On the moduli space of semistable sheaves

Author:

Matija Tapušković

Supervisor:

Prof. Qing Liu

Master Thesis

July 2016

1

Contents

Introduction 4

1 Moduli problems 7

1.1 The moduli functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Coarse moduli space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Fine moduli space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Pathological cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 The Quot scheme 14

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 The Quot functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 The Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Representability theorem of Altman and Kleiman . . . . . . . . . . . . . . . . . . . 22

2.5 Construction of the Quot scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Geometric Invariant Theory 30

3.1 Actions of algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Nice quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Hilbert’s 14th problem and Nagata’s theorem . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Constructing the projective GIT quotient . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 G-linearization of quasi-coherent OX -modules . . . . . . . . . . . . . . . . . . . . . . 39

3.6 The Hilbert-Mumford criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Stability of sheaves 47

4.1 Notions of stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Filtrations of sheaves and S-equivalence . . . . . . . . . . . . . . . . . . . . . . . . . 50

2

5 Construction of the moduli space of semistable sheaves 52

5.1 Boundedness of semistable sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 The moduli functor of semistable sheaves . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3 Semistable sheaves as points on a Quot scheme . . . . . . . . . . . . . . . . . . . . . 59

5.4 Preliminaries for the GIT construction . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.5 Linearizing the action of SL(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.6 Construction of the moduli space of semistable sheaves . . . . . . . . . . . . . . . . 64

5.6.1 Statement of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.6.2 Analysis of GIT semistability . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.6.3 Analysis of sheaf semistability - Le Potier’s theorem . . . . . . . . . . . . . 69

5.6.4 Proof of Theorem 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Conclusion 74

3

Introduction

Moduli problems are, in a certain sense, extended classification problems. Suppose we have

defined a certain class M of geometric objects that we wish to classify up to some defined

equivalence. We might ask if some geometric space, such as that of a scheme or a variety, might

exists, such that its points correspond toMmodulo the equivalence. Moreover, since in algebraic

geometry many objects of interest vary in algebraically defined families, it would be useful if

the geometry of this space was in some way natural with respect to these families. If we can

construct such a space, we may obtain additional information about the objects that we started

with by studying the global and local geometry of the space parametrizing them. One can bear

in mind some simple examples: the j-invariant classifying elliptic curves up to isomorphism, the

Grassmannian variety, which prametrizes linear subspaces with a given dimension of a vector

space V , or the Jacobian variety of a non-singular projective algebraic curve, which parametrizes

degree 0 line bundles on the given curve.

In fact, a more general version of this last example — the moduli space of vector bundles on

a curve of a given rank and degree (or equivalently, of a given first Chern class) — was studied

by Mumford in the early ’60s. This led him to the concept of stability of vector bundles and

Geometric Invariant Theory, which is still an invaluable tool in the theory of moduli. Attention

then naturally turned to vector bundles on surfaces, and to higher dimensional varieties. In fact,

the problem of determining which cohomology classes on a projective variety can be realized

as Chern classes of vector bundles is a profound and important problem in algebraic geometry.

It is natural then to try to describe the moduli space of vector bundles with a fixed rank and

Chern classes on an arbitrary projective variety.

As additional motivation, it is worth mentioning that the study of moduli spaces often

appears in the applications of algebraic geometry to number theory, such as in the work of

Faltings on the original proof of the Mordell Conjecture, as well as in Wiles’ proof of Fermat’s

Last Theorem. Moreover, a shining example of its use in connection to differential geometry

and mathematical physics is in the work of Donaldson, who used the moduli space of instantons

(certain connections in a principal bundle over a 4-dimensional Riemannian manifold) to define

4

certain invariants as the intersection numbers on the moduli space. These invariants depend on

the differentiable structure of the manifold, and can be used to construct homeomorphic smooth

4-manifolds which are not diffeomorphic.

If the previous discussion provides ample motivation to study moduli spaces, it also hints

at the vastness of the field. Therefore, we must focus in this text on a manageable selection of

topics. In Chapter 1, a general introduction to the theory of moduli spaces is made, along with

some simple examples. In Chapter 2, we first introduce some preliminaries, most importantly the

notion of flat families, and then turn to our first serious, and highly useful, example of a moduli

space — the Quot scheme of Grothendieck. The Quot scheme parametrizes quotients of a fixed

coherent sheaf on a fixed projective scheme. Its existence was demonstrated by Grothendieck in

his Séminarie Bourbaki talks, and has since been used extensively in constructing other moduli

spaces. Its special case is the Hilbert scheme, which parametrizes closed subschemes of a given

scheme. The Hilbert scheme is important in its own right, as illustrated for example by the work

of Beauville, who studied the Hilbert scheme of n points on a K3 surface, and proved that it is

an irreducible hyperkähler manifold, thus giving the first example of such a manifold in higher

dimensions ( dim > 2). This example also illustrates the common situation of moduli spaces

having a rich and interesting geometry. The Quot scheme plays a crucial role in the rest of the

text, as our main result is obtained by taking its quotient by a certain action of an algebraic

group. However, before we can get to that, we must turn to Mumford and his Geometric

Invariant Theory in Chapter 3, as an understanding of its inner workings is necessary for the

main construction. Next, in Chapter 4, the various notions of stability for sheaves are introduced

and briefly discussed. It is necessary to limit our attention to the family of semistable sheaves

on the fiexed projective variety, in order to obtain boundedness, which we do at the beginning

of Chapter 5, following the arguments of Le Potier [10] and Simpson [18]. This result enables us

to see a semistable sheaf as a point on a certain Quot scheme, and to follow Simpson’s approach

in constructing the moduli space.

The main sources for this thesis were N. Nitsure’s article on the construction of the Quot

scheme [16], Mumford’s book on GIT [13], and Huybrechts and Lehn’s book [8] on the construc-

tion of the moduli space of semistable sheaves. The latter contains all the prerequisites for the

main result, though in a compressed form, which is why the other two books proved very useful.

Additionally, Le Potier and Newstead’s books [15, 9] , which deal with vector bundles on an

algebraic curve (with a chapter on semistable sheaves on the projective plane in [9]), were helpful

in grasping the special case, so that one might more easily understand the general one, which is

the topic of this text. Finally, Liu’s book [11] was essential in grasping the necessary elements

5

of scheme theory, as it was the author’s companion through his first course on the topic, and

throughout the writing of this text.

6

Chapter 1

Moduli problems

Let us give some elementary definitions, in order to formalise the concepts from the introduction

in the language of category theory. We will then illustrate them with simple examples.

1.1 The moduli functor

Definition 1. Let M be a collection of objects, and ∼ an equivalence relation defined on M.

For any S-scheme T , let MT be the set of families parametrized by T , with an equivalence

relation ∼T defined onMT . Then, a moduli functor is a contravariant functor from the category

of S-schemes, which we will denote Sch/S to the category of sets (i.e. a presheaf on Sch/S)

M ∶ (Sch/S)o → Set

which to each S-scheme T associates the set of equivalence classes of families of elements ofM

parametrized by T , and to each morphism of S-schemes T → T ′ a pullback map f∗ ∶ M(T ′) →

M(T ).

Example 1. (The Grassmannian of a vector bundle) Let S be a scheme, V a vector bundle on

S and r a positive integer less then the rank of V . Define

Grass(S,V, r) ∶ (Sch/S)o → Set

be the contravariant functor that associates to an S-scheme T vector subbundles of T ×S V

of rank r, with the equivalence relation given by equality, and to a morphism f ∶ T → T ′ of

S-schemes associates the pullback f∗.

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1.2 Coarse moduli space

Definition 2. Given a moduli functor M we say that an S-scheme M is a coarse moduli scheme

for the functor M (or the class of objectsM) if there is a natural transformation of functors

Φ ∶M → HomS(−,M)

such that:

1. Φ(Speck) ∶ M(Speck) → HomS(Speck,M) is a bijection for every algebraically closed

field k,

2. For any S-scheme N and any natural transformation

Ψ ∶M → HomS(−,N)

there is a unique morphism of S-schemes e ∶M → N such that Ψ = he ○Φ

If only condition 2. is satisfied we say that M corepresents the functor M .

Remark 1. Let us unravel this categorical definition a bit. Consider the case when our base

scheme is simply a field k. Then condition 1. tells us that the elements of M, considered

as trivial families over Speck are in one-to-one correspondence with the rational points of M .

From the fact that Φ is a morphism of functors we get that for any family E ∈M(T ) its image

ϕE ∈ Homk(T,M) has the property that at each rational point t ∈ T , the point ϕE(t) ∈ M

corresponds to the equivalence class of the fiber Et ∈ M. We get this from the commutative

square 1.1 that is obtained when considering functoriality for the morphism q ∶ Speck → T

sending the point to t, combined with point 1. The condition 2. implies uniqueness up to

unique isomorphism of M if it exists: let (M,Φ) and (M ′,Φ′) be two coarse moduli spaces,

then by 2. we have morphisms f ∶ M → M ′ and f ′ ∶ M ′ → M . Since Φ = hf ′ ○ hf ○ Φ and

Φ = hidM ○ Φ we have by the uniqueness in 2. and the Yoneda lemma that f ′ ○ f = idM , and

similarly for f ○ f ′ = idM ′ .

M(T ) Homk(T,M)

M(Speck) Homk(Speck,M)

q∗ −○q (1.1)

Definition 3. IfM is a coarse moduli scheme for the moduli problemM, we define a tautological

family for M to be the family T such that for each k-point m, for any algebraically closed

field k, the fiber Tm is the element of M corresponding to m by the bijection M(Speck) →

HomS(Speck,M) above.

