Moduli Theory and Geometric Invariant Theory

November 29, 2017

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 betweenthem. The collection of objects of a category C is often denoted ob(C) and the collection of morphismsbetween A,B ∈ C is denoted HomC(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 ∈ HomC(C,C′)

a morphismF (f) : F (C)→ F (C ′)

in HomD(F (C), F (C ′)) such that F preserves identity morphisms and composition. A contravariant func-tor 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 ηC : F (C) → G(C) which is compatible withmorphisms f : C → C ′, i.e. we have a commutative square

F (C) //

��

F (C ′)

��

G(C) // G(C ′)

(1.1.1)

The contravariant functors from the category to sets are called presheaves.

Example 1.1.1. Let C be a category and (Sets) be the category of sets. For any A ∈ ob(C), we have acontravariant functor

hA : C → (Sets)

defined by hA(B) = Hom(B,A). This functor is called functor of maps to A, and it is also known as thefunctor of points. e.g. Let Sch be the category of schemes. Then the functor of points is a sheaf.

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Theorem 1. (Yoneda’s lemma) There is a bijection

Ψ : {natural transforms η : hC → F } ↔ F (C)

for any presheaf F on C. In particular, if hC and hC′ are the identified functors, then C and C ′ are canonicallyisomorphic.

Proof. The map is given by Ψ(η) = ηC(idC). Conversely, if s ∈ F (C), we can define a natural transformationη as follows. For A ∈ C, let ηA : hC(A)→ F (A) be the morphism ηA(f) = F (f)(s), where F (f) denotes themorphism F (C)→ F (A).

♣

Definition 1.1.2. A prefsheaf F is called representable if there exists an object C ∈ ob(C) and naturalisomorphism F ∼= hC .

Example 1.1.3. The global section functor

Γ : Sch→ (Sets)

X 7→ Γ(X,OX)

is representable by A1. The natural transform between Γ and hA1 is as below: the morphism ϕ : X → A1

corresponds to mapsϕ] : k[t]→ Γ(X,OX)

and ϕ](t) is the function on X.

Similarly, one can show that the functor

S 7→∏j

Γ(S,O⊕ajS )

is representable by∏j Aaj = A

∑aj .

Is every presheaf F of Sch 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 toclassify these objects up to equivalence.

Definition 1.2.1. A (naive) moduli problem (in algebraic geometry) is a collection M of objects (in algebraicgeometry) and an equivalence relation ∼ on M.

Example 1.2.2. 1. (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 P1 and ∼ be the equivalence relation given by thenatural action of the automorphism group PGL2 of P1.

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3. Let M be the set of hypersurfaces of degree d in Pn and ∼ can be chosen to be either equality or therelation given by projective change of coordinates (i.e. corresponding to the natural PGLn+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 1.2.3. Let M consist of 4 ordered distinct points (p1, p2, p3, p4) on P1. We want to classify thesequartuples up to the automorphisms of P1. We recall that the automorphism group of P1 is the projectivelinear group PGL2, which acts as Mobius transformations. We define our equivalence relation by

(p1, p2, p3, p4) ∼ (q1, q2, q3, q4)

if there exists an automorphisms f : P1 → P1 such that f(pi) = qi for i = 1, . . . , 4.

Note that for any 3 distinct points (p1, p2, p3) on P1, there exists a unique Mobius transformation f ∈ PGL2which sends (p1, p2, p3) to (0, 1,∞) and the cross-ratio of 4 distinct points (p1, p2, p3, p4) on P1 is given byf(p4) ∈ P1 − {0, 1,∞}, where f is the unique Mobius transformation that sends (p1, p2, p3) to (0, 1,∞).Therefore, we see that the set M/ ∼ is in bijection with the set of k-points in the quasi-projective varietyP1 − {0, 1,∞}.

We want more than this, we want a moduli space which encodes how these objects vary continuously infamilies; this information is encoded in a moduli functor. It is of the form

S 7→ {Families over S}

Definition 1.2.4. Let (M,∼) be a naive moduli problem. Then an extended moduli problem is given by

1. sets M(S) of families over S and an equivalence relation ∼S on M(S), for all schemes S,

2. pullback maps f∗ : M(S) → M(T ), for every morphism of schemes T → S, satisfying the followingproperties:

• (M(pt),∼pt) = (M,∼);

• for the identity Id : S → S and any family U over S, we have Id∗U = U ;

• for a morphism f : T → S and equivalent families U ∼S U ′ over S, we have f∗U ∼T f∗U ′;• for morphisms f : T → S and g : S → R, and a family W over R, we have an equivalence

(g ◦ f)∗W ∼T f∗g∗W .

1.3 Fine moduli space

Definition 1.3.1. Let M : C → (Sets) be a moduli functor; then M ∈ ob(C) is a fine moduli space for M ifit represents M.

Definition 1.3.2. Let M be a fine moduli space for M. Then the family U ∈M(M) corresponding to theidentity morphism on M is called the universal family.

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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

ηM ′((ηM )−1(IdM )) : M →M ′,

andηM ′((η

′M ′)−1(IdM ′)) : M ′ →M.

Definition 1.3.3. Let F and G be functors from Sch to (Sets). We say G is a subfunctor of F if G(S) ⊆ F (S)for all schemes S and G(S′)→ G(S) is the restriction of F (S′)→ F (S) for any S′ → S.

We say G is an open subfunctor if ∀S ∈ Sch and ξ ∈ F (S), there exists an open subscheme UGξ such that

for any f ∈ Hom(S′, S), we have f∗ξ ∈ G(S′) if and only if f factors through UGξ .

Definition 1.3.4. Let {Gi} be a collection of open subfunctors of F . We say that {Gi} form an opencovering of F if for all schemes S and ξ ∈ F (T ), the set {UGi

ξ } form an open cover of S.

Proposition 1. A functor F : Sch→ (Sets) is said to be a Zariski sheaf if it is compatible with gluing, i.e.the sequence

F (S)→∏

F (Uα)→→∏

F (Uα ∩ Uβ)

is exact for any open cover {Uα} of S. If F is a Zariski sheaf which admits an open covering by representablesubfunctors Gi then F is representable.

Proof. By assumption, each Gi ∼= hXi is represented by a scheme Xi. The functor Gi ×F Gj is an opensubfunctor of Gi and Gj , which allows us to glue Xi together to a scheme X.

Now, as hX and F are Zariski sheaves, and they coincide on an open covering. By sheaf axiom, they mustbe isomorphic, and F = hX . ♣

1.3.5 Grassmannian

Consider the moduli problem G(r, V ) of r-dimensional linear subspaces in a fixed n-dimensional vector spaceV , where a family over S is a rank r vector subbundle E of S × V and the equivalence relation is equality.In other words,

G(r, V ) : Sch→ (Sets)

G(r, V )(S) =

{subvector bundles of

rank r of S × V

}

Note that the trivial vector bundle corresponds to the free sheaf O⊕nS . Then subbundle E corresponds to asurjection O⊕nS → Q, where Q is a finite locally free sheaf of rank n− r. Thus we can regard G(r, V ) as thefunctor

G(r, n)(S) := {O⊕nS −→→ Q}

Theorem 2. The functor G(r, V ) is represented by a smooth projective scheme Grass(r, V ).

