(r, V ), let W Grass(r, V ) be the universal family, then r W is an line bundle whose sections defines a morphism ϕ r : Grass(r, V ) P( r V ) given by W [ r W ]. We claim that ϕ r is a closed embedding. For each I, we define P I P( r V ) to be the completement of the hyperplane [ r W I ] = 0. Then P I forms an open cover of P( r V ) and the map n G I P W is exactly the closed embedding A r(n r) r A Remark 2. The embedding ϕ r is called the Plücker embedding Hilbert scheme Definition Let F be a coherent sheaf on a projective scheme X P n. The Hilbert polynomial of F is defined to be the polynomial P F (d) = χ(f(d)) = ( 1) i h i (X, F(d)), where F(d) = F O X (d). We say P OX (z) to be the Hilbert polynomial of X. ome Remarks on Hilbert polynomial 1. For any homogenous ideal I k[x 0,..., x n ], the Hilbert function is given by h I (m) = dim(k[x 0,..., x n ]/I) m We can see that P OX (z) = H I (z), if X is the subscheme corresponding to I. More generally, we define the Hilbert function of F: h F (m) := h 0 (X, F(m)). 5

7 For m 0, one has H i (F O P n(1) m ) = 0 for i > 0 by erre vanishing theorem. Hence h F (m) = χ(f(m)) for m To see the function P F (z) is polynomial, one can proceed by induction on the dimension of n. Indeed, one has the twisted exact sequence 0 F(d 1) F(d) i G(d) 0 for some coherent sheaf G on a hyperplane section i : H = P n 1 P n. Then χ(f(d) χ(f(d 1))) = χ(g(d)). If χ(g(d)) is a polynomial, so is χ(f(d)). ( ) n + d Example The Hilbert polynomial of P n is χ(o(d)) =. n The Hilbert polynomial p(z) of a degree d hypersurface H = {f(x) = 0} in P n is ( ) ( ) n + z n + z d p H (z) = p P n(z) p P n(z d) = n n because of the exact sequence 0 O P n( d) f O P n i O H 0 Two well-known results are as below: Theorem 3. Let π : X be a projective morphism over a scheme. If F is a coherent sheaf on X and flat over, then the Euler characteristic of F is constant in fibers. In particular, the Hilbert polynomial of F is a locally constant as a function. Theorem 4. Let X be a projective scheme over. Let F be a coherent sheaf on X. Then for every polynomial p, there is a unique closed subscheme p with the following property: for any projective scheme T g A, the pullback F on T X is flat over T with Hilbert polynomial p, then g factors through p. An alternative proof of Theorem 2 (following Kollár) tep 1. Regard V as a vector bundle over a point. et E = r V, there is wedge product map φ : V E r+1 V. (1.3.1) et K = ker(φ) V E. Let G r E be the subscheme defined by all r r-subdetermininants of φ. Note that Lemma 1. Let V be a vector bundle on and E a rank r subbundle. Then E and r E are each others annihilators under the wedge product V r E r+1 E. By the Lemma 1, we have the corank of φ is at most r. Thus φ Gr has constant rank r, and its kernel K is free of rank r. Regard K as a sheaf on P(E). By Theorem 4, there is a locally closed subscheme H P(E) such that if f : Z P(E) is a morphism, then f K is free of rank r if and only if f factors as f : Z Grass(r, V ) P(E). 6

8 tep 2: Representability of G(r, V ). Let E V be a subbundle of rank r. We obtain a subline bundle r E r V which corresponds to a morphism ϕ : P(E) The pullback gives a map V r E r+1 V whose kernel is ϕ K. Thus ϕ factors through Grass(r, V ) P(E) by definition. On the other hand, if we have a morphism ρ : Gr(r, V ), this gives a subbundle ρ K V of rank r. Then r is the unique line bundle annihilating ρ K. This means ρ = K. Definition Given p(z) Q[z], we define the Hilbert functor as follows. For ob(ch k ), it associates a set Our goal is Hilb p (P n ) : ch k (ets) Hilb p (P n )() := {X P n flat and proper over, with Hilbert polynomial p }. Theorem 5. The Hilbert functor Hilb P (P n ) is representable by a projective scheme Hilb p (P n ), called the Hilbert scheme of P n with respect to p. Proof. Let X be a closed subscheme of P n with ideal sheaf I X. Choose N such that I Y (N) is generated by global sections. Then H 0 (P n, I X (N)) H 0 (P n, O P n(n)) determines X because I X (N) = Im(O P n H 0 (P n, I X (N)) O P n(n)) Let q(z) be the Hilbert polynomial of I X. Then we get an injective map { ubschemes of P n with Hilbert polynomial p} Grass(q(N), H 0 (P n, O P n(n))). (1.3.2) for N 0 by sending X to the vector space H 0 (P n, I X (N)). Let us make it precise. tep 1. For any scheme, we let X P n π be a closed subscheme, flat over with Hilbert polynomial p(z). Then the fiber of X is a closed subscheme of P n with Hilbert polynomial p(z). As we have an exact sequence 0 I X O P n i O X 0 After twisting O P n (N) on the exact sequence, we can get an inclusion π I X (N) π O P n (N) 7