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Example 2. Let us consider now a simple example of a coarse moduli space. LetM consist of

quadruples of ordered distinct points on the projective line. We recall that the automorphism

group of P1 is the projective linear group PGL2, and we define that two ordered quadruples of

distinct points (p1, p2, p3, p4) and (q1, q2, q3, q4) on P1 are equivalent ((p1, ..., p4) ∼ (q1, ..., q4)) if

there exists an automorphism f ∶ P1 → P1 such that f(pi) = f(qi) for all i = 1, ...,4. We recall

that for any 3 distinct points (p1, p2, p3) on P1, there exists a unique automorphism f ∈ PGL2which sends (p1, p2, p3) to (0,1,∞). Furthermore, we have a projective invariant associated to

any quadruple of distinct points on the projective line, the cross-ratio, defined by

(p1p2 ∶ p3p4) =(ξ1µ3 − ξ3µ1)(ξ2µ4 − ξ4µ2)(ξ1µ4 − ξ4µ1)(ξ2µ3 − ξ3µ2)

where pi = [ξi, µi], i = 1, ...,4. We can see that this is invariant by the action of PGL2 as

scaling any pair [ξi, µi] scales the numerator and the denominator by the same factor, and any

projective transformation⎛⎜⎝

ξi ξj

µi µj

⎞⎟⎠↦⎛⎜⎝

a b

c d

⎞⎟⎠

⎛⎜⎝

ξi ξj

µi µj

⎞⎟⎠

scales the cross terms (ξiµj − ξjµi) by the nonzero determinant ad − bc.

Now, as we said, we can move p1, p2, p3 by some element of f ∈ PGL2 to [1,0], [0,1], [1,1],

leaving the cross-ratio unchanged. Then

(p1p2 ∶ p3p4) = ([1,0][0,1] ∶ [1,1]f(p4)) =(1 ⋅ 1 − 0 ⋅ 1)(0 ⋅ µ̂4 − ξ̂4 ⋅ 1)(1 ⋅ µ̂4 − 0 ⋅ µ̂4)(0 ⋅ 1 − 1 ⋅ 1)

= ξ̂4µ̂4

where f(p4) = [ξ̂4, µ̂4]. Thus we get that the cross-ratio is given by f(p4) ∈ P1 − (0,1,∞).

Therefore, we see that the setM/ ∼ is in bijection with the set of k-points in the quasi-projective

variety P1 − (0,1,∞).

We can speak naturally about families of 4 distinct points on P1 over a scheme S. A family is

given by a proper flat morphism π ∶X → S, such that the fibers π−1(s) ≅ P1 are smooth rational

curves, and 4 disjoint sections (σ1, ..., σ4) of π. We say that two families (π ∶ X → S,σ1, ..., σ4)

and (π′ ∶ X ′ → S,σ′1, ..., σ′4) are equivalent if there is an isomorphism f ∶ X → X ′ over S (i.e.

π = π′ ○ f) such that f ○ σi = σ′i.

There exists a tautological family parametrized by P1−(0,1,∞). Let π ∶ P1−(0,1,∞)×P1 →

P1−(0,1,∞) be the projection map and choose sections (σ1(s) = 0, σ2(s) = 1, σ3(s) =∞, σ4(s) =

s). This family is in fact a universal family which makes P1 − (0,1,∞) a fine moduli space -

both to be defined shortly.

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1.3 Fine moduli space

Definition 4. If the functor M is isomorphic to a functor of the form hM = Hom(−,M), then

we say that M is a representable functor, or that it is represented by M . In the case of moduli

functors we call the scheme M that represents a moduli functor M a fine moduli space for M

(or forM).

Definition 5. If M → Hom(−,M) is an isomorphism, then in particular we have a bijection

M(M)→ Hom(M,M), and there is a family U parametrized by M corresponding to the identity

map idM ∈ Hom(M,M). U is called the universal family associated to the fine moduli space

M . This family has the following special property. As any family E ∈ M(T ) parametrized by

a S-scheme T corresponds to a unique morphism p ∶ T → M , called the classifying morphism,

via Φ, and as the families p∗U and E correspond to the same classifying morphism idM ○ p = p,

because Φ is a natural transformation of functors we have

p∗U = E

that is any family can be obtained by pulling back the universal family. Conversely, if there

exists a scheme M parametrizing a family U with such a property, then M represents the functor

M .

Remark 2. If a moduli functor has a fine moduli space, then it is also a coarse moduli space

for that functor. In fact, we have the bijection M(Speck) → Hom(Speck,M) for any al-

gebraically closed k, since M represents the functor M , and one easily checks the universal

property. Conversely, if we have a coarse moduli space (M,Φ) such that there exists a family

U over M such that ΦM(U) = idM and for any two families F ,G over some scheme S we have

F ∼S G ⇐⇒ ΦS(F) = ΦS(G), then M is a fine moduli space.

1.4 Pathological cases

Of course, a fine moduli space may fail to exist. This is why we weaken our requirements and

come to the notion of a coarse moduli space. However, sometimes even a coarse moduli space

may fail to exist for a problem. In this subsection, we will consider a couple of pathologies which

prevent a moduli functor from having a moduli space.

The first situation in which a coarse moduli space fails to exist is when we have a so called

jump phenomenon for our moduli problem. Consider the following lemma describing this situa-

tion.

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Lemma 1. Let M be a moduli functor and suppose there exists a family F ∈M(T ), where T is

an integral scheme of dimension at least 1 of finite type over k, such that Ft ∼ Ft1 for all t /= t0and Ft0 /∼ Ft1 (i.e. all fibers over T are isomorphic except one). Then for any scheme M and

natural transformation Φ ∶ M → hM we have that the classification morphism ΦT (F) ∶ T → M

is constant. In particular, there is no coarse moduli space for this problem.

Proof. By assumption we have a natural transformation Φ ∶M → hM , so we consider a morphism

f ∶ T → M given by ΦT (F). For any t ∶ Speck → T we have that f ○ t = ΦSpeck(Ft) by

functoriality, and Ft = Ft1 ∈ M(Speck), for t /= t0, so we have that f ∣T−{t0} is a constant map.

Let m ∶ Speck →M be the point corresponding to the equivalence class of Ft1 under Φ. Since the

M is integral of finite type over k all k-points are closed, so their preimages must also be closed.

Then, since T −{t0} ⊂ f−1(m), the closure T of T −{t0} must also be contained in f−1(m), so f

is a constant map, mapping T to m ∈M . In particular Φ(Speck) ∶M(Speck) → hM(Speck) is

not a bijection since Ft0 /∼ Ft1 ∈M(Speck) but they correspond to the same k-point m ∈M .

Example 3. Lets look at an example of a jump phenomenon. We consider the moduli problem

of classifying endomorphisms of an n-dimensional vector space. Our collection M consists of

pairs (V,ϕ) where V is an n-dimensional k-vector space and ϕ is an endomorphism of V . We

define an equivalence (V,ϕ) ∼ (V ′, ϕ′) when there exists an isomorphism h ∶ V → V ′ compatible

with the endomorphisms h ○ ϕ = ϕ′ ○ h. We extend the problem over any scheme of finite type

over k by defining a family over a scheme S to be a rank n vector bundle F over S with an

endomorphism ϕ ∶ F → F . Then we say that two families over S are equivalent if there exists

an isomorphism compatible with the endomorphisms as before.

For any n ≥ 2 we can construct a family which exhibits a jump phenomenon. Let n = 2 and

consider a family over A1 given by F = (O⊕2A1 , ϕ), where for s ∈ A1

ϕs =⎛⎜⎝

1 s

0 1

⎞⎟⎠

For any two s, t /= 0 these matrices are similar so we have an equivalence of families Ft ∼ Fs.

However F0 /∼ F1 since these matrices have distinct Jordan normal forms. Thus we have a jump

phenomenon and no coarse moduli space.

Apart from there being a jump phenomenon, a situation may occur that there is no family

over a scheme T which parametrises all objects in the moduli problem. In this case we say that

the problem is unbounded.

Example 4. Consider the moduli problem of vector bundles over P1 of rank 2 and degree 0.

Suppose there exists a family F over a scheme T such that for any vector bundle E on P1 of

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rank 2 and degree 0 there is a k-point t ∈ T such that Ft ≅ E . For each n ∈ N we have a rank 2

degree 0 vector bundle OP1(n)⊕OP1(−n). Moreover

dimH0(P1,OP1(n)⊕OP1(−n)) = dimk(k[x0, x1]n, k[x0, x1]−n) =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

2, n = 0

n + 1, n ≥ 1

Now consider the subschemes Tn ∶= {t ∈ T ∶ dimH0(P1,Ft) ≥ n} of T , which are closed by

the semicontinuity theorem (see [6] III Theorem 12.8). We obtain a decreasing chain of closed

subschemes

S = S2 ⫌ S3 ⫌ S4 ⫌ ...

each of which is distinct since OP1(n) ⊕ OP1(−n) ∈ Sn+1 − Sn+2. But we assumed that T is

Noetherian, so this chain should not exist. We’ve arrived at a contradiction and thus shown

that the moduli problem of rank 2 degree 0 vector bundles is unbounded.

Let us also consider a necessary condition for the existence of a fine moduli space for a given

moduli functor. It is based on the fact that for a scheme X, the functor hX satisfies descent

under faithfully flat quasi-compact coverings. Descent theory, developed by Grothendieck, is an

important topic that we are not able to cover here, but let us at least sketch the problem that

arises. Suppose that (fi ∶ Ui → U) is an open cover of U in faithfully flat quasi-compact topology,

or fpqc for short. What is meant here by "topology" is a Grothendieck topology in which we

consider a family of morphisms (fi ∶ Ui → U) to be a covering family for an affine scheme U if

they are jointly surjective with each Ui affine and each fi a flat morphism. For an arbitrary U ,

we say that (fi ∶ Ui → U) is a covering family in the fpqc topology if it is a covering family in

the previous sense after base changing to an open affine subset of U . Then, we say that a set

valued functor F satisfies faithfully-flat descent if for any covering family (fi ∶ Ui → U) in the

fpqc topology it satisfies the sheaf condition i.e. if the following sequence is exact

F (U)→∏i

F (Ui)→→∏i,j

F (Ui ×U Uj)

Let us now consider the problem that Mumford considered. Let C be a connected smooth

projective algebraic curve of genus g ≥ 2 over k = k̄ an algebraically closed field, and let S be a

k-scheme. Denote by

Fr,d(T ) ∶= {E a vector bundle on C ×k T of rank r and degree d}/ ≅

where ≅ is the isomorphism of vector bundles, the moduli functor we are trying to represent.