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Proof. For each subset I = {j1, . . . , jn−r} ⊆ {1, 2, . . . , n} of cardinality n− r, we let GI be the subfunctor ofG(r, n) corresponding to the surjection

O⊕nS −→→ Q

such that the induced map OIS =⊕ji

OS 〈eji〉 ∼= O⊕n−rS −→→ Q is an isomorphism. More precisely, GI associates

each scheme S the subset

GI(S) = {q : O⊕nS −→→ Q | q ◦ sI is surjective} ⊆ G(r, n)(S)

where sI : O⊕n−rS → O⊕nS is the direct sum of the natural coprojection (ji-th position) OS → O⊕nS .

However, the functor GI is isomorphic to the functor

S 7→ {q : O⊕nS −→→ OIS}

by composing sI . To determine q, we have to fix the images of q(ei) in Γ(S,O⊕n−rS ). So GI is isomorphic tothe functor

S 7→∏j

Γ(S,O⊕n−rS ).

This functor is representable by Ar(n−r). As G(r, V ) is a Zariski sheaf and GI forms an open covering as Ivaries, 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 ∧rW is an linebundle whose sections defines a morphism

ϕr : Grass(r, V )→ P(∧rV )

given by W 7→ [∧rW ]. We claim that ϕr is a closed embedding. For each I, we define PI ⊆ P(∧rV ) tobe the completement of the hyperplane [∧rWI ] = 0. Then PI forms an open cover of P(∧rV ) and the map

GI → PW is exactly the closed embedding Ar(n−r) → A

nr

−1

. ♣

Remark 2. The embedding ϕr is called the Plucker embedding.

1.3.6 Hilbert scheme

Definition 1.3.7. Let F be a coherent sheaf on a projective scheme X ⊆ Pn. The Hilbert polynomial of Fis defined to be the polynomial PF (d) = χ(F(d)) =

∑(−1)ihi(X,F(d)), where F(d) = F ⊗OX(d).

We say POX(z) to be the Hilbert polynomial of X.

Some Remarks on Hilbert polynomial

1. For any homogenous ideal I ⊆ k[x0, . . . , xn], the Hilbert function is given by

hI(m) = dim(k[x0, . . . , xn]/I)m

We can see that POX(z) = HI(z), if X is the subscheme corresponding to I. More generally, we define the

Hilbert function of F :hF (m) := h0(X,F(m)).

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For m� 0, one has H i(F ⊗OPn(1)⊗m) = 0 for i > 0 by Serre vanishing theorem. Hence hF (m) = χ(F(m))for m� 0.

2. To see the function PF (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 = Pn−1 ↪→ Pn. Then χ(F(d)−χ(F(d−1))) = χ(G(d)).If χ(G(d)) is a polynomial, so is χ(F(d)).

Example 1.3.8. • The Hilbert polynomial of Pn is χ(O(d)) =

(n+ dn

).

• The Hilbert polynomial p(z) of a degree d hypersurface H = {f(x) = 0} in Pn is

pH(z) = pPn(z)− pPn(z − d) =

(n+ zn

)−(n+ z − d

n

)because of the exact sequence

0→ OPn(−d)×f−−→ OPn → i∗OH → 0

Two well-known results are as below:

Theorem 3. Let π : X → S be a projective morphism over a scheme. If F is a coherent sheaf on X andflat over S, then the Euler characteristic of F is constant in fibers. In particular, the Hilbert polynomial ofF is a locally constant as a function.

Theorem 4. Let X be a projective scheme over S. Let F be a coherent sheaf on X . Then for every polynomialp, there is a unique closed subscheme Sp ↪→ S with the following property: for any projective scheme T

g−→ A,the pullback F on T ×S X is flat over T with Hilbert polynomial p, then g factors through Sp.

An alternative proof of Theorem 2 (following Kollar)

Step 1.

Regard V as a vector bundle over a point. Set E = ∧rV , there is wedge product map

φ : V ⊗ E → ∧r+1V. (1.3.1)

Set K = ker(φ) ⊆ V ⊗ E. Let Gr ⊆ E be the subscheme defined by all r × r-subdetermininants of φ.

Note that

Lemma 1. Let V be a vector bundle on S and E a rank r subbundle. Then E and ∧rE are each othersannihilators under the wedge product V ⊗ ∧rE → ∧r+1E.

By the Lemma 1, we have the corank of φ is at most r. Thus φGr has constant rank r, and its kernelK is free of rank r. Regard K as a sheaf on P(E). By Theorem 4, there is a locally closed subschemeH ⊆ 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 asf : Z → Grass(r, V )→ P(E).

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Step 2: Representability of G(r, V ).

Let E ⊆ V × S be a subbundle of rank r. We obtain a subline bundle ∧rE ⊆ ∧rV which corresponds to amorphism

ϕ : S → P(E)

The pullback gives a mapV ⊗ ∧rE → ∧r+1V

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 : S → Gr(r, V ),

this gives a subbundle ρ∗K ⊆ V of rank r. Then ∧r is the unique line bundle annihilating ρ∗K. This meansρ = K.

Definition 1.3.9. Given p(z) ∈ Q[z], we define the Hilbert functor

Hilbp(Pn) : Schk → (Sets)

as follows. For S ∈ ob(Schk), it associates a set

Hilbp(Pn)(S) := {X ⊆ PnS → S flat and proper over, with Hilbert polynomial p }.

Our goal is

Theorem 5. The Hilbert functor HilbP (Pn) is representable by a projective scheme Hilbp(Pn), called theHilbert scheme of Pn with respect to p.

Proof. Let X be a closed subscheme of Pn with ideal sheaf IX . Choose N such that IY (N) is generated byglobal sections. Then

H0(Pn, IX(N)) ⊆ H0(Pn,OPn(N))

determines X becauseIX(N) = Im(OPn ×H0(Pn, IX(N))→ OPn(N))

Let q(z) be the Hilbert polynomial of IX . Then we get an injective map

{ Subschemes of Pn with Hilbert polynomial p} → Grass(q(N), H0(Pn,OPn(N))). (1.3.2)

for N � 0 by sending X to the vector space H0(Pn, IX(N)). Let us make it precise.

Step 1. For any scheme S, we let X ⊆ PnSπ−→ S be a closed subscheme, flat over S with Hilbert polynomial

p(z). Then the fiber of X → S is a closed subscheme of Pn with Hilbert polynomial p(z). As we have anexact sequence

0→ IX → OPnS → i∗OX → 0

After twisting OPnS (N) on the exact sequence, we can get an inclusion

π∗IX (N) ↪→ π∗OPnS (N)

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as a subbundle of rank q(N).

Exercise: show that π∗IX (N) is locally free for N � 0.

In this way, it defines a natural transform between functors

Hilbp(Pn)→ Gr(q(N), S).

Step 2. Setπ1 : Grass(q(N), VN )× Pn → Grass(q(N), VN )

andπ2 : Grass(q(N), VN )× Pn → Pn

to be the natural projections. If E is the universal subbundle on Grass(q(N), VN ), we set F to be the cokernelof

π∗1E → π∗2OPn(N).

Applying Theorem 4 to the sheaf F(−N) with respect to π1, there is a largest subscheme i : Gp ↪→Grass(q(N), VN ) such that F is flat over Gp with Hilbert polynomial p. As i∗F(−N) is a quotient sheaf ofOGp×Pn , it defines a subscheme U ⊆ Gp × Pn, which is flat over Gp with Hilbert polynomial p(z).

