MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective schemeX does not come with a pre-specified embedding in a projective space. However, an ample line bundleL onX (or more precisely some power ofL) determines an embedding ofX into a projective space. More precisely, the projective schemeX and ample line bundleL, determine afinitely generated gradedk-algebra 0 r R(X, L) := H (X, L⊗ ). r 0 �≥ 0 r We can choose generators of thisk-algebra:s i H (X, L⊗ i ) fori=0, , ..n, wherer i 1. Then these sections determine a closed immersion ∈ ≥ X� P(r 0, . , rn) → into a weighted projective space, by evaluating each point ofX at the sectionss i. In fact, if m we replaceL byL ⊗ form sufficiently large, then we can assume that the generatorss i of the finitely generatedk-algebra m 0 mr R(X, L⊗ ) = H (X, L⊗ ) r 0 �≥ m all lie in degree 1. In this case, the sectionss i of the line bundleL ⊗ determine a closed immersion n 0 m X� P =P(H (X, L⊗ )∗) → given by evaluationx (s s(x)). Now suppose we have �→ an �→ action of an affine algebraic groupG onX; then we would like to do everything aboveG-equivariantly, by lifting theG-action toL such that the above embedding is equivariant and the action ofG onP n is linear. This idea is made precise by the notion of a linearisation. Definition 5.15. LetX be a scheme andG be an affine algebraic group acting onX via a morphismσ:G X X. Then a linearisation of theG-action onX is a line bundleπ:L X overX with an× isomorphism→ of line bundles → π∗ L=G L = σ∗L, X × ∼ whereπ X :G X X is the projection, such that the induced bundle homomorphism ˜σ:G L L de×fined→ by × → G L × ˜σ ∼= � σ∗L ��L idG π × π � � � G X �X. × σ induces an action ofG onL; that is, we have a commutative square of bundle homomorphisms idG ˜σ G G L × �G L × × × µ id ˜σ G× L � � G L �L. × ˜σ We say that a linearisation is (very) ample if the underlying line bundle is (very) ample. Let us unravel this definition a little. Since ˜σ:G L L is a homomorphism of vector bundles, we have × → i) the projectionπ:L X isG-equivariant, ii) the action ofG on→ thefibres ofL is linear: forg G andx X, the map on thefibres ∈ ∈ Lx L g x is linear. → · 44 VICTORIA HOSKINS Remark 5.16. (1) The notion of a linearisation can also be phrased sheaf theoretically: a linearisation of aG-action onX on an invertible sheaf is an isomorphism L Φ:σ ∗ π ∗ , L→ X L whereπ :G X X is the projection map, which satisfies the cocycle condition: X × → (µ id )∗Φ=π ∗ Φ (id σ) ∗Φ × X 23 ◦ G × whereπ 23 :G G X G X is the projection onto the last two factors. Ifπ:L X denotes the line× × bundle→ × associated to the invertible sheaf , then the isomorphism→ Φ determines a bundle isomorphism of line bundles overG XL: × Φ:(G X) L (G X) L × × πX ,X,π → × × σ,X,π and then we obtain ˜σ :=πX Φ. The cocycle condition ensures that ˜σ is an action. (2) The above notion of a linearisation◦ of aG-action onX can be easily modified to larger rank vector bundles (or locally free sheaves) overX. However, we will only work with linearisations for line bundles (or equivalently invertible sheaves). Exercise 5.17. For an action of an affine algebraic groupG on a schemeX, the tensor product of two linearised line bundles has a natural linearisation and the dual of a linearised line bundle also has a natural linearisation. By an isomorphism of linearisations, we mean an isomorphism of the underlying line bundles that isG-equivariant; that is, commutes with the actions ofG on these line bundles. We let PicG(X) denote the group of isomorphism classes of linearisations of aG-action onX. There is a natural forgetful mapα : Pic G(X) Pic(X). → Example 5.18. (1) Let us considerX = Speck with necessarily the trivialG-action. Then there is only one line bundleπ:A 1 Speck over Speck, but there are many linearisa- → tions. In fact, the group of linearisations ofX is the character group ofG. Ifχ:G G m → is a character ofG, then we define an action ofG onA 1 by acting byG A 1 A 1. × → Conversely, any linearisation is given by a linear action ofG onA 1; that is, by a group homomorphismχ:G GL 1 =G m. (2) For any schemeX with→ an action of an affine algebraic groupG and any character 1 χ:G G m, we can construct a linearisation on