Suppose that a k-scheme M represents this functor. Then we would have, by definition of fine

moduli space a bijection of sets

{ϕ ∶ T →M ∶ ϕ a k-morphism } = {E a vector bundle on C ×k T of rank r and degree d}/ ≅

12

To see why this cannot happen, we consider how the two sides of the above bijection behave

with respect to descent. Suppose we have a cover in fpqc topology (fi ∶ Ti → T ). For a k-scheme

M a k-morphism ϕ ∶ T → M is given by a k-morphism ϕi ∶ Ti → M for every i, satisfying the

condition ϕi∣Ti×kTj = ϕj ∣Ti×kTj for all i, j. For simplicity one can consider the Zariski topology

on T and note that the fiber product is the same as intersection in the cateogry of sets, so the

above is simply a generalization to fpqc topology of the situation where we have a covering of T

and ϕ is defined on the elements of the cover with the condition being that the maps coincide

on overlaps. Consider now vector bundles. A vector bundle E on C ×k T is given by:

- a vector bundle Ei on C ×k Ti for each i

- an isomorphism αi,j ∶ Ei ≅ Ej over C ×k Ti × Tj for each i, j

- satisfying the cocycle condition αi,k = αj,k ○ αi,j over C ×k Ti × Tj × Tk for all i, j, k

We can see from this explicitly the two sides of the above bijection behave differently under

"gluing", so F cannot be represented by any scheme M . We would have an exception to

this situation if we would consider isomorphism classes of objects without automorphisms. In

this case, the above isomorphisms αi,j are unique if they exist, so the cocycle condition is

automatically satisfied, so the sheaf condition for the fpqc topology is satisfied. This still doesn’t

guarantee representability, but in many cases it can be established. To deal with this problem

Mumford and Deligne introduced stacks, by following up on an idea that we should keep track

of both isomorphism classes as well as the isomorphisms themselves, even though we are only

interested in parametrizing isomorphisms classes, in order to understand "gluing". Therefore,

the moduli space should not be a set endowed with some geometric structure, but rather a

grupoid (of objects and their isomorphisms) i.e. a category in which every morphism is invertible,

endowed with some geometric structure. This is roughly the idea of stacks.

13

Chapter 2

The Quot scheme

The Quot scheme is a fine moduli space which generalises the Grassmannian in the sense that

it parametrises quotients of a fixed sheaf on a fixed projective scheme. Its existence was shown

by Grothendieck in [4, 5]. The Quot scheme, and its special case the Hilbert scheme, are of

fundamental importance in the theory of moduli spaces. We will use it in the sections to follow

to construct moduli spaces of semistable sheaves. In this section, we will define the moduli

problem that the Quot scheme represents and present its construction following [17].

2.1 Preliminaries

In this section, we will recall the notions from scheme theory that will be used often in what

follows. These notions are elementary and can be found in any textbook on the subject such

as Harshorne [6] and Liu [11]. We will use some elementary properties of these notions without

proof, and in those cases provide references to these textbooks. The choice for providing the def-

initions here was made for the purpose of completeness, and with a view towards an elementary

treatment of the subject matter.

Definition 6. Proper and locally closed morphisms Let f ∶ X → Y be a morphism of schemes.

f is said to be of finite type if f is quasi-compact (i.e. the inverse image of any affine open is

quasi-compact), and if for every affine open subset V ⊂ Y , and every affine open subset U of

f−1(V ), the canonical homomorphism OY (V )→ OX(U) makes OX(U) into a finitely generated

OY (V )-algebra. In fact, for a morphism to be of finite type, it is enough to be able cover Y

with finitely many affine open subsets such that their inverse images in X can again be covered

by a finite number of affine open subsets such that the same finiteness property of the canonical

morphism of rings is satisfied. We say that a scheme is of finite type if its structure morphism

14

is of finite type.

Now, define the diagonal morphism of f as (IdX , IdX) ∶ X → X ×Y X. Denote it by ∆X/Y .

We say that X is separated over Y if ∆X/Y is a closed immersion of schemes. We say that X is

separated if it is separated over Z.

Morphism f ∶X → Y is universally closed if it is closed and stays closed under base change.

A proper morphism of topological spaces is a morphism such that an inverse image of a

compact subset is compact. In the setting of schemes we generalize this to the following: we say

that a morphism f ∶X → Y is proper if it is of finite type, separated, and universally closed. We

say that a Y scheme is proper if the structural morphism is proper.

A morphism of schemes f ∶X → Y is called a locally closed immersion, or simply an immer-

sion, if it can be factored as j ○ i where i is a closed immersion and j is an open immersion.

Now we move on to some definitions related to sheaves.

Definition 7. An OX -module L is called invertible if for every x ∈X there exists an open neigh.

x ∈ U and an iso of OU -modules OX ∣U ≅ L∣U . If X is locally Noetherian this is the same as

saying L is coherent and Lx is of rank 1 over OX,x. A sheaf F is called free of rank n if it is

isomorphic to OnX . It is called locally free of finite rank if there exists a covering of X = ∪Xi and

integers ni such that F ∣Xi is free of rank ni for each i. We say it is locally free of rank n if ni = n

for all i. Locally free sheaves correspond to vector bundles, if one considers them as sheaves of

sections of the given vector bundle.

An OX -module F is called quasi-coherent if it has a local presentation, that is, if for any

x ∈X, there exists an open neighbourhood U of x and an exact sequence of OX -modules

O(J)X ∣U → O(I)X ∣U → F ∣U → 0

for some sets J and I, possibly infinite.

We say that F is finitely generated if for every x ∈ X, there exists an open neighbourhood

U ⊂ X, an integer n ≥ 1, and a surjective homomorphism OnX ∣U → F ∣U . We say that F is

coherent if it is finitely generated, and if for every open subset U ⊂X, and every homomorphism

OnX ∣U → F ∣U the kernel of the homomorphism is finitely generated.

On an affine scheme U = SpecA there is an equivalence of categories from A-modules to

quasi-coherent sheaves, taking a module M to its associated sheaf M̃ , where M̃ is defined as the

sheaf such that for any principal open subset D(f) ⊂X we have M̃(D(f)) =Mf (the localization

of M at f ∈ A). The inverse takes a quasi coherent sheaf F on U to the A-module F(U) of global

sections. Taking into account this equivalence of categories, a sheaf on an arbitrary scheme X is

quasi-coherent if any point x ∈X has an affine open neighbourhood such that the restriction of

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F to that neighbourhood is a sheaf associated to a module (see [11] Chapter 5. Theorem 1.7.).

Then, a quasi-coherent sheaf F on a locally Noetherian scheme X is coherent if and only if the

above modules can be taken to be finitely generated (see [11] Chapter 5. Proposition 1.11).

Quasi-coherent sheaves, and therefore coherent sheaves, on any scheme form an abelian

category.

We move on to flatness of a morphism. It is an algebraic notion used to guarantee some

continuity of objects, or certain important invariants of objects, attached to the fibers, and it is

of great use in moduli theory.

Definition 8. Flat sheaves A module M over a ring R is flat if for every monomorphism of

R-modules A→ B the induced map M ⊗RA→M ⊗RB is again a monomorphism. A morphism

of schemes f ∶X → Y is flat if for every point x ∈X the local ring OX,x, regarded as an OY,f(x)-

module via the map f#, is flat. An OX -module F is flat over Y if the stalk Fx is flat when

regarded as a OY,f(x)-module via the composite map OY,f(x) → OX,x → Fx where the first map

is f#x and the second is the one making F an OX -module.

To see what we mean by continuity consider what we call a family of closed subschemes of

a given scheme X over a base scheme T . This is a closed subscheme Z ⊂ T ×X, together with a

restriction to Z of the projection map T ×X → T . The fibers of Z over t ∈ T are then naturally

closed subschemes of the fibers Xt of T ×X over T . Now, let T = SpecR be a non-singular,

one-dimensional affine scheme (think of Speck[t]), 0 ∈ T be a closed point and T ∗ = T −{0}. Let

Z∗ ⊂ AnT ∗ = AnZ×T ∗ be any closed subscheme (a family of closed subschemes of the affine n-space

AnZ over T∗), and let π ∶ Z → T ∗ be the projection. Consider now the fibers Z∗t = π−1(t). We ask

what the limit of the fibers is, when t → 0? The only reasonable answer is to take the closure

of Z∗ in AnT and take the limit of the schemes Zt when t → 0 to be the fiber Z0 of Z∗ over the

point 0 ∈ T . This construction can, however, yield some unexpected results, such as embedded

points in the limit fiber, even in simple cases. For examples of these pathological situations, as

well as a great geometrically intuitive introduction to the notion of flatness see [2] II §3.4. There

one finds the proof of the following proposition which illustrates the importance of flatness in a

very special, yet geometrically intuitive, case:

Proposition 1. Let T and Z be as above. Then the following conditions are equivalent:

(i) π is flat over 0

(ii) The fiber Z0 = π−1(0) is the limit of the fibers Zt = π−1(t) as t→ 0

(iii) No irreducible component or embedded component of Z is supported on Z0

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One can generalize this statement to higher dimensional base schemes T which are reduced

and over a field, and when Z is of finite type over T (see [2] Lemma II-30). In fact, in this case,

the morphism π ∶ Z → T is flat at p ∈ Z if, under any embedding of a neighbourhood of p in

affine space AnT , the fiber Z0 = π−1(0) over π(p) = 0 ∈ T is (an open subset of) the limit of the

fibers Zt as t ∈ T approaches 0 along any one-parameter family.