Step 3. Gp represents the functor Hilp(Pn).

If X ⊆ PnS is a closed subscheme, flat over S with Hilbert polynomial P , then we have a rank q(N) subbundle

π∗IX (N) ↪→ π∗OPnS (N) = VN × S

This corresponds to a morphismρ : S → Grass(q(N), VN )

such thatρ∗(U ↪→ VN ×Grass(q(N), VN )) = π∗IX (N) ↪→ π∗OPnS (N)

So π∗1ρ∗(F) ∼= OX (N) and ρ∗F(−N) is flat over S with Hilbert polynomial p(z). Then ρ factors through Gp.

♣

1.4 Coarse moduli space

In general, the moduli functors are not representable due to the existence of non-trivial automorphisms.

Definition 1.4.1. A coarse moduli space for a moduli functor M is a scheme M and a natural transformationof functors

η : M→M (1.4.1)

such that

• Spec k : M(Spec k)→ hM (Spec k) is bijective.

• For any scheme N and natural transformation ν : M→ hN , there exists a unique morphism of schemesf : M → N such that ν = hf ◦ η, where hf : hM → hN is the corresponding natural transformation ofpresheaves.

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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 hf , hf ′ fit into a commutative diagram

hM

hf��

M

η′}}η!!

oo // hM ′

hf ′��

hM ′ hM

(1.4.2)

As η = hf ′ ◦ hf ◦ η and η = hid ◦ η, then f ◦ f ′ = idM and f ◦ f ′ = idM ′ by Yoneda Lemma. ♣

Proposition 2. Let M be a moduli problem and suppose there exists a family F over A1 such that Fs ∼ F1

for all s 6= 0 and F0 is not equivalent to F1. Then for any scheme M and natural transformation η : M → hM ,we have ηA1(F) : A1 →M is constant. In particular, there is no coarse moduli space for this moduli problem.

Proof. Suppose we have a natural transformation

η : M→ hM ,

then η sends the family F over A1 to a morphism f : A1 → M . For any s : Spec k → A1, we havef ◦ s = η(Fs). By assumption, the restriction f |A1−{0} is a constant map. Then f is a constant map as well.However, the map

ηSpec k : M(Spec k)→ hM (Spec k) = M

is not a bijection as F0 6= F1 in M(Spec k), but these non-equivalent objects correspond to the same pointin M .

♣

Example 1.4.2. Consider the moduli functor

Mn(S) = {(E → S 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 regardMn as the moduli problem of classifying endomorphisms of a n-dimensional vector space.

Then Mn does not admit a coarse moduli space by Proposition 2. For instance, when n = 2, consider thevector bundle O⊕2

A1 over A1 and ϕ is given by the matrix

ϕ =

(1 s0 1

)Then ϕt ∼ ϕs if s, t 6= 0 and ϕ0 � ϕ1.

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 ascheme M .

Remark 4. If we consider the moduli problemMg of curves of genus g, then it is not a fine moduli problemand it only has coarse moduli space.

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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 2.1.1. An algebraic group over k is a scheme G over k with morphisms

e : Spec 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×Gm×id//

id×m��

G×Gm��

G×G m // G

; (identity) Spec k ×G //

&&

G×Gm��

G× Spec koo

xxG

and

(inverse) G(i,id)

//

��

G×Gm

��

G(id,i)oo

��

Spec ke // G Spec k

eoo

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 thegroup 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.

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Remark 5. The comultiplication, coinversion and co identity form a finitely generated Hopf algebra. Thereis a bijection between the set of affine algebraic groups and finitely generated Hopf algebra.

Example 2.1.2.

1. The additive group Ga = Spec k[t] is an affine algebraic group, whose group structure is the additiondefined by

m∗(t) = t⊗ 1 + 1⊗ t, i∗(t) = −t.

2. The multiplicative group Gm = Spec k[t, t−1] is an affine algebraic group. whose group action is givenby the comultiplication

m∗(t) = t⊗ t, i∗(t) = t−1.

3. The general linear group GLn is an open subvariety of An2, which is the completement of closed subset

defined by the determinant equationdet(xij) = 0

The comultiplication is defined by

m∗(xij) =∑t

xit ⊗ xtj

4. A linear algebraic group G is a subgroup of GLn defined by polynomial equations.

e.g. SOn and Un are linear algebraic groups.

Definition 2.1.3. An algebraic (linear) representation of an algebraic group G on a vector space V is linearmap µV : V → V ⊗O(G) satisfying

1. the composition VµV−−→ V ⊗O(G)

1⊗e∗−−−→ V is the identity.

2. the diagram

VµV //

µV��

V ⊗O(G)

µV ⊗id

��

V ⊗O(G)id⊗m∗

// V ⊗O(G)⊗O(G)

commutes.

Remark 6. The definition above is equivalent to the usual definition of a representation as a homomorphismof group valued functors

ρ : G→ GL(V ).

The equivalence is given byfij = ρ∗xij ∈ O(G)

where xij is the coordinates of GLn. So one can define µV (ei) =∑ej ⊗ fji.

Proposition 3. Every representation of G is locally finite dimensional, i.e. each point x ∈ V is containedin a finite dimensional subrepresentation.

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Proof. Write µ(x) =∑xi ⊗ fi for some xi ∈ V and linearly independent elements fi ∈ O(G). Then x is

contained in the subspace U = 〈x1, . . . xm〉. To see U is a subrepresentation, it suffices to show µ(xi) ∈ U .Now as

µ(x) =∑

µ(xi)fi =∑

xim∗(fi),

and fi are linearly independent, we get µ(xi) ∈ 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. Thevector space W spanned by a choice of generators for O(G) as a k-algebra is finite dimensional. One canview 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 andcontains W . For a basis f1, . . . , fn of V , we have

m∗(fi) ∈ O(G)⊗ V.

Write m∗(fi) =n∑j=1

aij ⊗ fj for some functions aij ∈ O(G). The action ρ : G→ GL(V ) can be described as

ρ(g)(fi) =∑

aij(g)fj .

This defines a k-algebra homomorphism

ρ∗ : O(GL(V ))→ O(G), xij 7→ aij .

To show that the corresponding morphism of affine schemes ρ : G→ GL(V ) is a closed embedding, we needto show ρ is surjective. Note that V is contained in the image of ρ∗ as

fi = (IdO(G) ⊗ e∗)m∗(fi) = (IdO(G) ⊗ e∗)∑i

aij ⊗ fj =∑i

e∗(fj)aij .

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 diagramcommutes

O(GL(V ))m∗V //

��

O(GL(V ))⊗O(GL(V ))

��

O(G) // O(G)⊗O(G)

This is equivalent to show that

m∗(aij) = m∗(ρ∗(xij)) = (ρ∗ ⊗ ρ∗)(m∗V (xij) = (ρ∗ ⊗ ρ∗)(∑

xit ⊗ xtj)

This is clear as∑ait ⊗ atj ⊗ fj =

∑m∗V (aij)⊗ fj by associativity. ♣

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Representations of torus

We say G is a torus if G = Gnm for some n. Let us start with G = Gm. The representations of G is easy todescribe. Observe that for each a ∈ Z and a vector space V , one can get a representation defined by

V → V ⊗ k[t, t−1], v 7→ v ⊗ ta,

called representation of weight a.