9 as a subbundle of rank q(n). Exercise: show that π I X (N) is locally free for N 0. In this way, it defines a natural transform between functors tep 2. et and Hilb p (P n ) Gr(q(N), ). π 1 : Grass(q(N), V N ) P n Grass(q(N), V N ) π 2 : Grass(q(N), V N ) P n P n to be the natural projections. If E is the universal subbundle on Grass(q(N), V N ), we set F to be the cokernel of π 1E π 2O P n(n). Applying Theorem 4 to the sheaf F( N) with respect to π 1, there is a largest subscheme i : G p Grass(q(N), V N ) such that F is flat over G p with Hilbert polynomial p. As i F( N) is a quotient sheaf of O Gp P n, it defines a subscheme U G p P n, which is flat over G p with Hilbert polynomial p(z). tep 3. G p represents the functor Hil p (P n ). If X P n is a closed subscheme, flat over with Hilbert polynomial P, then we have a rank q(n) subbundle This corresponds to a morphism such that π I X (N) π O P n (N) = V N ρ : Grass(q(N), V N ) ρ (U V N Grass(q(N), V N )) = π I X (N) π O P n (N) o π 1 ρ (F) = O X (N) and ρ F( N) is flat over with Hilbert polynomial p(z). Then ρ factors through G p. 1.4 Coarse moduli space In general, the moduli functors are not representable due to the existence of non-trivial automorphisms. Definition A coarse moduli space for a moduli functor M is a scheme M and a natural transformation of functors η : M M (1.4.1) such that pec k : M(pec k) h M (pec k) is bijective. For any scheme N and natural transformation ν : M h N, there exists a unique morphism of schemes f : M N such that ν = h f η, where h f : h M h N is the corresponding natural transformation of presheaves. 8

10 Lemma 2. The coarse moduli space for M is unique up to a unique isomorphism. Proof. If (M, η ) is another coarse moduli space for M, then there exists unique morphisms f : M M, f : M M such that h f, h f fit into a commutative diagram h M M h M (1.4.2) h f h M η η As η = h f h f η and η = h id η, then f f = id M and f f = id M by Yoneda Lemma. h M h f Proposition 2. Let M be a moduli problem and suppose there exists a family F over A 1 such that F s F 1 for all s 0 and F 0 is not equivalent to F 1. Then for any scheme M and natural transformation η : M h M, we have η A 1(F) : A 1 M is constant. In particular, there is no coarse moduli space for this moduli problem. Proof. uppose we have a natural transformation η : M h M, then η sends the family F over A 1 to a morphism f : A 1 M. For any s : pec k A 1, we have f s = η(f s ). By assumption, the restriction f A 1 {0} is a constant map. Then f is a constant map as well. However, the map η pec k : M(pec k) h M (pec k) = M is not a bijection as F 0 F 1 in M(pec k), but these non-equivalent objects correspond to the same point in M. Example Consider the moduli functor M n () = {(E vector bundle of rank n; ϕ : E E an endomorphism)}/ Here (E, ϕ) (E, ϕ ) if there exists an isomorphism f : E E such that f ϕ = ϕ f. One can regard M n as the moduli problem of classifying endomorphisms of a n-dimensional vector space. Then M n does not admit a coarse moduli space by Proposition 2. For instance, when n = 2, consider the vector bundle O 2 over A 1 and ϕ is given by the matrix A 1 ( ) 1 s ϕ = 0 1 Then ϕ t ϕ s if s, t 0 and ϕ 0 ϕ 1. Remark 3. When does M admits a coarse moduli space? A very important result is given by Keel-Mori. (Deligne-Mumford stack) Theorem 6 (Keel-Mori). If M is a separated Deligne-Mumford stack, then M is coarsely representable by a scheme M. Remark 4. If we consider the moduli problem M g of curves of genus g, then it is not a fine moduli problem and it only has coarse moduli space. 9

11 Chapter 2 Geometric Invariant Theory The construction of many moduli spaces could follows from Mumford s Geometric Invariant Theory. 2.1 Affine algebraic groups and algebraic representation Let k be a field. Definition An algebraic group over k is a scheme G over k with morphisms e : pec k G (identity element); m : G G G(group law) and i : G G (group inversion) such that we have commutative diagrams (associativity) G G G m id G G ; (identity) pec k G G G G pec k id m G G m G m G m and (inverse) G (i,id) G G (id,i) G pec k m e e G pec k We say G is an affine algebraic group if the underlying scheme is affine. A homomorphism of algebraic groups is a morphism of schemes, which is also compatible with the group operations. If G is affine, then the multiplication m : G G G corresponds to a k-algebra homomorphism m : O(G) O(G) O(G), called the comultiplication, and similarly, the coinversion i : O(G) O(G) and coidentity e : O(G) k. 10

12 Remark 5. The comultiplication, coinversion and co identity form a finitely generated Hopf algebra. There is a bijection between the set of affine algebraic groups and finitely generated Hopf algebra. Example The additive group G a = pec k[t] is an affine algebraic group, whose group structure is the addition defined by m (t) = t t, i (t) = t. 2. The multiplicative group G m = pec k[t, t 1 ] is an affine algebraic group. whose group action is given by the comultiplication m (t) = t t, i (t) = t The general linear group GL n is an open subvariety of A n2, which is the completement of closed subset defined by the determinant equation det(x ij ) = 0 The comultiplication is defined by m (x ij ) = t x it x tj 4. A linear algebraic group G is a subgroup of GL n defined by polynomial equations. e.g. O n and U n are linear algebraic groups. Definition An algebraic (linear) representation of an algebraic group G on a vector space V is linear map µ V : V V O(G) satisfying 1. the composition V µ V V O(G) 1 e V is the identity. 2. the diagram V µ V V O(G) commutes. µ V µ V id V O(G) id m V O(G) O(G) Remark 6. The definition above is equivalent to the usual definition of a representation as a homomorphism of group valued functors ρ : G GL(V ). The equivalence is given by f ij = ρ x ij O(G) where x ij is the coordinates of GL n. o one can define µ V (e i ) = e j f ji. Proposition 3. Every representation of G is locally finite dimensional, i.e. each point x V is contained in a finite dimensional subrepresentation. 11