For us, the more important property that flat families, or fibers of a flat morphism, have

under suitable assumptions is the invariance of the Hilbert polynomial of coherent OX -modules

(see Definition 12). In fact,

Proposition 2. Let T be a locally Noetherian and connected scheme, with the generic point ν

and a closed point t. Let f ∶ Z → T be a projective morphism (see Remark 6), and F a coherent

sheaf on Z that is flat over T . Then we have the equality

χk(t)(Ft) = χk(ν)(Fν)

For the proof of the case when T is the spectrum of a discrete valuation ring, as well as

the reference to the more general case, see [11] Chapter 5. Proposition 3.28. If moreover T is

reduced, we have the other implication i.e. if the Hilbert polynomial on the fiber Ft is indepen-

dent of t ∈ T then F is flat over T (see [6] III Theorem 9.9). If we consider the structure sheaf

OX in the previous statement, and the definition of the Hilbert polynomial, we can conclude

that for a flat family Z ⊂ PrT of closed subschemes of a projective space over a locally Noetherian

connected base T the dimension and the degree of the fiber Zt are independent of t.

Next, we define sheaves generated by their global sections, ample and very ample sheaves.

Definition 9. We say that an OX -module F is generated by its global sections at x ∈ X if the

canonical evaluation morphism F(X)OX,x → Fx is surjective. We say that F is generated by

its global sections, or globally generated, if it is true at every point x ∈ X. Equivalently, F is

generated by its global sections if and only if there exists a set I and a surjective homomorphism

of OX -modules O(I)X → F .

Let f ∶ X → SpecA be a scheme over a ring A. Let i ∶ X → PnA be an immersion. Let

OX(n) = i∗(OPnA(n)). This is an invertible sheaf on X and it depends on i. The sheaf OX(1) =

i∗(OPnA(1)) is called a very ample sheaf. An invertible sheaf L on a quasi-compact scheme X is

ample if for any finitely generated quasi-coherent sheaf F on X there exists an integer n0 ≥ 1

such that for every n ≥ n0 we have that F ⊗ L⊗n is generated by its global sections. Theorem

1.27 in [11] Chapter 5. says that very ample implies ample. Theorem 1.34 in the same chapter

says that a sufficiently high tensor power of ample is very ample for f if f ∶ X → SpecA is of

17

finite type and X is Noetherian or f is separated. We say that a line bundle L is f -ample, for

a projective morphism f ∶X → S if its restriction to any fiber Xs is ample.

As we will mainly be dealing with projective schemes over an algebraically closed field, we

will use the notions of (Weil) divisors, linear systems, as well as the correspondence between

divisors and invertible sheaves (equivalently line bundles) in this case. For an introduction to

the language of divisors see [6] II §6 or [11] Chapter 7. §7.1 and §7.2.

2.2 The Quot functor

Definition 10. Let S be a noetherian scheme. Let X be a finite type S-scheme and E a coherent

sheaf on X. For any locally noetherian S-scheme T let ET ∶= π∗X(E) be the pullback of E to

T ×SX =∶XT via the projection πX ∶ T ×SX →X. A family of quotients of E parametrized by T

is a pair (F , q) where F is a coherent sheaf on X ×S T such that its schematic support is proper

over T , F is flat over T , and q is a surjective OX×ST -linear morphism of sheaves q ∶ ET → F . We

say that two families (F , q) and (F ′, q′) are equivalent if and only if ker q = ker q′, or equivalently

if there is an isomorphism F → F ′ compatible with q and q′. Let (Sch/S) be the category of

all locally noetherian S-schemes. Then we define:

QuotE/X/S ∶ (Sch/S)o → Set

to be the functor sending an S-scheme T to the set of equivalence classes of families parametrised

by T . Since properness and flatness are preserved by base change ([11] Chapter 3. Proposition

3.16 (c) and Chapter 4. Proposition 3.3 (b)), for a morphism g ∶ T ′ → T of S-schemes we can

define QuotE/X/S(g) to be the map sending ET → F to ET ′ → g∗XF , where gX ∶ T ′×SX → T ×SX.

Definition 11. We define a special case of the Quot functor - the Hilbert functor. If we set

E = OX in the previous definition then the functor QuotOX/X/S associates to T the set of

all closed subschemes Y ⊂ T ×S X that are proper and flat over T . We denote this functor by

HilbX/S ∶= QuotOX/X/S . The special case of the Quot scheme that represents the Hilbert functor

is called the Hilbert scheme. We also denote HilbPnZ ∶=HilbPnZ /SpecZ.

Definition 12. For a coherent sheaf E over a projective scheme X equipped with a fixed ample

invertible sheaf L, the Hilbert polynomial of E with respect to L is a polynomial Φ(m) ∈ Q[t]

such that for m sufficiently large

Φ(m) = χ(E(m)) = ΣdimEi=0 (−1)i dimH i(X,E ⊗L⊗m)

where dimE is the dimension of the support of E . We refer to the leading coefficient of the

Hilbert polynomial as the multiplicity of E .

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Serre’s vanishing theorem (see [6] III thm. 5.2) states that since L is ample, and E is coherent,

we have that for m sufficiently large all higher cohomology groups of E ⊗L⊗m vanish. Thus for

m sufficiently large we have Φ(m) = dimH0(X,E ⊗ L⊗m). For a proof that such a polynomial

exists see [8] Lemma 1.2.1. However, in the special case when X is a smooth projective curve,

and E is a locally free sheaf, or equivalently a vector bundle, we can write down the Hilbert

polynomial explicitly by using the Riemann-Roch theorem. First we recall the notion of degree

of locally free and coherent sheaves.

Definition 13. The Grothendieck group of X, denoted K0(X), is the free group generated by

classes [E], for E a coherent sheaf on X, modulo the relations [E]− [F]+ [G] = 0 for short exact

sequences 0→ E → F → G → 0.

There exists a homomorphism

(det, rk) ∶K0(X)→ Pic⊕Z (2.1)

which sends a locally free sheaf E to (det(E) ∶= ∧rkEE , rk(E)). We can extend this map to coher-

ent sheaves when X is smooth projective by taking a finite locally free resolution of the given

sheaf E (for the existence of the locally free resolution see [8] Proposition 2.1.10). Then by the

relations defining K0(X) we have [E] = Σi(−1)i[Ei], where Ei are the locally free sheaves in the

resolution of E . Therefore, we get a well defined line bundle det(E) ∶= det([E]). We recall that

the degree of a line bundle is defined to be the degree of the corresponding Weil divisor (for X

smooth the Picard group of isomorphism classes of line bundles and the group of Weil divisors

modulo linear equivalence are isomorphic - for proof see [6] II §6). Now we use the previous

definition of the determinant line bundle to define the degree of any coherent sheaf on a smooth

projective variety deg(E) ∶= deg(det(E)).

More generally we can define the rank of a coherent sheaf on a scheme X by rk(E) ∶= αd(E)αd(OX) ,

where αi are the coefficients of the Hilbert polynomial. Then we define deg(E) ∶= αd−1(E)−rk(E)⋅

αd−1(OX). The equivalence of this definition to the previous one in the projective non-singular

case follows from the Hirzebruch-Riemann-Roch theorem. Note that αdimX(OX) is exactly the

degree of X with respect to OX(1).

Now we return to a simple example of a Hilbert polynomial.

Example 5. Let X be a smooth projective curve of genus g. Fix a line bundle L = OX(1). For

a vector bundle E over X of rank n and degree d, the twist E(m) ∶= E ⊗ L⊗m has rank n and

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degree d +mdeg(X)n. The Riemann-Roch theorem then gives

χ(E(m)) = h0(X,E(m)) − h1(X,E(m)) =

= deg(E(m)) + rank(E(m))(1 − g)

= d +mdeg(X)n + n(1 − g)

(2.2)

So E has Hilbert polynomial Φ(m) =mdeg(X)n+d+n(1−g) of degree 1 with leading coefficient

given by the rank.

Definition 14. Let X be a finite type scheme over a scheme S, and L a line bundle on X. We

have a decomposition:

QuotE/X/S(T ) = ∐Φ∈Q[t]

QuotΦ,LE/X/S(T )

where for any polynomial Φ ∈ Q[t] the functor QuotΦ,LE/X/S associates to an S-scheme T the

equivalence classes of families (F , q), such that at each t ∈ T the Hilbert polynomial of the

restriction FT ∣(XT )t , calculated with respect to the pull-back of L, is Φ.

Remark 3. We will take X to be projective over S and L a relatively very ample line bundle.

Then the Quot functor will be representable by schemes, but not in general.

Remark 4. From the previous definitions it is clear that we can think of the Quot scheme as

parametrising the coherent subsheaves of F up to equality rather then quotients of F up to the

equivalence given in the definition of the Quot functor. However, the quotient perspective will

be more useful in the rest of this text.

2.3 The Grassmannian

Example 6. A special case of the Quot functor is the Grassmannian functor defined as follows.

Quotr,OZ⊕nOZ/Z/Z=∶ Grass(n, r), a functor associating to T /Z the set of equivalence classes of

quotients ⊕nOT → F where F is a locally free sheaf on T of rank d, n ≥ r ≥ 1.