Proposition 4. Every representation V of Gm is a direct sum V =∑Va, where Va is a subrepresentation

of weight a.

Proof. Define Va := {v ∈ V | µ(v) = v⊗ ta}. Then Va is a subrepresenation of V of weight a. For any v ∈ V ,one can write µ(v) =

∑va ⊗ ta. One has to get v =

∑va from the 1st condition in Definition 2.1.3. Next,

from the 2nd condition in Definition 2.1.3, we have∑µ(va)t

a =∑

va ⊗ ta ⊗ ta

As ta are linearly independent, then µ(va) = va ⊗ ta. This implies va ∈ Va and proves the assertion. ♣

Definition 2.1.4. Let G be an affine algebraic group. A character of G is a function χ ∈ O(G) satisfyingm∗(χ) = χ⊗ χ and i∗(χ)(χ) = 1.

Let V be a representation of G. Then

Vχ = {v ∈ V |µ(x) = x⊗ χ}

is a subrepresentation of V of weight χ.

e.g. G = GLn, 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 subrepresentationsof weight χ

V =⊕

Vχ.

2.2 Group actions

Definition 2.2.1. An algebraic action of an affine algebraic group G on a scheme X is a morphism ofschemes σ : G×X → X such that the following diagram commute

Spec k ×X e×id//

''

G×X

��

X

; G×G×X //

��

G×X

��

G×X // X

Suppose we have an affine algebraic group G acting on two schemes X and Y . Then a morphism f : X → Yis G-equivariant if the following diagram commutes

G×X //

��

G× Y

��

X // Y

If G acts on Y trivially, we say f is a G-invariant morphism.

13

Example 2.2.2. Consider the action of Gm on An by the multiplication

Gm × An → An

(t, (a1, . . . , an)) 7→ (ta1, . . . , tan)(2.2.1)

Example 2.2.3. Let Hp(Pn) be the Hilbert scheme parametrizing closed subschemes in Pn with Hilbertpolynomial p(z). Then the group PGL(n) acts on Hp(Pn) by changing coordinates via linear transforms.

PGL(n)×Hp(Pn)→ Hp(Pn)

(f : Pn → Pn, X) 7→ f(X)(2.2.2)

Orbits and stabilisers

Definition 2.2.4. Let G be an affine algebraic group acting on a scheme X by σ : G×X → X and let x bea k-point of X.

1. The orbit G · x of x is the (set-theoretic) image of the morphism σx = σ(−, x) : G(k)→ X(k) given byg 7→ g · x.

2. The stabiliser Gx of x is the fibre product of σx : G → X and x : Spec k → X. The stabiliser Gx ofx is a closed subscheme of G (as it is the preimage of a closed subscheme of X under σx : 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 arelocally closed subsets of X. The boundary of an orbit G · x − G · x is a union of orbits of strictly smallerdimension. 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 σx and hence it isconstructible. i.e., there exists a dense open subset U of G · x with U ⊂ G · x ⊂ G · x. Because G actstransitively on G · x through σx, this implies that every point of G · x is contained in a translate of U . Thisshows 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 Gred on G·x which is transitiveon k-points. So it makes sense to talk about its dimension. The boundary of an orbit G ·x is invariant underthe action of G and so is a union of G-orbits. Since G · x is locally closed, the boundary G · x−G · x, beingthe complement of a dense open set, is closed and of strictly lower dimension than G · x. This implies thatorbits of minimum dimension are closed and so each orbit closure contains a closed orbit.

♣

Definition 2.2.5. We say the action of G on X is closed if all G-orbits are closed.

Example 2.2.6.

• Consider the action of Gm on An by the multiplication

Gm × An → An

(t, (a1, . . . , an)) 7→ (ta1, . . . , tan)(2.2.3)

The orbits are punctured lines through the origin and the origin.

14

• The action of Gm on A2 is given by t · (x, y) = (tx, t−1y). The orbits of this action are

(a) conics (x, y) : xy = a for a ∈ A1 − 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 theirorbit closures.

Proposition 7. With the assumption as above, then

1. dimG = dimGx + dim(G · x)

2. the function d : X → Z given by d(x) = dimGx is upper semi-continuous, i.e. the set

{x ∈ X : dimGx ≥ n}

is closed in X.

Proof. For (1), since the dimension is a topological invariant of a scheme, we can assume G and X arereduced. The orbit G · x, as a locally closed subscheme of X, is reduced by definition. This implies that themorphism

σx : 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 locusof σ, there exists a dense open set U such that

σ−1x (U)→ U

is flat. Using the transitive action of G on G · x, we deduce that σx is flat. Moreover, by definition, the fibreof σx at x is the stabiliser Gx. The assertion follows from the dimension formula for fibres of a flat morphism.

For (2), we can consider the map Γ : G×X (π2,σ)−−−−→ X ×X of the action and the fiber product

Pϕ

//

��

X

∆��

G×X Γ // X ×X

(2.2.4)

via the diagonal map ∆ : X → X ×X. We define the function h : P → Z by

h(g, x) = dimϕ−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

2.3 Affine GIT quotient

Categorical quotient

Let G be an affine algebraic group acting on a scheme X over k.

Definition 2.3.1. A categorical quotient for the action of G on X is a G-invariant morphism ϕ : X → Ywhich satisfying the universal property, for every G-invariant morphism X → Z, it factors uniquely throughϕ. 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 followsthat G · x = G · x = ϕ−1(x) is closed. ♣

Example 2.3.2. Consider the action of Gm on An as scalar multiplication, as the origin is in the closure ofevery single orbit, any G-invariant morphism

An → Y

must be a constant morphism. Therefore, we claim that the categorical quotient is the structure mapϕ : An → Spec k to the point Spec k. This morphism is clearly G-invariant and any other G-invariantmorphism f : An → Z is a constant morphism. Therefore, there is a unique morphism z : Spec k → Z suchthat f = z ◦ ϕ.

Good quotient and geometric quotient

Definition 2.3.3. 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 OY (U) → OX(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 W1 and W2 are disjoint G-invariant closed subsets of X, then ϕ(W1) ∩ ϕ(W2) = ∅.

(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 morphismϕ : X → Y satisfying properties i), iii), iv) and v) in the definition of good quotient. Then ϕ is a categoricalquotient. 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 allG-invariant morphisms from X.

Let f : X → Z be a G-invariant morphism. Taking a finite affine open cover Ui of Z, we set

16

• Wi := X − f−1(Ui) to a G-invariant closed subset in X;

• the image ϕ(Wi) ⊆ Y is closed by iv)

• Vi := Y − ϕ(Wi) be the open complement;

By construction, we have an inclusionϕ−1(Vi) ⊆ f−1(Ui).

As Ui cover Z, the intersection⋂iWi is empty. By property v) of the good quotient ϕ, we have

⋂ϕ(Wi) = ∅;

that is, Vi is an open cover of Y . Since f is G-invariant, the homomorphism

f∗ : OZ(Ui)→ OX(f−1(Ui))

has image in OX(f−1(Ui))G. Therefore, there is a unique morphism h∗i which makes the following square

commute

OZ(Ui)h∗i //

��

OY (Vi)

∼=��

OX(f−1(Ui))G // OX(ϕ−1(Vi))

G

where the isomorphism on the right hand side of this square is given by property iii) of the good quotient ϕ.Since Ui is affine, the k-algebra homomorphism OZ(Ui)→ OY (Vi) corresponds to a morphism

hi : Vi → Ui.