13 Proof. Write µ(x) = x i f i for some x i V and linearly independent elements f i O(G). Then x is contained in the subspace U = x 1,... x m. To see U is a subrepresentation, it suffices to show µ(x i ) U. Now as µ(x) = µ(x i )f i = x i m (f i ), and f i are linearly independent, we get µ(x i ) U O(G) for each i. Theorem 7. Any affine algebraic group G over k is a linear algebraic group. Proof. As G is an affine scheme, the ring of regular functions O(G) is a finitely generated k-algebra. The vector space W spanned by a choice of generators for O(G) as a k-algebra is finite dimensional. One can view O(G) as a representation of G via the comultiplication m : O(G) O(G) O(G). By Proposition 3, there is a finite dimensional subspace V of O(G) which is preserved by the G-action and contains W. For a basis f 1,..., f n of V, we have m (f i ) O(G) V. Write m (f i ) = n a ij f j for some functions a ij O(G). The action ρ : G GL(V ) can be described as j=1 This defines a k-algebra homomorphism ρ(g)(f i ) = a ij (g)f j. ρ : O(GL(V )) O(G), x ij a ij. To show that the corresponding morphism of affine schemes ρ : G GL(V ) is a closed embedding, we need to show ρ is surjective. Note that V is contained in the image of ρ as f i = (Id O(G) e )m (f i ) = (Id O(G) e ) i a ij f j = i e (f j )a ij. It follows that ρ is surjective and ρ is a closed immersion. Next, to show ρ : G GL(V ) is a group homomorphism, we only need to check the following diagram commutes O(GL(V )) m V O(GL(V )) O(GL(V )) This is equivalent to show that O(G) O(G) O(G) m (a ij ) = m (ρ (x i j)) = (ρ ρ )(m V (x ij ) = (ρ ρ )( x it x tj ) This is clear as a it a tj f j = m V (a ij) f j by associativity. 12

14 Representations of torus We say G is a torus if G = G n m for some n. Let us start with G = G m. The representations of G is easy to describe. Observe that for each a Z and a vector space V, one can get a representation defined by called representation of weight a. V V k[t, t 1 ], v v t a, Proposition 4. Every representation V of G m is a direct sum V = V a, where V a is a subrepresentation of weight a. Proof. Define V a := {v V µ(v) = v t a }. Then V a is a subrepresenation of V of weight a. For any v V, one can write µ(v) = v a t a. One has to get v = v a from the 1st condition in Definition Next, from the 2nd condition in Definition 2.1.3, we have µ(va )t a = v a t a t a As t a are linearly independent, then µ(v a ) = v a t a. This implies v a V a and proves the assertion. Definition Let G be an affine algebraic group. A character of G is a function χ O(G) satisfying m (χ) = χ χ and i (χ)(χ) = 1. Let V be a representation of G. Then is a subrepresentation of V of weight χ. V χ = {v V µ(x) = x χ} e.g. G = GL n, the characters of G are powers of the determinant. Proposition 5. Let G be a torus. Then every representation of G is the direct sum of its subrepresentations of weight χ V = V χ. 2.2 Group actions Definition An algebraic action of an affine algebraic group G on a scheme X is a morphism of schemes σ : G X X such that the following diagram commute pec k X e id G X ; G G X G X X G X X uppose we have an affine algebraic group G acting on two schemes X and Y. Then a morphism f : X Y is G-equivariant if the following diagram commutes G X X G Y If G acts on Y trivially, we say f is a G-invariant morphism. 13 Y

15 Example Consider the action of G m on A n by the multiplication G m A n A n (t, (a 1,..., a n )) (ta 1,..., ta n ) (2.2.1) Example Let H p (P n ) be the Hilbert scheme parametrizing closed subschemes in P n with Hilbert polynomial p(z). Then the group P GL(n) acts on H p (P n ) by changing coordinates via linear transforms. P GL(n) H p (P n ) H p (P n ) (f : P n P n, X) f(x) (2.2.2) Orbits and stabilisers Definition Let G be an affine algebraic group acting on a scheme X by σ : G X X and let x be a k-point of X. 1. The orbit G x of x is the (set-theoretic) image of the morphism σ x = σ(, x) : G(k) X(k) given by g g x. 2. The stabiliser G x of x is the fibre product of σ x : G X and x : pec k X. The stabiliser G x of x is a closed subscheme of G (as it is the preimage of a closed subscheme of X under σ x : G X). Furthermore, it is a subgroup of G. Proposition 6. Let G be an affine algebraic group acting on a scheme X. The orbits of closed points are locally closed subsets of X. The boundary of an orbit G x G x is a union of orbits of strictly smaller dimension. In particular, each orbit closure contains a closed orbit (of minimal dimension). Proof. Let x X(k). Then the orbit G x is the set-theoretic image of the morphism σ x and hence it is constructible. i.e., there exists a dense open subset U of G x with U G x G x. Because G acts transitively on G x through σ x, this implies that every point of G x is contained in a translate of U. This shows that G x is open in G x, which means that G x is locally closed. With the corresponding reduced scheme structure of G x, there is an action of G red on G x which is transitive on k-points. o it makes sense to talk about its dimension. The boundary of an orbit G x is invariant under the action of G and so is a union of G-orbits. ince G x is locally closed, the boundary G x G x, being the complement of a dense open set, is closed and of strictly lower dimension than G x. This implies that orbits of minimum dimension are closed and so each orbit closure contains a closed orbit. Definition We say the action of G on X is closed if all G-orbits are closed. Example Consider the action of G m on A n by the multiplication G m A n A n (t, (a 1,..., a n )) (ta 1,..., ta n ) (2.2.3) The orbits are punctured lines through the origin and the origin. 14