An elementary construction of the Grassmannian scheme is given in [17] 5.1.6 along with a

tautological quotient u ∶ ⊗nOGrass(n,r) → U , the proof of properness of π ∶ Grass(n, r) → SpecZ,

and a closed embedding Grass(n, r) ↪ P(π∗ detU) = PmZ , where m = (nr) − 1. It is constructed

by gluing together affine patches U I ∶= Spec(Z[XI]), where Z[XI] is the polynomial ring in the

variables xIp,q which we get as follows: let M be an r×n matrix and I ⊂ 1, ..., n with #I = r. MIis the Ith minor of M meaning a r × r minor given by columns indexed by I. Consider now the

r × n matrix XI which has XII given by the identity matrix 1r×r and the remaining entries are

the free variables xIp,q. Then Z[XI] is the polynomial ring in these variables. Let J be another

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size r subset of {1, ..., n}. Let P IJ be the polynomial in the variables xI obtained by taking the

determinant of XIJ and let UIJ be the open subset of U

I where P IJ is non-zero. We define the

gluing map: ΘI,J ∶ U IJ → UJI via the ring homomorphism θI,J ∶ Z[XJ ,1/P JI ] → Z[XI ,1/P IJ ]

where the images of xJp,q are given by the entries of the matrix θI,J(XJ) = (XIJ)−1XI . In

particular θI,J(P JI ) = 1/P IJ , so the map extends to Z[XJ ,1/P JI ]. One checks that for any three

subsets I, J,K the cocycle condition θI,K = θI,JθJ,K is satisfied. Therefore these affine patches

glue to form a finite type scheme Grass(n, r) over SpecZ ([11] Chapter 2. Lemma 3.33.).

The universal quotient vector bundle and quotient map are constructed using the same affine

patches as in the construction of the Grass(n, r). The main idea is that on each patch, elements

of the Grassmannian are naturally represented by a unique matrix. Let U ∣UI = ⊕rOUI and

define the map qUI ∶ ⊕nO∣UI → ⊕rOUI via the matrix XI . Define the transition morphisms as

ρI,J ∶ ⊕rOUIJ = U ∣UIJ → ⊕rOUJI = U ∣UJI via the matrix (X

IJ)−1 ∈ GLr(U IJ). One checks that they

satisfy the cocycle condition for glueing sheaves ([11] Chapter 2. Exercise 2.8.).

We can show that the Grassmannian scheme Grass represents the functor Grass(n, r) de-

scribed at the beginning of the example as follows. We need to show the bijectionGrass(n, r)(Y ) ≅

HomZ(Y,Grass(n, r)) for any Y /Z. Take a morphism f ∶ Y → Grass(n, r). Then consider

f∗(u) ∶ OnY → f∗U . Since f∗ is right-exact this is an epimorphism so f∗(u) ∈ Grass(n, r)(Y ).

Therefore we get a map in one direction. Take now a surjection q ∶ OnY → F with F locally free of

rank r. This is given by choosing n ordered global sections of F which generate F at each point.

Let Y I be the open subset of Y where the global sections in a set I of degree I generate F i.e.

over Y I we have F ≅ OrY I

. Then over Y I we have that the map q is given by an r×n matrix, call

it M I with values in OY (Y I). Now define a map Y I → U I by sending xIp,q to the entries of the

matrix M I . This map will glue together to give a map Y → Grass(n, r) because the transition

matrix on Y I∩Y J is precisely (M IJ)−1 - switching from the basis of the I global sections to the ba-

sis of J global sections.This gives a map in the opposite direction inverse to the one given before.

Remark 5. We can generalise the previous construction as follows. Instead of considering the

locally free sheaf ⊕nOZ over SpecZ we can consider any locally free sheaf, or equivalently a vector

bundle, V over a scheme S. We get a functor Quotr,OSV /S/S =∶ Grass(V, r). It is representable by

a scheme Grass(V, d) and called the relative Grassmannian of V over S. If π ∶ Grass(V, r) →

S is the projection, and π∗V → U the universal quotient then the determinant bundle ∧rU

is very ample over S and it gives a closed embedding Grass(V, r) ↪ P(π∗ ∧r U) ⊂ P(∧rV ).

The representability of this more general functor can be obtained by observing that the group

scheme GLn,Z acts naturally on Grass(n, r) from the previous example, since it acts on the

21

free module ⊕nZ, and thus in turn on the locally free sheaf ⊕nOZ. The projective scheme

representing Grass(V, r) is then obtained by modding out Grass(n, r) by this action. Actions of

group schemes and the role of Geometric Invariant Theory in constructing moduli spaces will be

discussed in detail later. For now we mention one more generalisation of the Grassmannian. We

may replace the vector bundle V with a coherent sheaf E on S and obtain a functor Quotr,OSE/S/S =∶

Grass(E , r) which is still representable by a projective scheme over S. This is obtained from

the previous case by locally expressing a coherent sheaf as a quotient of a vector bundle.

2.4 Representability theorem of Altman and Kleiman

Definition 15. Let A be a ring and M an A-module. Let Tn(M) be the tensor product of

M with itself n times. Then T (M) ∶= ⊕d≥0T d(M) is a non-commutative A-algebra called the

tensor algebra of M . We define the symmetric algebra S(M) ∶= ⊕d≥0Sd(M) to be the quotient

of T (M) by the two-sided ideal generated by all expression x⊗ y − y ⊗ x for all x, y ∈M . As an

example note that for M a free A-module of rank rS(M) ≅ A[x1, ..., xr]. We define the exterior

algebra ∧(M) = ⊕d≥0 ∧d (M) as the quotient of T (M) by all expressions x⊗ x for all x ∈M .

Now, for a scheme (X,OX), and F a sheaf of OX -modules we define the tensor algebra,

symmetric algebra, and exterior algebra of F by taking the sheaves associated to the presheaf

which to each open U ∈ X assigns the corresponding tensor operation applied to F(U) as an

OX(U)-module. The results are OX -algebras and their components in each degree are OX -

modules.

Definition 16. Let V be a locally free sheaf on S. Then P(V ) ∶= Proj(Sym(V )) is called the

associated projective space bundle, where Sym(V ) is the symmetric algebra of V . S(V ) is a

graded OS-algebra, so Proj(Sym(V )) is defined as the unique S-scheme f ∶ Proj(Sym(V )) → S

such that for any affine open U ⊂ S we have f−1(U) ≅ Proj(Sym(V )(U)). Proj(Sym(V )(U))

is endowed with a sheaf O(1) = (Sym(V )(U)(1))∼. These sheaves glue to an invertible sheaf

OP(V )(1) ( see [11] Chapter 8. Remark 1.9.).

We will prove the following representability theorem by Altman and Kleiman:

Theorem 1. Let X be a closed subscheme of P(V ), L = OP(V )(1)∣X , W a locally free sheaf

on S, E a coherent quotient sheaf of π∗(W )(n), where π ∶ X → S and n ∈ Z. Let Φ ∈ Q[t].

Then the functor QuotΦ,LE/X/S is representable by a closed subscheme of Grass(V′, n′), where

V ′ =W ⊗OS SymrV , n′ = Φ(r).

Remark 6. In Grothendieck’s version of the theorem a weaker version of projectivity is used.

Recall that Grothendieck defines a morphism π ∶ X → S to be projective if there exists a

22

coherent sheaf E on S together with a closed embedding of X into P(E) = Proj SymOS E . We get

Altman and Kleiman’s notion of strong projectivity appearing in Theorem 1 by requiring that

E be a locally free sheaf on S. The generalization is simple and given in [17] 5.5.8. We should

mention here the strongest notion of projectivity in use: a morphism π ∶ X → S is projective in

the strongest sense if X admits a closed embedding into PnS , and π factors through this closed

embedding followed by the canonical morphism PnS → S.

2.5 Construction of the Quot scheme

Definition 17. Let F,G ∶ (Sch/S)o → Set be two functors, and let F be a subfunctor of G

i.e. there exists a natural transformation ν ∶ F → G which is injective when evaluated on any

object of (Sch/S). We say that F is a closed subfunctor (resp. locally closed) if for every T /S,

and any natural transformation γ ∶ hT → G from the functor of points of a scheme T , there

exists a closed (resp. locally closed) subscheme T ′/S such that the functor F ×G HomS(−, T ) is

represented by T ′ i.e. F ×G HomS(−, T ) ≅ hT ′ , where the fiber product is given by

(F ×G HomS(−, T ))(Y ) = {(H ∈ F (Y ), f ∶ Y → T ∈ hT (Y )) ∶ γY (f) = νY (H) ∈ G(Y )}

Remark 7. The previous definition just says that if G is represented by some T there exists

a closed (resp. locally closed) subscheme T ′ of T that represents F , because in that case

Hom(−, T ′) ≅ F ×GHomS(−, T ) ≅ F ×GG ≅ F . Moreover, in the case when F and G are moduli

functors, and G is representable by T , there exists a canonical choice for the map hT → G - the

map corresponding to the tautological quotient of G, which is just the map corresponding to

the identity on T .

We present the proof of Theorem 1 in several steps:

Step 1. Reduce to the case of QuotΦ,Lπ∗(W )/P(V )/S via the following lemma:

Lemma 2. (a) Tensoring by L−n gives an isomorphism QuotΦ,LE/X/S → QuotΨ,LE(−n)/X/S where

Ψ(m) = Φ(m − n)

(b) Let ψ ∶ E → G be en epimorphism of coherent sheaves on X. Then the corresponding natural

transformation QuotΦ,LG/X/S → QuotΦ,LE/X/S is a closed embedding.

(a) gets rid of the twist in the theorem of Altman and Kleiman. Furthermore, since we

assumed that E is a quotient sheaf of π∗(W ) and that X is a closed subscheme of P(V ) we

get an epimorphism of sheaves π∗(W ) → E . Hence, by (b) if we can represent the functor

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QuotΦ,Lπ∗(W )/P(V )/S by a scheme T , then our original functor will be represented by a closed

subscheme of T .

Proof. (of (b)) We need the vanishing scheme V (ϕ) of a morphism of coherent sheaves ϕ ∶ F → F ′

on X/S, which is a scheme with the universal property that any morphism f ∶ Z → X factors

through V (ϕ) if and only if f∗ϕ ∶= f∗F → f∗F ′ = 0. We use the following result:

Proposition 3. Let S be a Noetherian scheme and π ∶X → S a projective morphism. Let F and

F ′ be coherent sheaves on X with F ′ flat over S. Then, the contravariant functor hom(F ,F ′),

which assigns to each T /S the set Hom(FXT ,F ′XT ) is represented by some Z = Spec(SymQ)

where Q is a coherent sheaf on S. In addition, the closed subscheme Z0 defined by the vanishing

of the ideal sheaf generated by Q is the closed subscheme where the universal homomorphism

vanishes (corresponding to the identity on Z).