the trivial line bundleX A X by → × → g (x, z)=(g x,χ(g)z). · · More generally, for any linearisation ˜σ onL X, we can twist the linearisation by a → χ characterχ:G G m to obtain a linearisation ˜σ. (3) Not every linearisation→ on a trivial line bundle comes from a character. For example, 1 1 considerG=µ 2 = 1 acting onX=A 0 by ( 1) x=x − . Then the {±1 } −{ } 1− · linearisation onX A X given by ( 1) (x, z)=(x − , xz) is not isomorphic to a linearisation given× by a character,→ as over− the· fixed points +1 and 1 inX, the action − of 1 µ 2 on thefibres is given byz z andz z respectively. − ∈ �→ �→n − (4) The natural actions of GLn+1 and SLn+1 onP inherited from the action of GLn+1 on n+1 A by matrix multiplication can be naturally linearised on the line bundle n (1). To O P see why, we note that the trivial rankn+1-vector bundle onP n has a natural linearisation n n+1 of GLn+1 (and also SLn+1). The tautological line bundle Pn ( 1) P A is preserved by this action and so we obtain natural linearisationsO − of⊂ these actions× on n Pn ( 1). However, the action of PGLn+1 onP does not admit a linearisation on Pn (1) O ± n O (see Exercise Sheet 9), but we can always linearise anyG-action onP to Pn (n + 1) as this is isomorphic to thenth exterior power of the cotangent bundle, andO we can lift any action onP n to its cotangent bundle. Lemma 5.19. LetG be an affine algebraic group acting on a schemeX viaσ:G X X and let˜σ:G L L be a linearisation of the action on a line bundleL overX. Then× → there is a natural× linear→ representationG GL(H 0(X, L)). → MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 45 0 0 Proof. We construct the co-moduleH (X, L) (G) k H (X, L) defining this representation by the composition →O ⊗ σ H0(X, L) ∗ �H0(G X,σ L) = H0(G X,G L) = H0(G, ) H 0(X, L) × ∗ ∼ × × ∼ O G ⊗ where thefinal isomorphism follows from the K¨unneth formula and the middle isomorphism is defined using the isomorphismG L = σ L. × ∼ ∗ � Remark 5.20. Suppose thatX is a projective scheme andL is a very ample linearisation. Then the natural evaluation map H0(X, L) L ⊗ k OX → isG-equivariant. Moreover, this evaluation map induces aG-equivariant closed embedding 0 X� P(H (X, L)∗) → such thatL is isomorphic to the pullback of the Serre twisting sheaf (1) on this projective space. In this case, we see that we have an embedding ofX as aO closed subscheme of a 0 projective spaceP(H (X, L)∗) such that the action ofG onX comes from a linear action ofG 0 onH (X, L)∗. In particular, we see that a linearisation naturally generalises the setting ofG acting linearly onX P n. ⊂ 5.4. Projective GIT with respect to an ample linearisation. LetG be a reductive group acting on a projective schemeX and letL be an ample linearisation of theG-action onX. Then consider the gradedfinitely generatedk-algebra 0 r R :=R(X, L) := H (X, L⊗ ) r 0 �≥ r of sections of powers ofL. Since each line bundleL ⊗ has an induced linearisation, there is 0 r an induced action ofG on the space of sectionsH (X, L⊗ ) by Lemma 5.19. We consider the graded algebra ofG-invariant sections G 0 r G R = H (X, L⊗ ) . r 0 �≥ The subalgebra of invariant sectionsR G is afinitely generatedk-algebra and ProjR G is projec- G G tive overR 0 =k =k following a similar argument to above. Definition 5.21. For a reductive groupG acting on a projective schemeX with respect to an ample line bundle, we make the following definitions. 1) A pointx X is semistable with respect toL if there is an invariant sectionσ 0 ∈r G ∈ H (X, L⊗ ) for somer> 0 such thatσ(x)= 0. 2) A pointx X is stable with respect toL if dim� G x = dimG and there is an invariant ∈ 0 r G · sectionσ H (X, L⊗ ) for somer> 0 such thatσ(x)= 0 and the action ofG on X := x∈ X:σ(x)= 0 is closed. � σ { ∈ � } We letX ss(L) andX s(L) denote the open subset of semistable and stable points inX respec- tively. Then we define the projective GIT quotient with respect toL to be the morphism Xss X// G := ProjR(X, L) G → L associated to the inclusionR(X, L) G � R(X, L). → Exercise 5.22. We have already defined notions of semistability and stability when we have a linear action ofG onX P n. In this case, the action can naturally be linearised using the line ⊂ bundle n (1).