16 The action of G m on A 2 is given by t (x, y) = (tx, t 1 y). The orbits of this action are (a) conics (x, y) : xy = a for a A 1 0 (b) the punctured x-axis, (c) the punctured y-axis, (d) the origin. The origin and the conic orbits are closed whereas the punctured axes both contain the origin in their orbit closures. Proposition 7. With the assumption as above, then 1. dim G = dim G x + dim(g x) 2. the function d : X Z given by d(x) = dim G x is upper semi-continuous, i.e. the set is closed in X. {x X : dim G x n} Proof. For (1), since the dimension is a topological invariant of a scheme, we can assume G and X are reduced. The orbit G x, as a locally closed subscheme of X, is reduced by definition. This implies that the morphism σ x : G G x is flat at every generic point of G x (every k-scheme is flat over k). Hence, by the openess of the flat locus of σ, there exists a dense open set U such that σx 1 (U) U is flat. Using the transitive action of G on G x, we deduce that σ x is flat. Moreover, by definition, the fibre of σ x at x is the stabiliser G x. The assertion follows from the dimension formula for fibres of a flat morphism. For (2), we can consider the map Γ : G X (π 2,σ) X X of the action and the fiber product P ϕ X (2.2.4) G X Γ X X via the diagonal map : X X X. We define the function h : P Z by h(g, x) = dim ϕ 1 (ϕ(g, x)). This is upper semi-continuous, which means {h(g, x) n} is closed in P. As X = {(e, x)} is a closed subscheme of P, we conclude the assertion by restricting h to X. 15

17 2.3 Affine GIT quotient Categorical quotient Let G be an affine algebraic group acting on a scheme X over k. Definition A categorical quotient for the action of G on X is a G-invariant morphism ϕ : X Y which satisfying the universal property, for every G-invariant morphism X Z, it factors uniquely through ϕ. It is called an orbit space if the preimage of each point in Y is a single orbit. Lemma 3. The categorical quotient ϕ : X Y is an orbit space only if the action of G on X is closed. Proof. If ϕ is a orbit space, as ϕ has to be constant on orbit closures as it is constant on the orbit, it follows that G x = G x = ϕ 1 (x) is closed. Example Consider the action of G m on A n as scalar multiplication, as the origin is in the closure of every single orbit, any G-invariant morphism A n Y must be a constant morphism. Therefore, we claim that the categorical quotient is the structure map ϕ : A n pec k to the point pec k. This morphism is clearly G-invariant and any other G-invariant morphism f : A n Z is a constant morphism. Therefore, there is a unique morphism z : pec k Z such that f = z ϕ. Good quotient and geometric quotient Definition A morphism ϕ : X Y is a good quotient for G acting on X if (i) ϕ is G-invariant. (ii) ϕ is surjective. (iii) for any open subset U Y, the morphism O Y (U) O X (ϕ 1 (U)) is an isomorphism onto the G- invariant part. (iv) If W X is a G-invariant closed subset of X, ϕ(w ) is closed in Y. (v) If W 1 and W 2 are disjoint G-invariant closed subsets of X, then ϕ(w 1 ) ϕ(w 2 ) =. (vi) ϕ is affine. We say ϕ is a geometric quotient if ϕ is a good quotient and the fiber of ϕ is a single orbit. Proposition 8. Let G be an affine algebraic group acting on a scheme X and suppose we have a morphism ϕ : X Y satisfying properties i), iii), iv) and v) in the definition of good quotient. Then ϕ is a categorical quotient. In particular, any good quotient is a categorical quotient. Proof. By i), we know ϕ is G-invariant and so we only need to prove that it is universal with respect to all G-invariant morphisms from X. Let f : X Z be a G-invariant morphism. Taking a finite affine open cover U i of Z, we set 16