For the proof see [17] Theorem 5.8. A corollary is that the vanishing scheme of a morphism

ϕ ∶ F → F ′ on X is given by f−1(Z0), where f is the image of ϕ ∈ hom(F ,F ′)(X) under the

bijection hom(F ,F ′)(X) ≅ hom(X,Z).

Back to the proof of (b): We want to prove that if a scheme E/S represents the functor

QuotΦ,LE/X/S , then there exists a closed subscheme G ⊂ E that represents QuotΦ,LG/X/S . By the

previous remark about closed subfunctors of moduli functors, this is equivalent to showing that

for a given T /S, and a family (F , q) ∈ QuotΦ,LE/X/S(T ), the functor

QuotΦ,LG/X/S ×QuotΦ,LE/X/S HomS(−, T ) is represented by a closed subscheme T′ ⊂ T . Given any

coherent sheaf (F , q) ∈ QuotΦ,LE/X/S(T ), by the Yoneda lemma (Nat(hA,G) ≅ G(A)), we get a

natural transformation hom(−, T )→ F ∈ QuotΦ,LE/X/S , therefore a commutative diagram (of sets):

QuotΦ,LG/X/S ×QuotΦ,LE/X/S HomS(−, T ) HomS(−, T )

QuotΦ,LG/X/S QuotΦ,LE/X/S

Applying all of the above to a scheme Y /S, we take a look at what the elements of

QuotΦ,LG/X/S(Y ) ×QuotΦ,LE/X/S HomS(Y,T ) are. Let ((F′, q′), f) be such an element, where F ′ is a

sheaf on XY . Then by the proof of Yoneda lemma (we send idT ↦ F , for any morphism of

schemes p we have Quot(p) is the pullback by f) we get f ↦ (f∗F , f∗q) by the map on the right

(where f is used both for f ∶ Y → T and the induced map f ∶XY →XT , since F is defined on XT ).

The map on the bottom, induced by our map of coherent sheaves ψ is composition (on the right)

by ψY (i.e. since q′ ∶ GXY → F ′, and ψ is epi, then ψY ∶ EXY → GXY is epi, so q′ ○ψY ∶ EXY → F ′

24

makes it a quotient of EXY ). Therefore we should have (F ′, q′ ○ ψY ) = (f∗F , f∗q). Now this is

true if kerψ ⊂ ker q′. So we get the set:

{((F ′, q′), f) ∶ f ∈ HomS(Y,T ),F ′ = f∗F , q′ = f∗q,kerψ ⊂ ker q′}(∗∗)

We need a closed subscheme T ′ ⊂ T such that a morphism Y → T factors through T ′ if and only

if f∗q∣kerψ = 0 thereby establishing a bijection between morphisms Y → T ′ and the elements of

the set (**). But then T ′ = V (f∗q∣kerψ), the vanishing scheme of f∗q∣kerψ, which exists by the

previous proposition.

Step 2. Let us now construct an injective natural transformation from the Quot mod-

uli functor to the Grassmannian moduli functor. To do this we will use the following result

that is based on Mumford-Castelnuovo regularity, which will be discussed and used again later

(Definition 45 and Lemma 11), and a flat base change result. For proof we refer to [17] 5.5.4.

Proposition 4. There exists an integer m depending only on the rank of V , the rank of W and

Φ s.t. for all r ≥ m, for all schemes T /S and for all T -flat coherent quotients F of EXT with

kernel G, the following facts hold:

1. πT∗(F(r)), πT∗(G(r)), πT∗(EXT (r)) are locally free are locally free sheaves of ranks deter-

mined by the rank of V , rank of W , r and Φ, and in particular the rank of πT∗(F(r)) is Φ(r).

Additionally, all higher direct image sheaves RiπT∗(F(r)),RiπT∗(G(r)),RiπT∗(EXT (r)) vanish,

2. In particular, the following is a commutative diagram of locally free sheaves on XT in which

the rows are exact and the vertical maps are surjective:

0 π∗TπT∗(G(r)) π∗TπT∗(EXT (r)) π∗TπT∗(F(r)) 0

0 G(r) EXT (r) F(r) 0

Fix r ≥m and consider a scheme T /S and a sheaf (F , q) ∈ QuotΦ,LE/X/S(T ). Twisting it by Lr

and applying πT∗ we get an epimorphism:

πT∗(q(r)) ∶ πT∗(EXT (r))→ πT∗(F(r))

i.e. applying the functor πT∗ to the exact sequence of sheaves 0 → G(r) → EXT (r) → F(r) → 0,

we get that the map above is surjective since the higher direct images vanish by the previous

proposition. Direct image functor is also left-exact (the right adjoint to the inverse image func-

tor), so it commutes with ker. Therefore if we have another element (F ′, q′) ∈ QuotΦ,LE/X/S(T ) s.t.

ker q = ker q′ then kerπT∗q = πT∗ ker q = πT∗ ker q′ = kerπT∗q′. So π∗ is a natural transformation:

QuotΦ,LE/X/S(T )→ QuotΦ(r),OSπ∗(π∗(W )⊗Lr)/S/S(T )

25

[q(r) ∶ π∗X(π∗(W )) = EXT (r)→ F(r)]↦ [πT∗(q(r)) ∶ πT∗(π∗X(π∗(W )⊗Lr))→ πT∗(F(r))]

On the right hand side we have a Quot functor that assigns to T /S quotients of the pullback

to T of π∗(π∗(W )⊗Lr) which is a sheaf on S. The fact that πT∗(q(r)) ∶ πT∗(π∗X(π∗(W )⊗Lr))→

πT∗(F(r)) is such a quotient follows from the fact that in the following diagram:

XT T

X S

πT

πX p

π

we have that πT∗(π∗X(π∗(W )⊗Lr)) ≅ p∗(π∗(π∗(W )⊗Lr)) (special case when X is PnS can

be found in [17] Lemma 5.4).

The fact that the sheaf πT∗(q(r)) is locally free of rank Φ(r) comes from the previous

proposition fact 1. Now we remark that the Quot functor on the right above is actually the

Grassmannian functor, and after applying the projection formula ([6] II Exercise 5.1. d))we get

QuotΦ(r),OSπ∗(π∗(W )⊗Lr)/S/S = Grass(π∗(π

∗(W )⊗Lr),Φ(r)) = Grass(W ⊗ Symr V,Φ(r))

We show now that this natural transformation is injective i.e. that we can recover q from

πT∗(q(r)). q(r) is the cokernel of the map G(r)→ EXT (r), by definition of G. Since the vertical

maps in the commutative diagram in Proposition 2. are surjective we can precompose with

the first one of them and we will not change the kernel i.e. q(r) is the cokernel of the map

π∗TπT∗(G(r))→ G(r)→ EXT (r). By the commutativity of the diagram this is the same map as

π∗TπT∗(G(r))→ π∗TπT∗(EXT (r))→ EXT (r) (2.3)

Now the second map is canonical and the first map is just the pullback of the inclusion of

the kernel of πT∗(q(r)).

Step 3. To prove thatQuotΦ,LE/X/S is a locally closed subfunctor ofGrass(W⊗Symr V ),Φ(r)),

i.e. that it is represented by a locally closed subscheme of the Grassmannian scheme, we need

to use flattening stratification.

Theorem 2. Let S be a Noetherian scheme and let F be a coherent sheaf on X/S, where X

is a closed subscheme of P(V ) for some locally free sheaf V on S. Then, the set I of Hilbert

polynomials of restrictons of F to the fibers of the map from X to S is finite. Moreover, for each

Φ ∈ I, there exists a locally closed subscheme SΦ of S such that the following three properties

hold:

26

1. Point-set: The underlying set ∣SΦ∣ consists of all points s ∈ S above which the Hilbert polyno-

mial of F is Φ.

2. Universal Property: Let S′ be the scheme theoretic disjoint union (coproduct) of the SΦ over

all Φ ∈ I and let f ∶ S′ → S be the natural surjection. Then, the sheaf f∗(F) on P(f∗(V ))

is flat over S′. Moreover f ∶ S′ → S has the universal property that a moprhism g ∶ T → S

factors through f if and only if g∗F is flat over T (we use g to denote the base changed morphism

g ∶XT = T ×S X → T ). The subscheme SΦ is thus uniquely determined by Φ.

3. Closure of Strata: Let the set I be given the total order by putting Φ < Φ′ if for sufficiently

large n ∈ Z Φ(n) < Φ′(n). Then, the closure of SΦ consists of the union of all SΦ′ for Φ′ ≥ Φ.

Proof. See [17] Theorem 5.13.