By construction, we havef |ϕ−1(Vi) = hi ◦ ϕ : ϕ−1(Vi)→ Ui

and hi = hj on Vi ∩ Vj ; therefore, we can glue the morphisms hi to obtain a morphism h : Y → Z such thatf = h ◦ ϕ. Since the morphisms hi are unique, it follows that h is also unique. ♣

Example 2.3.4. Consider the action of Gm on A2 by (λ, (x1, x2)) = (λx1, λ−1x2). As the origin is in the

closure of the punctured axes, all three orbits will be identified as a point by the categorical quotient. Thesmooth conic orbits {(x, y) : xy = a 6= 0} are closed. These conic orbits are parametrised by A1 − {0}.Therefore, we may naturally expect that

ϕ : A2 → A1

given by (x, y) 7→ xy is a categorical quotient.

(1). This morphism is clearly G-invariant and surjective, which shows parts i) and ii).

(2). Consider the morphismϕ∗ : k[z]→ k[x, y]

given by z 7→ xy. We claim that this is an isomorphism onto the ring of Gm-invariant functions.

The action of λ ∈ Gm on k[x, y] is given by

λ · x = λx, λ · y = λ−1y

Therefore, the invariant subalgebra is k[x, y]G = k[xy] as required. This verifies part iii). Also, note vi) holdstrivially.

17

(3). Note that any G-invariant closed subvariety in A2 is either a finite union of orbit closures or the entirespace A2. 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 · x1 ∩G · x2 6= ∅ if and only if ϕ(x1) = ϕ(x2)

2. for each y ∈ Y , the preimage ϕ−1(y) contains a unique closed orbit. In particular, if the action isclosed, ϕ is a geometric quotient.

Warm up: |G| is finite

Let A be a finitely generated regular k-algebra and let X = Spec A be a smooth affine variety. Let G be afinite group acting on X. We will define X/G algebraically, i.e. X/G will has a scheme structure.

Note that G acts on OX and therefore on the ring A = Γ(X,OX). Let AG be the subring generatedG-invariant classes and we define

X//G = Spec AG (2.3.1)

Theorem 8. Suppose p - |G|. The ring AG 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 AG is finitely generated. Let us assume that A = k[x1, . . . , xn] is a polynomialring and G acts on A preserving the grading. Then A =

⊕d≥0

Ad and AG =⊕AG ∩Ad.

Consider the ideal AG+ generated by AG ∩ A+, as A is Noetherian, AG+ is finitely generated by polynomialsf1, . . . , fm ∈ AG+. Claim: AG is generated by fi.

To show h ∈ AG is a polynomial of fi, we can proceed by induction on deg(h). If deg(h) > 0, we can write

h =∑

hifi

for some hi ∈ A and deg(hi) < deg(h). As |G| <∞, there is an average operator

h =1

|G|∑g∈G

g · h =1

|G|∑i

∑g∈G

(g · hi)fi

and 1|G|∑g∈G

(g ·hi) ∈ AG with degree strictly less than h. By induction, we get h can be written as a polynomial

of fi.

In general, if A is generated by a1, . . . , an, we can find a finite dimensional subspace V ⊆ A containinga1, . . . an and let b1, . . . , bm be a basis of V . So one can define a homomorphism

k[x1, . . . , xm]→ A (2.3.2)

by sending xi to bi. Then we know the pullback of the action on A preserve the grading of k[x1, . . . , xm]and k[x1, . . . , xm]G is finitely generated. The assertion then follows from the fact k[x1, . . . , xm]G → AG issurjective.

18

For the 2nd assertion, the condition is i), ii), vi) in the definition are automatically satisfied. We only needto verify for any two disjoint G-invariant closed subsets W1 and W2 in X, there is a function f ∈ O(X)G

such that f(W1) = 0 and f(W2) = 0. If such f exists, as O(X)G = O(X//G), one can regard f as a regularfunction on X//G such that f(ϕ(W1)) = 1 and f(ϕ(W2)) = 0. This implies ϕ(W1) ∩ ϕ(W2) = ∅.

To find such f , note that Wi are disjoint and closed, we have

A = I(W1 ∩W2) = I(W1) + I(W2),

and we can write 1 = f1 + f2 for some fi ∈ I(Wi). Then f1(W1) = 0 and f1(W2) = 1. The function

fG =1

|G|∑

g · f1

is a G-invariant function and fG(W1) = 0, fG(W2) = 1. This proves the assertion. ♣

General case: G is reductive

Definition 2.3.5. Let G be an affine algebraic group over k. An element g is semisimple (resp. unipo-tent) if there is a faithful linear representation ρ : G→ GLn 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-invariantvector.

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 2.3.6.

1. GL(n), SL(n) and PGL(n) are reductive algebraic groups.

2. Gm is reductive.

3. Ga is not reductive since it is unipotent.

Definition 2.3.7. An algebraic group G is said to be

• linearly reductive if for every epimorphism φ : V → W of G representations, the induced map onG-invariants φG : V G →WG is surjective.

• geometrically reductive if for every finite dimensional representation ρ : G → GL(V ) and everyG-invariant vector v ∈ V , there is a G-invariant non-constant homogenous polynomial f such thatf(v) 6= 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 V1 ⊆ Vadmits a G-stable complement (i.e. there is a subrepresentation V2 ⊆ V such that V = V1 ⊕ V2).

19

3. Every finite representation of G is completely reducible, i.e. it decomposes into a direct sum of irre-ducible representations.

4. For any finite dimensional linear representation ρ : G→ GL(V ) and every non-zero G-invariant pointv ∈ V , there is a G-invariant linear form f : V → k such that f(v) 6= 0.

Proof. The equivalence (2)⇔ (3) is clear.

For (2)⇒ (1), if φ : V →W is an epimorphism and V1 = kerφ. Then V1 has a G-stable complement V2∼= W

and φG : V G = V G1 ⊕ V G

2 →WG is surjective.

For (1)⇒ (2), let ρ : G→ GL(V ) be a finite dimensional linear representation and V1 a G-invariant subspace.Consider the sujrective map of G-representations

Hom(V, V1)→ Hom(V1, V1).

By assumption, we get a lift of the identity id : V1 → V1 ∈ Hom(V1,V1)G to a G-equivariant homomorphismφ : V → V1. Then V1 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-invariantvector. Then v determines a G-invariant linear form φ : V ∨ → k. By letting G act trivially on k, we canview φ as a surjection of G-representations. Applying (2), we get the existence of f .

(4)⇒ (2) is easy. ♣

Example 2.3.8. 1. Gm is linear reductive as every finite representation is completely reducible.

2. If cha(k) > 0, GL(n), SL(n) and PGL(n) are not linear reductive when n > 1. For instance, ifcha(k) = 2, the representation of SL(2)

ρ(a bc d

) =

1 ac bd0 a2 b2

0 c2 d2

is not completely reducible.

Proposition 10. Every reductive group G over C is linearly reductive.