18 W i := X f 1 (U i ) to a G-invariant closed subset in X; the image ϕ(w i ) Y is closed by iv) V i := Y ϕ(w i ) be the open complement; By construction, we have an inclusion ϕ 1 (V i ) f 1 (U i ). As U i cover Z, the intersection W i is empty. By property v) of the good quotient ϕ, we have ϕ(w i ) = ; i that is, V i is an open cover of Y. ince f is G-invariant, the homomorphism f : O Z (U i ) O X (f 1 (U i )) has image in O X (f 1 (U i )) G. Therefore, there is a unique morphism h i commute O Z (U i ) h i O Y (V i ) which makes the following square O X (f 1 (U i )) G = O X (ϕ 1 (V i )) G where the isomorphism on the right hand side of this square is given by property iii) of the good quotient ϕ. ince U i is affine, the k-algebra homomorphism O Z (U i ) O Y (V i ) corresponds to a morphism By construction, we have h i : V i U i. f ϕ 1 (V i ) = h i ϕ : ϕ 1 (V i ) U i and h i = h j on V i V j ; therefore, we can glue the morphisms h i to obtain a morphism h : Y Z such that f = h ϕ. ince the morphisms h i are unique, it follows that h is also unique. Example Consider the action of G m on A 2 by (λ, (x 1, x 2 )) = (λx 1, λ 1 x 2 ). As the origin is in the closure of the punctured axes, all three orbits will be identified as a point by the categorical quotient. The smooth conic orbits {(x, y) : xy = a 0} are closed. These conic orbits are parametrised by A 1 {0}. Therefore, we may naturally expect that ϕ : A 2 A 1 given by (x, y) xy is a categorical quotient. (1). This morphism is clearly G-invariant and surjective, which shows parts i) and ii). (2). Consider the morphism ϕ : k[z] k[x, y] given by z xy. We claim that this is an isomorphism onto the ring of G m -invariant functions. The action of λ G m on k[x, y] is given by λ x = λx, λ y = λ 1 y Therefore, the invariant subalgebra is k[x, y] G = k[xy] as required. This verifies part iii). Also, note vi) holds trivially. 17

19 (3). Note that any G-invariant closed subvariety in A 2 is either a finite union of orbit closures or the entire space A 2. One can directly check the conditions iv) and (v) holds for disjoint orbit closures. Finally, ϕ is not a geometric quotient, as φ 1 (0) is a union of 3 orbits. Corollary 1. If ϕ : X Y is a good quotient, then 1. G x 1 G x 2 if and only if ϕ(x 1 ) = ϕ(x 2 ) 2. for each y Y, the preimage ϕ 1 (y) contains a unique closed orbit. In particular, if the action is closed, ϕ is a geometric quotient. Warm up: G is finite Let A be a finitely generated regular k-algebra and let X = pec A be a smooth affine variety. Let G be a finite group acting on X. We will define X/G algebraically, i.e. X/G will has a scheme structure. Note that G acts on O X and therefore on the ring A = Γ(X, O X ). Let A G be the subring generated G-invariant classes and we define X//G = pec A G (2.3.1) Theorem 8. uppose p G. The ring A G is finitely generated. In particular, X//G is an affine k-scheme. Moreover, the natural morphism ϕ : X X//G is a good quotient. Proof. We first show that A G is finitely generated. Let us assume that A = k[x 1,..., x n ] is a polynomial ring and G acts on A preserving the grading. Then A = d 0 A d and A G = A G A d. Consider the ideal A G + generated by A G A +, as A is Noetherian, A G + is finitely generated by polynomials f 1,..., f m A G +. Claim: A G is generated by f i. To show h A G is a polynomial of f i, we can proceed by induction on deg(h). If deg(h) > 0, we can write h = h i f i for some h i A and deg(h i ) < deg(h). As G <, there is an average operator and 1 G of f i. h = 1 G g G g h = 1 G (g h i )f i (g h i ) A G with degree strictly less than h. By induction, we get h can be written as a polynomial g G In general, if A is generated by a 1,..., a n, we can find a finite dimensional subspace V A containing a 1,... a n and let b 1,..., b m be a basis of V. o one can define a homomorphism i g G k[x 1,..., x m ] A (2.3.2) by sending x i to b i. Then we know the pullback of the action on A preserve the grading of k[x 1,..., x m ] and k[x 1,..., x m ] G is finitely generated. The assertion then follows from the fact k[x 1,..., x m ] G A G is surjective. 18

20 For the 2nd assertion, the condition is i), ii), vi) in the definition are automatically satisfied. We only need to verify for any two disjoint G-invariant closed subsets W 1 and W 2 in X, there is a function f O(X) G such that f(w 1 ) = 0 and f(w 2 ) = 0. If such f exists, as O(X) G = O(X//G), one can regard f as a regular function on X//G such that f(ϕ(w 1 )) = 1 and f(ϕ(w 2 )) = 0. This implies ϕ(w 1 ) ϕ(w 2 ) =. To find such f, note that W i are disjoint and closed, we have A = I(W 1 W 2 ) = I(W 1 ) + I(W 2 ), and we can write 1 = f 1 + f 2 for some f i I(W i ). Then f 1 (W 1 ) = 0 and f 1 (W 2 ) = 1. The function f G = 1 g f1 G is a G-invariant function and f G (W 1 ) = 0, f G (W 2 ) = 1. This proves the assertion. General case: G is reductive Definition Let G be an affine algebraic group over k. An element g is semisimple (resp. unipotent) if there is a faithful linear representation ρ : G GL n such that ρ(g) is diagonalisable (resp. unipotent). We say G is unipotent if every non-trivial linear representation ρ : G GL(V ) has a non-zero G-invariant vector. We say G is reductive if it is smooth and every smooth unipotent normal algebraic subgroup of G is trivial. (this is equivalent to the unipotent radical of G is trivial) Example GL(n), L(n) and PGL(n) are reductive algebraic groups. 2. G m is reductive. 3. G a is not reductive since it is unipotent. Definition An algebraic group G is said to be linearly reductive if for every epimorphism φ : V W of G representations, the induced map on G-invariants φ G : V G W G is surjective. geometrically reductive if for every finite dimensional representation ρ : G GL(V ) and every G-invariant vector v V, there is a G-invariant non-constant homogenous polynomial f such that f(v) 0. Proposition 9. For an affine algebraic group G, the following statements are equivalent. 1. G is linearly reductive. 2. For any finite dimensional linear representation ρ : G GL(V ), any G-invariant subspace V 1 V admits a G-stable complement (i.e. there is a subrepresentation V 2 V such that V = V 1 V 2 ). 19