Again, as in a previous case of proving that a functor is a closed subfunctor (now locally),

given any T /S and (F , q) ∈ Grass(W ⊗ Symr V,Φ(r))(T ) we need to show that there exists a

locally closed subscheme T ′ → T representing

QuotΦ,LE/X/S ×Grass(W⊗Symr V ),Φ(r)) Hom(−, T )

Let us define T ′ as follows. Define G to be the kernel of

q ∶ πT∗(EXT (r)) = (π∗(E(r)))T → F

(again using π∗X ○ πT∗ = p∗ ○ π∗ from the fiber product square from Step 2.). Let h be the

composite map:

π∗T (G)→ π∗TπT∗(EXT (r))→ EXT (r)

Let J be the cokernel of h and define T ′ to be the subscheme obtained in the flattening strati-

fication for the sheaf J corresponding to the polynomial Φ. Let us show that T ′ is the scheme

we are looking for:

For any Y /S we have that by the universal property of the stratification that a morphism

f ∶ Y → T factors through T ′ if and only if f∗J is flat with Hilbert polynomial Φ (again f is

used for the base changed morphism f ∶ XY → XT ). Rephrasing the last property we remark

the following: f∗J = f∗(cokerh) = coker(f∗h), since both f∗ and coker are colimits. This gives

us, writing out h:

f∗J = coker(f∗π∗T (ker q)→ f∗π∗TπT∗(EXT (r))→ f∗EXT (r))

Because πT∗ and πY ∗ don’t have higher direct images on the relevant sheaves by the Mumford-

Castelnuovo regularity result stated before we have that f∗π∗T = π∗Y f∗ and f∗πT∗ = πY ∗f∗. This

gives us:

f∗J = coker(π∗Y f∗(ker q)→ π∗Y πY ∗f∗(EXT (r))→ f∗EXT (r))

27

On the other hand f lives in the image of the above functor if and only if f∗q comes from some

(F ′, q′) ∈ QuotΦ,LE/X/S(Y ), as can be seen from a similar diagram to a previous one:

QuotΦ,LG/X/S(Y ) ×Grass(W⊗Symr V ),Φ(r))(Y ) HomS(Y,T ) HomS(Y,T )

QuotΦ,LG/X/S(Y ) Grass(W ⊗ Symr V ),Φ(r))(Y )

By looking at equation (1) which gives us a way to recover an element of the Quot from an

element of the Grass, we have that there exists such a q′ if and only if

coker(π∗Y ker(f∗q)→ π∗Y f∗π∗T (EXT (r))→ f∗(EXT ))

is flat (then the q′ we need is equal to it). We can use the identities from before to get:

coker(π∗Y ker(f∗q)→ π∗Y π∗Y f∗(EXT (r))→ f∗(EXT ))

Since f∗ is right exact we can exchange ker(f∗q) and f∗ ker(q) without changing the cokernel.

Thus we get that the last sheaf is equal to f∗J .

So we can conclude that a morphism f ∶ Y → T factors through T ′ if and only if f∗J is flat and

has a Hilbert polynomial Φ which is true if and only if

((f∗F , f∗q), f) ∈ QuotΦ,LG/X/S(Y ) ×Grass(W⊗Symr V ),Φ(r))(Y ) HomS(Y,T )

thus establishing the needed bijection. From Theorem 2 we get the required locally closed sub-

scheme of Grass.

Step 4. To finish the proof of Theorem 1 we need to show that the QuotΦ,LE/X/S functor, as

defined in 10, is actually represented by a closed subscheme of the Grassmannian. To do this we

will check the valuative criterion for properness (see [11] Chapter 3. Theorem 3.25.). Namely,

if R is a DVR together with a given morphism SpecR → S and field of fractions K, we need to

show that the restriction map

QuotΦ,LE/X/S(SpecR)→ QuotΦ,LE/X/S(SpecK)

is bijective. So, take an element (F , q) in QuotΦ,LE/X/S(SpecK)

q ∶ EXK → F

where XK ∶= X ×S SpecK. Let j ∶ XK → XR be the natural map given by the inclusion of the

generic point SpecK in SpecR. Now let F̄ be the image of the composition of EXR → j∗EXK

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with j∗q. Let q̄ ∶ EXR → F̄ be the epimorphism obtained by looking only at the image. If we

show that F̄ is flat over SpecR we will have that

(F̄ , q̄) ∈ QuotΦ,LE/X/S(SpecR)

and therefore we will have proven the surjectivity of the restriction map because applying j∗

makes the first map of the composition the identity map, and the second map q. But on a DVR

being torsion free is the same as being flat. Now, F is torsion free over SpecK since it is flat

over it, and hence j∗(F) is a torsion free sheaf over SpecR, thus so is F̄ .

Injectivity of the restriction map follows immediately from the valuative criterion for sep-

aratedness ([11] Chapter 3. Remark 3.12.) because we know that the scheme QuotΦ,LE/X/S is

separated as locally closed subschemes of separated schemes are separated (see [11] Chapter 3.

Proposition 3.9, and [17] 5.1.6 for separatedness of the Grassmannian).

Therefore, we have shown Theorem 1.

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Chapter 3

Geometric Invariant Theory

In this section we will present another crucial tool in the theory of moduli spaces - Mumford’s

Geometric Invariant Theory.

3.1 Actions of algebraic groups

Definition 18. An algebraic group over a field k is a scheme G over k of finite type with

morphisms of k-schemes e ∶ Speck → G, m ∶ G × G → G and i ∶ G → G defining the identity

element, multiplication and inversion, and satisfying the usual axioms for groups. A more general

notion is that of a group scheme, which is the generalisation of the above definition to working

over an arbitrary ring R. We say that G is an affine algebraic group if the underlying scheme is

affine. A homomorphism of algebraic groups is a morphism of k-schemes which commutes with

the multiplication morphisms.

Remark 8. The functor of points hG ∶= Homk(−,G) of an algebraic group factors through the

category of groups i.e. for any k-scheme X Hom(X,G) is endowed with a group structure and

every map hG(f) ∶ hG(X)→ hG(Y ) is a morphism of group with this structure. One can show,

using the Yoneda lemma, that there is an equivalence of categories between the category of

algebraic groups and the category of functors F ∶ Sch/k→Grp such that the composition with

the forgetful functor Grp→ Set is representable.

Considering OG(G), the k-algebra of regular functions on G, the morphisms e,m, i above

correspond to k-algebra homomorphisms m∗ ∶ OG(G)→ OG(G)×OG(G), i∗ ∶ OG(G)→ OG(G)

and OG(G) → k, called comultiplication, coinverse and counit respectively. These define a

structure of a Hopf algebra on OG(G). There is a bijection between Hopf algebras and affine

algebraic groups ([12] II Theorem 5.1)

By a theorem of Cartier every affine algebraic group over a field k of characteristic 0 is

30

smooth ([12] VI Theorem 9.3).

Example 7. Let us see some examples of algebraic groups that will be used later.

1. The additive group Ga = Speck[t], whose underlying scheme is the affine line A1 and the

group structure is given by addition

m∗(t) = t⊗ 1 + 1⊗ t and i∗(t) = −t

For a k-algebra R we have Ga(R) = (R,+) which justifies the name.

2. The multiplicative group Gm = Speck[t, t−1] whose underlying scheme is A1 − {0} and

whose group action is given by multiplication

m∗(t) = t⊗ t and i∗(t) = t−1

For a k-algebra R we have Gm(R) = (R×, ⋅), hence the name.

3. The general linear group GLn over k is a subvariety of An2

cut out by the condition that the

determinant is non-zero. It is an affine variety whose coordinate ring is k[x11, x12, ..., xnn, t]/(tdet(xij)−

1). The morphisms are given by

m∗(xij) =n

∑k=1

xik ⊗ xkj and i∗(xij) = (xij)−1ij

where the (xij)−1ij is the (i, j)-th entry in the inverse matrix.

4. If V is a finite-dimensional vector space over k, there is an affine algebraic group GL(V )

non-canonically isomorphic to GLdimV , with the isomorphism given by choosing a basis.

For a k-algebra R GL(V )(R) = AutR(V ⊗k R)

Remark 9. A linear algebraic group is a closed subgroup of GLn. Thus, any linear algebraic

group is affine. The converse is also true, any affine algebraic group is linear (see [1] I Proposition

1.10.).

We want to consider actions of algebraic groups, and in particular we want to know when

they will provide us with "nice" quotients.

Definition 19. An action of an algebraic group G on a k-scheme X is a morphism σ ∶X×G→X

which satisfies the usual associativity rules with respect to the multiplication, identity and

inverse morphisms of the algebraic group. Let X,Y be two k-schemes with actions of G. A

morphism of k-schemes f ∶ X → Y is called G-equivariant if it commutes with the actions i.e.

f(g ⋅ x) = g ⋅ f(x). If the action on Y is trivial and the morphism is G-equivariant we call it

G-invariant i.e. f(g ⋅ x) = f(x).

31

As we mentioned before, we are interested in "nice" quotients, so let us define what we will

consider a "nice" quotients. It may happen that the orbit space X/G = G ⋅ x∣x ∈X admits a

scheme structure, though this does not happen often. Let us look at some weaker notions of

quotients.

3.2 Nice quotients

Definition 20. A cateogrical quotient for the action of G on X is a scheme Y and a G-invariant

morphism ϕ ∶ X → Y of schemes with the following universal property: every other G-invariant

morphism f ∶ X → Z factors uniquely through ϕ i.e. there exists a unique morphism h ∶ Y → Z

such that f = ϕ ○ h. Furthermore, if the preimage of each k-point in Y is a single orbit we say

that Y is an orbit space.

Equivalently, a categorical quotient is a scheme Y that represents the functor

X/G ∶ (Sch/k)o → Set, T ↦X(T )/G(T )

The morphism ϕ above is obtained as the image of [idX] in X(X)/G(X).

Remark 10. The categorical quotient has nice functorial properties: if ϕ ∶X → Y is G-invariant

and if there is a cover Y = ∪iUi such that ϕ−1(Ui)→ Ui is a categorical quotient for each i then

ϕ is a categorical quotient.

Example 8. Let Gm, as defined earlier, act on An by t ⋅ (a1, ..., an) = (ta1, ..., tan). In this case

there are two types of orbits: punctured lines through the origin, and the origin itself. The origin

is the only closed orbit and each orbit has the origin in its closure. Therefore, any G-invariant

morphism (i.e. constant on orbits) An → Y must be constant on An. Clearly, the categorical

quotient in this case is the structure map ϕ ∶ An → Speck since any other map f ∶ An → Z is

constant i.e. maps everything to a point z ∈ Z so we may choose the morphism h ∶ Speck → Z

to be the one sanding the one point to z ∈ Z to get f = h ○ ϕ.

This example illustrates how non-closed orbits can create problems, in that the categorical

quotient we constructed is far from an orbit space. If we were to remove the origin, and then

take the quotient of An − {0} by Gm with the same action as before, we would get Pn−1, which

is what we intuitively expect from this action. So we need a new notion that is closer to our

intuitive idea of a quotient.