Proof. We let K ⊆ G be a maximal compact subgroup.

Step 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-invariantsubspace W ⊆ V has a K-stable complement. Note that there is a K-invariant Hermitian inner product onV , as we can take any Hermitian inner product h on V and integrate over K using a Haar measure dµ on Kto obtain a K-invariant Hermitian inner product

hK(v1, v2) :=

∫K

(g · v1, g · v2)dµ

Then, we define the K-stable complement of W ⊆ V to be the orthogonal complement of W with respect tothis K-invariant Hermitian inner product.

20

Step 2. For G reductive and a maximal compact subgroup K ⊆ G, the elements of K are Zariski dense inG (one may use Lie algebra and Identity theorem).

Step 3. For any finite dimensional linear representation ρ : G→ GL(V ), we claim that V G = V K , where Kis a maximal compact of G. Clearly, we have V G ⊆ V K . To prove the reverse inclusion, let v ∈ V K and weconsider the morphism

σ : G→ V

given by g → ρ(g)(v). Then σ−1(v) ⊆ G is Zariski closed. Since v ∈ V K , we have K ⊆ σ−1(v) andK ⊆ σ−1(v). Note that K ⊆ G is Zariski dense, it follows that G ⊆ σ−1(v); that is, v ∈ V G as required.

Step 4. By Proposition 4.14, it suffices to show for every epimorphism φ : V → W , the induced homo-morphism φG on invariant subspaces is also surjective. By Step 3, this is equivalent to showing that φK issurjective. ♣

Indeed, we have the following theorem towards the relation between various definitions.

Theorem 9. For smooth affine algebraic group schemes, we have

Linearly reductive ⇒ Geometrically reductive ⇔ Reductive.

All three notions coincide in characteristic zero.

Reynolds operator

Definition 2.3.9. For a group G acting on a k-algebra A, a linear map RA : A→ AG is called a Reynoldsoperator if it is a projection onto AG and for a ∈ AG and b ∈ A, we have RA(ab) = aRA(b).

Lemma 4. Let G be a linearly reductive group acting rationally on a finitely generated k-algebra A; thenthere exists a Reynolds operator RA : A→ AG.

Proof. Since A is finitely generated, it has a countable basis. Therefore, we can write A as an increasingunion of finite dimensional G-invariant vector subspaces An ⊆ A constructed as below:

Let a1, a2, . . . be the basis. Then we can iteratively construct the subsets An by letting An be the finitedimensional G-invariant subspace containing a1, . . . , an, a basis of An−1 and aj∆An−1 for j = 1, . . . , n. ThenA =

⋃n≥1

An. Since G is linearly reductive and each An is a finite dimensional G-representation, we can write

An = AGn ⊕A′n

where A′n is the direct sum of all non-trivial irreducible G-subrepresentations of An. Set

Rn : An → AGn

be the canonical projection onto the direct factor AGn . For m > n, we have a commutative square

AnRn //

��

AGn

��

AmRm // AGm

21

as we have A′n ⊆ A′m and AGn ⊂ AGm. So we obtain a linear map

RA : A→ AG

given by the compatible projections Rn : An → AGn for each n.

It remains to check that for a ∈ AG and b ∈ A, we have RA(ab) = aRA(b). Pick n such that a, b ∈ An andpick m ≥ n such that a(An) ⊆ Am. Then consider the homomorphism

la : An → Am

of G−representations given by left multiplication by a. Write

A′n = W1 ⊕ · · · ⊕Wrn

as a direct sum of non-trivial irreducible subrepresentations Wi ⊆ An. Since G acts by algebra homomor-phisms and a ∈ AG, we have la(A

Gn ) ⊆ AGm. Moreover, the image of each irreducible Wi under la is either

zero or isomorphic to Wi. Therefore, we have la(Wi) ⊆ A′m. This implies that la(A′n) ⊆ A′m. In particular,

if we write b = bG + b′ for bG ∈ AGn , b′ ∈ A′n, then

ab = la(b) = la(bG) + la(b

′) = abG + la(b) ∈ AGm ⊕A′m

and RA(ab) = Rm(ab) = abG = aRA(b) as required. ♣

Using the Reynolds operator, the same argument gives

Theorem 10. Suppose G is linear reductive. The ring AG is finitely generated. In particular, X//G is anaffine 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 AG for reductive groups. Thereductivity is the optimal condition for AG 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 generatedk-algebra A, AG is finitely generated.

Stable points and geometric quotient

Definition 2.3.10. We say x ∈ X is stable if its orbit is closed in X and dimGx = 0 (or equivalently,dimG · x = dimG). We let Xs denote the set of stable points.

Example 2.3.11. Consider Gm acting on X = A2 via automorphisms. Then Xs = {xy 6= 0}.

Theorem 12. Suppose a reductive group G acts on an affine scheme X and let φ : X → Y = X//G be theaffine GIT quotient. Then Xs ⊆ X is an open and G-invariant subset, Y s = φ(Xs) is an open subset of Yand Xs = φ−1(Y s). Moreover, Xs → Y s is a geometric quotient.

Proof. First, we show that Xs is open. For any x ∈ Xs, by semicontinuity, the set

X+ := {x ∈ X : dim(G · x) > 0}

22

of points with positive dimensional stabilisers is a closed subset of X. As we proved in Theorem 10, there isa function f ∈ O(X)G such that

f(X+) = 0, f(G · x) = 1.

So we have x ∈ Xf . We claim that Xf ⊆ Xs. Since all points in Xf have stabilisers of dimension zero, itremains to check that their orbits are closed.

Suppose z ∈ Xf has a non-closed orbit and w /∈ G · z belongs to the orbit closure of z. Then w ∈ Xf as fis G-invariant and so w must have stabiliser of dimension zero. However, we know that the boundary of theorbit G · z is a union of orbits of strictly lower dimension and so the orbit of w must be of dimension strictlyless than that of z which contradicts the fact that w has zero dimensional stabiliser. Hence, Xs is an opensubset of X and is covered by open subsets Xf . ♣

2.4 Projective GIT quotient

Construction of Proj

A Z-graded ring is a ring

S =⊕n

Sn,

where multiplication respects the grading, i.e. sends Sm×Sn to Sm+n. Clearly S0 is a subring, each Sn is anS0-module, and S is a S0-algebra. An ideal I of S is a homogeneous ideal if it is generated by homogeneouselements.

Definition 2.4.1 (Proj construction.). Write S+ :=⊕d>0

Sd. If S+ is a finitely generated ideal, We say S

is finitely generated graded ring over S0. Then X = Proj(S) is the collection of homogenous primes idealsof S not containing the irrelevant ideal S+. The closed subsets on X are the projective vanishing set V (I),where I ⊆ S+. We call this the Zariski topology on Proj(S).

To give the ringed space structure, we take the projective distinguished open set Xf = Proj(S)\V (f) (theprojective distinguished open set) be the complement of V (f). One can show that Xf is isomorphic toSpec((Sf )0). Hence it gives arise a natural ringed space structure on X.

Projective quotient

Definition 2.4.2. 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→ GLn+1 and aG-equivariant projective embedding X → Pn, or equivalently we say the G-action on X is linear.