21 3. Every finite representation of G is completely reducible, i.e. it decomposes into a direct sum of irreducible representations. 4. For any finite dimensional linear representation ρ : G GL(V ) and every non-zero G-invariant point v V, there is a G-invariant linear form f : V k such that f(v) 0. Proof. The equivalence (2) (3) is clear. For (2) (1), if φ : V W is an epimorphism and V 1 = ker φ. Then V 1 has a G-stable complement V 2 = W and φ G : V G = V1 G V 2 G W G is surjective. For (1) (2), let ρ : G GL(V ) be a finite dimensional linear representation and V 1 a G-invariant subspace. Consider the sujrective map of G-representations Hom(V, V 1 ) Hom(V 1, V 1 ). By assumption, we get a lift of the identity id : V 1 V 1 Hom(V 1, V 1 ) G to a G-equivariant homomorphism φ : V V 1. Then V 1 has a G-stable completement ker(φ). For (2) (4), let V be a finite dimensional linear G-representation and v V G be a non-zero G-invariant vector. Then v determines a G-invariant linear form φ : V k. By letting G act trivially on k, we can view φ as a surjection of G-representations. Applying (2), we get the existence of f. (4) (2) is easy. Example G m is linear reductive as every finite representation is completely reducible. 2. If cha(k) > 0, GL(n), L(n) and PGL(n) are not linear reductive when n > 1. For instance, if cha(k) = 2, the representation of L(2) is not completely reducible. ρ( a b c d ) = 1 ac bd 0 a 2 b 2 0 c 2 d 2 Proposition 10. Every reductive group G over C is linearly reductive. Proof. We let K G be a maximal compact subgroup. tep 1. Claim: every finite dimensional representation of K is completely reducible. Let ρ : K GL(V ) be a finite dimensional representation of K. It suffices to prove that every K-invariant subspace W V has a K-stable complement. Note that there is a K-invariant Hermitian inner product on V, as we can take any Hermitian inner product h on V and integrate over K using a Haar measure dµ on K to obtain a K-invariant Hermitian inner product h K (v 1, v 2 ) := (g v 1, g v 2 )dµ K Then, we define the K-stable complement of W V to be the orthogonal complement of W with respect to this K-invariant Hermitian inner product. 20

22 tep 2. For G reductive and a maximal compact subgroup K G, the elements of K are Zariski dense in G (one may use Lie algebra and Identity theorem). tep 3. For any finite dimensional linear representation ρ : G GL(V ), we claim that V G = V K, where K is a maximal compact of G. Clearly, we have V G V K. To prove the reverse inclusion, let v V K and we consider the morphism σ : G V given by g ρ(g)(v). Then σ 1 (v) G is Zariski closed. ince v V K, we have K σ 1 (v) and K σ 1 (v). Note that K G is Zariski dense, it follows that G σ 1 (v); that is, v V G as required. tep 4. By Proposition 4.14, it suffices to show for every epimorphism φ : V W, the induced homomorphism φ G on invariant subspaces is also surjective. By tep 3, this is equivalent to showing that φ K is surjective. Indeed, we have the following theorem towards the relation between various definitions. Theorem 9. For smooth affine algebraic group schemes, we have All three notions coincide in characteristic zero. Linearly reductive Geometrically reductive Reductive. Reynolds operator Definition For a group G acting on a k-algebra A, a linear map R A : A A G is called a Reynolds operator if it is a projection onto A G and for a A G and b A, we have R A (ab) = ar A (b). Lemma 4. Let G be a linearly reductive group acting rationally on a finitely generated k-algebra A; then there exists a Reynolds operator R A : A A G. Proof. ince A is finitely generated, it has a countable basis. Therefore, we can write A as an increasing union of finite dimensional G-invariant vector subspaces A n A constructed as below: Let a 1, a 2,... be the basis. Then we can iteratively construct the subsets A n by letting A n be the finite dimensional G-invariant subspace containing a 1,..., a n, a basis of A n 1 and a j A n 1 for j = 1,..., n. Then A = A n. ince G is linearly reductive and each A n is a finite dimensional G-representation, we can write n 1 A n = A G n A n where A n is the direct sum of all non-trivial irreducible G-subrepresentations of A n. et R n : A n A G n be the canonical projection onto the direct factor A G n. For m > n, we have a commutative square A n R n A G n A m R m A G m 21

23 as we have A n A m and A G n A G m. o we obtain a linear map R A : A A G given by the compatible projections R n : A n A G n for each n. It remains to check that for a A G and b A, we have R A (ab) = ar A (b). Pick n such that a, b A n and pick m n such that a(a n ) A m. Then consider the homomorphism l a : A n A m of G representations given by left multiplication by a. Write A n = W 1 W rn as a direct sum of non-trivial irreducible subrepresentations W i A n. ince G acts by algebra homomorphisms and a A G, we have l a (A G n ) A G m. Moreover, the image of each irreducible W i under l a is either zero or isomorphic to W i. Therefore, we have l a (W i ) A m. This implies that l a (A n) A m. In particular, if we write b = b G + b for b G A G n, b A n, then ab = l a (b) = l a (b G ) + l a (b ) = ab G + l a (b) A G m A m and R A (ab) = R m (ab) = ab G = ar A (b) as required. Using the Reynolds operator, the same argument gives Theorem 10. uppose G is linear reductive. The ring A G is finitely generated. In particular, X//G is an affine k-scheme. Moreover, the natural morphism ϕ : X X//G is a good quotient. Remark 7. More generally, Nagata has proved the finite generation of A G for reductive groups. The reductivity is the optimal condition for A G being finitely generated. This is due to the following result. Theorem 11. An affine algebraic group G is reductive iff for every rational G-action on a finitely generated k-algebra A, A G is finitely generated. table points and geometric quotient Definition We say x X is stable if its orbit is closed in X and dim G x = 0 (or equivalently, dim G x = dim G). We let X s denote the set of stable points. Example Consider G m acting on X = A 2 via automorphisms. Then X s = {xy 0}. Theorem 12. uppose a reductive group G acts on an affine scheme X and let φ : X Y = X//G be the affine GIT quotient. Then X s X is an open and G-invariant subset, Y s = φ(x s ) is an open subset of Y and X s = φ 1 (Y s ). Moreover, X s Y s is a geometric quotient. Proof. First, we show that X s is open. For any x X s, by semicontinuity, the set X + := {x X : dim(g x) > 0} 22

24 of points with positive dimensional stabilisers is a closed subset of X. As we proved in Theorem 10, there is a function f O(X) G such that f(x + ) = 0, f(g x) = 1. o we have x X f. We claim that X f X s. ince all points in X f have stabilisers of dimension zero, it remains to check that their orbits are closed. uppose z X f has a non-closed orbit and w / G z belongs to the orbit closure of z. Then w X f as f is G-invariant and so w must have stabiliser of dimension zero. However, we know that the boundary of the orbit G z is a union of orbits of strictly lower dimension and so the orbit of w must be of dimension strictly less than that of z which contradicts the fact that w has zero dimensional stabiliser. Hence, X s is an open subset of X and is covered by open subsets X f. 2.4 Projective GIT quotient Construction of Proj A Z-graded ring is a ring = n n, where multiplication respects the grading, i.e. sends m n to m+n. Clearly 0 is a subring, each n is an 0 -module, and is a 0 -algebra. An ideal I of is a homogeneous ideal if it is generated by homogeneous elements. Definition (Proj construction.). Write + := d>0 d. If + is a finitely generated ideal, We say is finitely generated graded ring over 0. Then X = Proj() is the collection of homogenous primes ideals of not containing the irrelevant ideal +. The closed subsets on X are the projective vanishing set V (I), where I +. We call this the Zariski topology on Proj(). To give the ringed space structure, we take the projective distinguished open set X f = Proj()\V (f) (the projective distinguished open set) be the complement of V (f). One can show that X f is isomorphic to pec(( f ) 0 ). Hence it gives arise a natural ringed space structure on X. Projective quotient Definition Let X be a projective variety and let G be an affine algebraic group. A linear G-equivariant projective embedding of X is a group homomorphism G GL n+1 and a G-equivariant projective embedding X P n, or equivalently we say the G-action on X is linear. For X P n, we set R(X) = k[x 0,..., x n ]/I(X). If G acts linearly on X P n, we let R(X) G be the G-invariant subalgebra. Define the nullcone N to be the closed subscheme of X defined by the homogeneous ideal R(X) G + in R(X). We define the semistable set X ss = X N to be the open subset of X given by the complement to the nullcone. x X ss is called semistable if there exists a G-invariant homogeneous function f R(X) G r for r > 0 such that f(x) 0. 23