• For X ⊆ Pn, we set R(X) = k[x0, . . . , xn]/I(X). If G acts linearly on X ⊆ Pn, we let R(X)G bethe G-invariant subalgebra. Define the nullcone N to be the closed subscheme of X defined by thehomogeneous ideal R(X)G+ in R(X).

• We define the semistable set Xss = X −N to be the open subset of X given by the complement to thenullcone. x ∈ Xss is called semistable if there exists a G-invariant homogeneous function f ∈ R(X)Grfor r > 0 such that f(x) 6= 0.

23

• We call the morphism Xss → X//G := Proj(R(X)G) the GIT quotient of this action.

Theorem 13. The GIT quotientφ : Xss → X//G

is a good quotient of the G-action on the open subset Xss of semistable points in X. Furthermore, X//G isa 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 Spec k. As R(X)G is a finitely generated k-algebra, we canpick generators f1, . . . , fr in degrees d1, . . . , dr. Let d = d1 · · · dr; then (R(X)G)(d) =

⊕l≥0

R(X)Gdl is finitely

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 Yf ⊆ Y form a basis ofthe open sets on Y . Since f ∈ R(X)G+ ⊆ R(X)+, we consider the open affine subset Xf ⊆ X and we haveφ−1(Yf ) = Xf . Then

O(Yf ) = ((R(X)G)f )0 = ((R(X)f )0)G = (O(Xf )0)G ∼= O(Xf )G

and so the corresponding morphism of affine schemes φf : Xf → Yf = Spec O(Xf )G is an affine GIT quotient,and so also a good quotient by Theorem 10. The morphism φ : Xss → Y is obtained by gluing the goodquotients φf : Xf → Yf . As being a good quotient is a local property, we can conclude that φ is a goodquotient as well.

♣

Definition 2.4.3. Consider a linear action of a reductive group G on a closed subscheme X ⊆ Pn. Then apoint x ∈ X is

1. stable if dimGx = 0 and there is a G-invariant homogeneous polynomial f ∈ R+(X)G such thatx ∈ Xf and the action of G on Xf 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 suchthat φ−1(Y s) = Xs and that the GIT quotient restricts to a geometric quotient φ : Xs → Y s.

Proof. Let Yc be the union of Yf for f ∈ R(X)G+ where the G-action on Xf is closed. Let Xc be the unionof Xf such that Xc = φ−1(Yc). Then φ : Xc → Yc is constructed by gluing

φf : Xf → Yf

for f ∈ R(X)G+. Each φf is a good quotient and as the action on Xf is closed, then φf is also a geometricquotient by Corollary 1. It follows that φ : Xc → Yc is a geometric quotient.

By definition, Xs is the open subset of Xc consisting of points with zero dimensional stabilisers and we letY s = φ(Xs) ⊆ Yc. As φ : Xc → Yc is a geometric quotient and Xs is a G-invariant subset of X, then Y s isopen in Yc. So we can conclude that Y s ⊆ Y is open. Finally, the geometric quotient φ : Xc → Yc restrictsto a geometric quotient φ : Xs → Y s.

♣

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Remark 8. In general, a geometric quotient (i.e. orbit space) does not exist because the action is notnecessarily closed. For finite groups G, every good quotient is a geometric quotient as the action of a finitegroup is always closed.

Lemma 5. Let G be a reductive group acting linearly on X ⊆ Pn. A k-point x ∈ X is stable if and only ifx is semistable, its orbit G · x is closed in Xss and its stabiliser Gx is zero dimensional.

Proof. Suppose x is stable and x′ ∈ G · x ∩ Xss; then φ(x′) = φ(x) and x′ ∈ Xs. As G acts on Xs withzero-dimensional stabiliser, this action must be closed as the boundary of an orbit is a union of orbits ofstrictly lower dimension. Therefore, x′ ∈ G · x and so the orbit G · x is closed in Xss.

Conversely, we suppose x is semistable with closed orbit in Xss and zero dimensional stabiliser. As x issemistable, there is a homogeneous f ∈ R(X)G+ such that x ∈ Xf . As G · x is closed in Xss, it is also closedin the open affine set Xf ⊆ Xss. By semi-continuity, the G-invariant set

Z := {z ∈ Xf | dimGz > 0}

is closed in Xf . Since Z is disjoint from G · x and both sets are closed in the affine scheme Xf , there existsh ∈ O(Xf )G such that h(Z) = 0 and h(G · x) = 1.

Write h = h′

fr for some G-invariant homogeneous polynomial h′ ∈ R(X)G+. Then x ∈ Xfh′ and Xfh′ is disjointfrom Z. Then all the orbits in Xfh′ are closed and hence x is stable. ♣

Definition 2.4.4. A k-point x ∈ X is said to be polystable if it is semistable and its orbit is closed in Xss.We say two semistable k-points are S-equivalent if their orbit closures meet in Xss. One can see that everystable k-point is polystable.

Proposition 11. Let x ∈ X be a semistable k-point; then its orbit closure G · x contains a unique polystableorbit. 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 S-equivalence.

Proof. As φ is constant on the closure of the orbits, φ−1(φ(x)) contains a unique closed orbit of a polystablepoint xp. If x is not stable, then dimG · xp < dimG · x. This implies xp can not be stable. ♣

Linearisations

There is a more general theorem for constructing GIT quotients of reductive group actions on quasi-projectiveschemes with respect to linearisations.

Definition 2.4.5. Let L be a line bundle on X and we set

R(X,L) :=⊕r≥0

H0(X,L⊗r)

to be the section ring.

Suppose G acts linearly on X via σ. A linearisation of the G-action on X is a line bundle π : L → X overX with an isomorphism of line bundles

π∗XL∼= σL

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where πX : G×X → X is the projection, such that the diagram commutes

G×G× Lµ×id��

idG×σ// G× L

σ��

G× L σ // L

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(H0(X,L)).

Proof. We consider the mapH0(X,L)→ O(G)⊗H0(X,L)

via the composition

H0(X,L)σ∗−→ H0(G×X,σ∗L) ∼= H0(G×X,G× L) ∼= H0(G,OG)⊗H0(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-invariantsections

R(X,L)G =⊕r≥0

H0(X,L⊗r)G

Definition 2.4.6. Let X be a projective scheme with an action by a reductive group G and let L be anample linearisation.

(i) A point x ∈ X is semistable with respect to L if there is an invariant section σ ∈ H0(X,L⊗r)G forsome r > 0 such that σ(x) 6= 0 and Xσ = {x ∈ X : σ(x) 6= 0} is affine.

(ii) A point x ∈ X is stable with respect to L if dimG ·x = dimG and σ ∈ H0(X,L⊗r)G for some r > 0such that σ(x) 6= 0, Xσ = {x ∈ X : σ(x) 6= 0} is affine and the action of G on Xσ is closed.

The open subsets of stable and semistable points with respect to L are denotedXs(L) andXss(L) respectively.Then we can define the projective GIT quotient with respect to L to be the map

Xss(L)→ Proj(R(X,L)G)

Theorem 15. The GIT quotient Xss(L) → X//LG is a good quotient and X//LG is a projective schemewith a natural ample line bundle L′. Furthermore, there is an open subset Y s ⊆ X//LG such that φ−1(Y s) =Xs(L) and φ : Xs(L)→ Y s is a geometric quotient.

2.5 Hilbert-Mumford Criteria

Definition 2.5.1. Let G be a reductive group acting linearly on a projective scheme X with respect to anample linearisation L.

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• A 1-parameter subgroup (1-PS) of G is a non-trivial group homomorphism

λ : Gm → G.

• If a 1-PS λ : Gm → X extends to a morphism A1 → X, then the image of 0 ∈ A1 is called the limit ofλ as t→ 0, denoted by lim

t→0λ(t) · x.

• The Hilbert-Mumford weight of the action of the 1-PS λ on x ∈ X with respect to L is

µL(x, λ) = r

where r is the weight of the λ(Gm) on the fibre Ly over y = limt→0

λ(t) ·x, where λ(Gm) acts by a character

t 7→ tr.

Theorem 16. With the assumption as above,

x ∈ Xss(L)⇔ µL(x, λ) ≥ 0 for all 1-PS λ of G,

x ∈ Xs(L)⇔ µL(x, λ) > 0 for all 1-PS λ of G.

For the proof, we divide it into three parts.

1. (Reduction to X = Pn) Proof. Ex.

2. (Topological criterion, reduced to An)The action of G on Pn natural lifts to the affine cone An+1.

Theorem B. Let x ∈ Pn and choose a non-zero lift x ∈ An+1. Then

1. x is semistable iff 0 /∈ G · x

2. x is stable iff dimGx = 0 and G · x is closed.

In particular, if G = λ is a 1-PS, x is semistable if and only if µL(x, λ) ≥ 0 and µL(x, λ−1) ≥ 0 x is stableif and only if µ(x, λ) > 0 and µ(x, λ−1) > 0.

Proof.

• If x is semistable, we can find a homogeneous polynomial f ∈ k[x0, . . . , xn]G such that f(x) 6= 0. As f isconstant on G · x, we know f(G · x) 6= 0. As f(0) = 0, we get 0 /∈ G · x.

Conversely, if 0 /∈ G · x, then there exists a function f ∈ k[x0, . . . , xn]G such that f(G · x) = 1 and f(0) = 0.Write f =

∑fi for homogeneous polynomials fi, then fi is G-invariant as the action of G is linear. Then

there exists some fi such that fi does not vanish on G · x. This means x is semistable.

• If x is stable, then as Gx ⊆ Gx, we have dimGx = 0. Since there exists a homogeneous polynomialf ∈ k[x0, . . . , xn]G such that f(x) 6= 0, we can consider the closed subscheme

Z = {z ∈ An+1| f(z) = f(x)}.

The projection π : Z → Pnf is a surjective and finite morphism. Note G · x ⊆ Z, it suffices to show G · x is

closed in Z. As G · x is closed, π−1(G · x) is closed and it is a finite union of G-orbits of dimension G. Itforces all these orbits to be closed.

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Conversely, if dimGx = 0 and G · x is closed, then x semistable and there is a non-constant G-invariantpolynomial f such that f(x) 6= 0. The morphism π : Z → Pnf is finite and surjective. Then

dimG · x = dimG · x.

At last, G · x = π(G · x) is closed in Pnf and hence in (Pn)ss.

• For the last assertion, we pick a basis {ei} of An+1 such that λ(t) · ei = triei. Write x =∑aiei, then we

have

0 /∈ λ(Gm) · x iff limt→0

∑iait

riei 6= 0 and limt→∞

∑iait

riei 6= 0

This is equivalent to µL(x, λ) ≥ 0 and µL(x, λ−1) ≥ 0.

For x being stable, as the orbit is closed iff the boundary is empty. This means the limits limt→0

∑ait

riei and

limt→∞

∑ait

riei do not exist. This is equivalent to µL(x, λ) > 0 and µL(x, λ−1) > 0.

3. (Fundamental theorem)

Suppose the results below holds, then x is semistable ⇔ 0 /∈ G · x ⇔ there does not exist 1-PS λ such thatlimt→0

λ(t) · z = 0 ⇔ µL(x, λ) ≥ 0 for all λ.

And x is stable ⇔ dimGx = 0, G · x is closed ⇔ for all 1-PS λ, the limit limt→0

λ(t) · z does not exist.

Theorem C. If G acts linearly on An and z ∈ An with y ∈ G · z, then there exists a 1-PS such thatlimt→0

λ(t) · z = y.

Proof. For simplicity, we assume y = 0.

1. Suppose 0 ∈ G · z. By Bertini theorem, there is an irreducible curve C1 ⊆ G · z, which contains 0 in itsclosure.

2. Consider the morphismσz : G→ An

given by g → g ·z and we can find a curve C2 in G which dominates C1 under σz. Let C be a projectivecompletion of the normalisation C2 → C2; then we have a rational map

ϕ : C 99K G

defined by the morphism C2 → C2 → G. As the morphism C2 → C1 is dominant, it extends to theirsmooth projective completions and as 0 lies in the closure of C1, we can take a preimage c ∈ C of zerounder this extension. Then we have

limx→c

ϕ(x) · z = 0

3. Since C is a smooth proper curve, the completion of the local ring OC,c is isomorphic to the formalpower series ring k[[t]], whose field of fractions is the field of Laurent series k((t)). We fix the notations

R = k[[t]]

K = k((t))

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As the rational map ϕ : C 99K G is defined in a punctured neighbourhood of c and it induces amorphism

ψ : Spec K → Spec R→ G(R)

such that limt→0

ψ(t) · z = 0.

4. By definition, G(R) is a subgroup of G(K). And we have a specialization map G(R)→ G(k) by lettingt goes to 0. There is a morphism

Spec K → Gm = Spec k[s, s−1]

induced by the ring homomorphism s 7→ t. For a 1-PS λ, we define 〈λ〉 ∈ G(K) to be the compositionSpec K → Gm → G(K).

5. (Iwahori Theorem) Every double coset in G(K) for the subgroup G(R) is represented by a Laurentseries expansion 〈λ〉 for some 1-PS λ of G.

We prove it for the case G = GLn. A K-point of G(K) is a matrix A with entries in K. Write A = trAwith A ∈ G(R). Since R is PID (or by linear algebra), we can diagonalize A. This means that we canfind M1,M2 ∈ G(R) such that M1AM2 is a diagonal matrix diag[tr1 , . . . , trn ]. The 1-PS of G is thengiven by

λ : Gm → G

t 7→ diag[tr1 , . . . , trn ].

♣Now, as ψ ∈ G(K), there exists li ∈ G(R) for i = 1, 2 and a 1-PS λ of G such that

l1 · ψ = 〈λ〉 · l2.

6. Let gi = li(0) ∈ G, then we have

0 = g1 · 0 = limt→0

l1(t) · lim(ψ(t) · z) = limt→0

[(〈λ〉 · l2)(t) · z].

The action of the 1-PS λ on An decomposes into weight spaces. Since l2 ∈ G(R) and g2 = limt→0

l2(t), we

can writel2(t) · z = g2 · z + ε(t),

where ε(t) only contains strictly positive powers of t. Then with respect to the weight space decompo-sition, we have

g2 · z + ε(t) =∑r

(g2 · z)r + ε(t)r, r ∈ Z.

It follows that (g2 · z)r = 0 for all r ≤ 0 and limt→0

λ(t) · g2 · z = 0. Then the 1-PS λ′ := g−12 λg2 satisfies

limt→0

λ′(t) · z = 0.

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