25 We call the morphism X ss X//G := Proj(R(X) G ) the GIT quotient of this action. Theorem 13. The GIT quotient φ : X ss X//G is a good quotient of the G-action on the open subset X ss of semistable points in X. Furthermore, X//G is a projective scheme. Proof. By construction, X//G is the projective spectrum of the finitely generated graded k-algebra R(X) G. We claim that Proj(R(X) G ) is projective over pec k. As R(X) G is a finitely generated k-algebra, we can pick generators f 1,..., f r in degrees d 1,..., d r. Let d = d 1 d r ; then (R(X) G ) (d) = R(X) G dl is finitely l 0 generated by (R(X) G ) (d) 1 as a k-algebra and so Proj(R(X)G) (d) is projective. As X//G = Proj(R(X) G ) = Proj(R(X) G ) (d), we can conclude that X//G is projective. For f R(X) G, the open affine subsets Y f Y form a basis of the open sets on Y. ince f R(X) G + R(X) +, we consider the open affine subset X f X and we have φ 1 (Y f ) = X f. Then O(Y f ) = ((R(X) G ) f ) 0 = ((R(X) f ) 0 ) G = (O(X f ) 0 ) G = O(Xf ) G and so the corresponding morphism of affine schemes φ f : X f Y f = pec O(X f ) G is an affine GIT quotient, and so also a good quotient by Theorem 10. The morphism φ : X ss Y is obtained by gluing the good quotients φ f : X f Y f. As being a good quotient is a local property, we can conclude that φ is a good quotient as well. Definition Consider a linear action of a reductive group G on a closed subscheme X P n. Then a point x X is 1. stable if dim G x = 0 and there is a G-invariant homogeneous polynomial f R + (X) G such that x X f and the action of G on X f is closed. 2. unstable if it is not semistable. Theorem 14. Let Y = X//G be the projective GIT quotient. There is an open subscheme Y s Y such that φ 1 (Y s ) = X s and that the GIT quotient restricts to a geometric quotient φ : X s Y s. Proof. Let Y c be the union of Y f for f R(X) G + where the G-action on X f is closed. Let X c be the union of X f such that X c = φ 1 (Y c ). Then φ : X c Y c is constructed by gluing φ f : X f Y f for f R(X) G +. Each φ f is a good quotient and as the action on X f is closed, then φ f is also a geometric quotient by Corollary 1. It follows that φ : X c Y c is a geometric quotient. By definition, X s is the open subset of X c consisting of points with zero dimensional stabilisers and we let Y s = φ(x s ) Y c. As φ : X c Y c is a geometric quotient and X s is a G-invariant subset of X, then Y s is open in Y c. o we can conclude that Y s Y is open. Finally, the geometric quotient φ : X c Y c restricts to a geometric quotient φ : X s Y s. 24

26 Remark 8. In general, a geometric quotient (i.e. orbit space) does not exist because the action is not necessarily closed. For finite groups G, every good quotient is a geometric quotient as the action of a finite group is always closed. Lemma 5. Let G be a reductive group acting linearly on X P n. A k-point x X is stable if and only if x is semistable, its orbit G x is closed in X ss and its stabiliser G x is zero dimensional. Proof. uppose x is stable and x G x X ss ; then φ(x ) = φ(x) and x X s. As G acts on X s 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 orbit G x is closed in X ss. Conversely, we suppose x is semistable with closed orbit in X ss and zero dimensional stabiliser. As x is semistable, there is a homogeneous f R(X) G + such that x X f. As G x is closed in X ss, it is also closed in the open affine set X f X ss. By semi-continuity, the G-invariant set Z := {z X f dim G z > 0} is closed in X f. ince Z is disjoint from G x and both sets are closed in the affine scheme X f, there exists h O(X f ) G such that h(z) = 0 and h(g x) = 1. Write h = h f for some G-invariant homogeneous polynomial h R(X) G +. Then x X r fh and X fh is disjoint from Z. Then all the orbits in X fh are closed and hence x is stable. Definition A k-point x X is said to be polystable if it is semistable and its orbit is closed in X ss. We say two semistable k-points are -equivalent if their orbit closures meet in X ss. One can see that every stable k-point is polystable. Proposition 11. Let x X be a semistable k-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. Hence X//G is the set of polystable points on X modulo -equivalence. Proof. As φ is constant on the closure of the orbits, φ 1 (φ(x)) contains a unique closed orbit of a polystable point x p. If x is not stable, then dim G x p < dim G x. This implies x p can not be stable. Linearisations There is a more general theorem for constructing GIT quotients of reductive group actions on quasi-projective schemes with respect to linearisations. Definition Let L be a line bundle on X and we set R(X, L) := r 0 H 0 (X, L r ) to be the section ring. uppose G acts linearly on X via σ. A linearisation of the G-action on X is a line bundle π : L X over X with an isomorphism of line bundles π XL = σl 25

27 where π X : G X X is the projection, such that the diagram commutes G G L id G σ G L µ id G L σ L σ where σ : G L L is the induced morphism via pullback of σ. Lemma 6. If L is a linearisation, there is a natural linear representation G GL(H 0 (X, L)). Proof. We consider the map via the composition H 0 (X, L) O(G) H 0 (X, L) H 0 (X, L) σ H 0 (G X, σ L) = H 0 (G X, G L) = H 0 (G, O G ) H 0 (X, L) This defines the linear representation. Using the result above, we get an induced action of G on R(X, L). Consider the graded algebra of G-invariant sections R(X, L) G = H 0 (X, L r ) G r 0 Definition Let X be a projective scheme with an action by a reductive group G and let L be an ample linearisation. (i) A point x X is semistable with respect to L if there is an invariant section σ H 0 (X, L r ) G for some r > 0 such that σ(x) 0 and X σ = {x X : σ(x) 0} is affine. (ii) A point x X is stable with respect to L if dim G x = dim G and σ H 0 (X, L r ) G for some r > 0 such that σ(x) 0, X σ = {x X : σ(x) 0} is affine and the action of G on X σ is closed. The open subsets of stable and semistable points with respect to L are denoted X s (L) and X ss (L) respectively. Then we can define the projective GIT quotient with respect to L to be the map X ss (L) Proj(R(X, L) G ) Theorem 15. The GIT quotient X ss (L) X// L G is a good quotient and X// L G is a projective scheme with a natural ample line bundle L. Furthermore, there is an open subset Y s X// L G such that φ 1 (Y s ) = X s (L) and φ : X s (L) Y s is a geometric quotient. 2.5 Hilbert-Mumford Criteria Definition Let G be a reductive group acting linearly on a projective scheme X with respect to an ample linearisation L. 26