The action of an algebraic group G on a k-scheme X induces an action on the k-algebra

O(X) of regular functions on X by

g ⋅ f(x) = f(g−1 ⋅ x)

32

and we denote the subalgebra of invariant functions by

O(X)G ∶= {f ∈ O(X) ∶ g ⋅ f = f for all g ∈ G}

If U ⊂ X is an open set that is invariant under the action of G i.e. for all u ∈ U and g ∈ G we

have u ⋅g ∈ U , then G also acts on OX(U) and we denote by OX(U)G the subalgebra of invariant

functions.

Definition 21. A morphism ϕ ∶X → Y is a good quotient for the action of G on X if

1. ϕ is G-invariant

2. ϕ is surjective, and U ⊂ Y is open if and only if ϕ−1(U) ⊂X is open

3. If U ⊂ Y is an open subset the morphism OY (U)→ OX(ϕ−1(U))G is an isomorphism

4. If W is any G-invariant closed subset of X then ϕ(W ) is closed in Y . If W1 and W2 are

disjoint G-invariant closed subsets then ϕ(W1) and ϕ(W2) are also disjoint

5. ϕ is affine i.e. the preimage of every affine open subset is affine.

If, moreover, the geometric fibers of ϕ are orbits of geometric points of X then ϕ is a geometric

quotient. Finally, ϕ is a universal good (resp. geometric) quotient if for any morphism Y ′ → Y

of k-schemes we have that the base change Y ′ ×Y X → Y ′ is a good (resp. geometric) quotient.

A good quotient is a categorical quotient (see [13] Proposition 0.1).

Example 9. Consider an action of Gm on A2k, where k is an algebraically closed field, given by

t ⋅ (x, y) = (tx, t−1y). The orbits of this action are

• conics {(x, y) ∶ xy = λ} for λ ∈ A1 − {0}

• the punctured x-axis

• the punctured y-axis

• the origin

The origin and the conics are closed orbits, while the punctured axes contain the origin in their

closure. Therefore the categorical quotient will identify all orbits except for the conics which are

parametrised by A1 − {0}. We may thus expect the morphism ϕ ∶ A2 → A1 given by (x, y)↦ xy

to be the categorical quotient. In fact, it is also a good quotient. Let us quickly check that fact.

ϕ is obviously surjective and G-invariant. Now let U ⊂ A1 be an open subset and consider

ϕ∗ ∶ OA1(U)→ OA2(ϕ−1(U))

33

For U = A1 we have ϕ∗ ∶ k[t]→ k[x, y] given by t↦ xy. Let us show that this is an isomorphism

on the ring of invariants. The action on O(A2) = k[x, y] is given by

t ⋅ (Σi,jaijxiyj) = (Σi,jaijtj−ixiyj)

as discussed above. Therefore the invariant subalgebra is

k[x, y]k∗ = {Σi,jaijxiyj ∶ aij = 0 for i /= j} = k[xy]

as we wanted. Now take a proper open subset U ⊂ A1. It is of the form U = A1 − {a1, ..., an}

and OA1(U) = k[t]f where f(t) = (t − a1)⋯(t − an). Then ϕ−1(U) is the non-vanishing locus of

F (x, y) ∶= f(xy) ∈ k[x, y] so we have OA2(ϕ−1(U)) = k[x, y]F . In particular,

OA2(ϕ−1(U))GM = (k[x, y]F )GM = (k[x, y]GM )F = k[xy]F ≅ k[t]f = OA1(U)

Furthermore, closed G-invariant subsets of A2 are either a finite union of orbit closures or the

entire space A2. Therefore it is enough to prove condition 4. for being a good quotient for

W1 = G ⋅ p1 and W2 = G ⋅ p2 disjoint. As we have the list of all orbits we know that for W1 and

W2 to be disjoint we must have one of the following two cases

• both p1 and p2 distinct and not lying on the axes of A2 so that their orbits are disjoint

conics (x, y) ∶ xy = αi, therefore ϕ(W1) = α1 /= α2 = ϕ(W2)

• one point lies on an axis, say p1 so ϕ(W1) = 0, and the other does not so ϕ(W2) /= 0

The condition that the morphism is affine is satisfied since any morphism of affine varieties is

affine.

Additionally, this morphism is not a geometric quotient since ϕ−1(0) is the union of three

orbits (all except the conics).

Lemma 3. Let G be an algebraic group acting on a scheme X and let ϕ ∶ X → Y be a good

quotient. Then:

1. G ⋅ x1 ∩G ⋅ x2 /= ∅ 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 is closed (i.e. all orbits are closed), then ϕ is a geometric quotient.

Proof. For the first part it is clear that ϕ(x1) = ϕ(x2) if G ⋅ x1 ∩G ⋅ x2 since we have said that ϕ

is G-invariant and thus constant on orbit closures. The other direction follows from property 4.

of the good quotient. For the second part, suppose that converse is true i.e. there are two closed

orbits W1,W2 in a preimage ϕ−1(y). But then the fact that we have two closed G-invariant

subsets that have the same image contradicts the property 4.

34

Moreover, it is easy to see that good and geometric quotients restrict for any open U ⊂ Y to

ϕ∣ ∶ ϕ−1(U)→ ϕ(U) to good and geometric quotients respectively of the action of G on ϕ−1(U).

3.3 Hilbert’s 14th problem and Nagata’s theorem

Remark 11. The story of constructing the nice quotients we’ve been discussing begins with the

14th item on the list of problems that David Hilbert proposed in 1900. Namely, any G-invariant

morphism ϕ ∶ X → Y , which is the first property that we ask of the categorical, and therefore

any "nice" quotient, induces a homomorphism ϕ∗ ∶ O(Y ) → O(X) whose image is contained in

the subalgebra of G-invariant regular functions O(X)G.

Definition 22. First off, a rational representation of an algebraic group G is a morphism

h ∶ G→ GL(V ), for a finite dimensional k-vector space V , of algebraic groups i.e. a morphism of

groups that is at the same time a morphism of varieties. This automatically gives as an action

of G on V , which we call a linear action. Moreover, we say that an action of G on a k-algebra

A is rational if the map a ↦ g ⋅ a is an automorphism of A for any g ∈ G, and every element

of R is contained in a finite-dimensional G-invariant linear subspace of A on which G acts by a

rational representation.

Now, a formulation of Hilbert’s 14th problem follows: For a rational action of an algebraic

group on a finitely generated k-algebra A, is the algebra of G-invariants AG finitely generated?

In general, this question has a negative answer. Nagata provided a counterexample (see

[14]), as well as a theorem proving that for G geometrically reductive Hilbert’s problem has a

positive answer (for proof see [15] Theorem 3.4.).

We give the definition of geometrically reductive groups, even though we will not prove

Nagata’s theorem, and will thus only care about the fact that the groups GLn,SLn and PGLn

are in fact geometrically reductive, so that we can apply the theory that follows to taking

quotients by their actions.

Definition 23. A linear algebraic group G is called geometrically reductive if, for any rational

representation h ∶ G→ GLn, and every point v /= 0 ∈ kn invariant by the induced linear action of

G on kn, there exists an invariant homogeneous polynomial f in n variables of degree ≥ 1 such

that f(v) /= 0.

In fact, there exist notions of a reductive group - a group whose radical is isomorphic to a

torus (i.e. direct product of multiplicative groups), and that of a linearly reductive group - a

group whose every rational representation is completely reducible i.e. expressible as a direct

35

sum or representations possessing no proper invariant subspaces. Mumford’s theory originally

used his proof that Hilbert’s 14th problem had a positive answer for linearly reductive groups

in all characteristics, and it was proven that linearly reductive is equivalent to reductive in

characteristic 0, but Nagata provided an example in positive characteristic of a reductive but

not linearly reductive group. This motivated the search for an appropriate weaker notion, and

the proof by Haboush in 1974, that geometrically reductive is equivalent to reductive in all

characteristics (see [13] Theorem A.1.0.).

3.4 Constructing the projective GIT quotient

Let X be a projective scheme with an action of an affine algebraic group G. A linear G-

equivariant projective embedding of X is a rational representation h ∶ G → GL(n + 1) and a

G-equivariant projective embedding X ↪ Pn. G then acts on Pn via the homomorphism h, and

on X by restriction. The action of G on Pn lifts to an action on the affine cone An+1 over Pn,

and since the projective embedding of X ⊂ Pn is G-equivariant there is an induced action of G

on the affine cone X̃ ⊂ An+1 over X ⊂ Pn. We have:

O(An+1) = k[x0, ..., xn] =⊕r≥0

k[x0, ..., xn]r =⊕r≥0

H0(Pn,OX(r))

and if X ⊂ Pn is the closed subscheme associated to a homogeneous ideal I(X) ⊂ k[x0, ..., xn]

then the affine cone X̃ = Spec(R(X)) where R(X) = k[x0, ..., xn]/I(X).

The homogeneous algebras O(An+1) and R(X) are graded by homogeneous degree and, since

the G-action on An+1 is linear it preserves the graded pieces, so that the invariant subalgebra:

O(An+1)G =⊕r≥0

k[x0, ..., xn]Gr

is a graded algebra and so is R(X)G = ⊕r≥0R(X)Gr . By Nagata’s theorem R(X)Gr is finitely

generated when G is reductive (as we assume in these notes). The inclusion of finitely generated

graded k-algebras R(X)G ↪ R(X) determines a rational morphism of projective schemes:

X ⇢ ProjR(X)G

whose indeterminacy locus is the closed subscheme of X defined by the homogeneous ideal

R(X)G+ ∶=⊕r>0R(X)Gr . This motivates the definition of semistable points as follows:

Definition 24. For an action of a reductive group G on a projective scheme X as in the

previous discussion, we define the nullcone N of X to be the closed subscheme of X defined by

the homogeneous ideal R(X)G+ . We define the set of semistable points on X to be Xss ∶=X −N .

We can thus define x ∈ X to be a semistable point if there exists a G-invariant homogeneous

polynomial f ∈ R(X)Gr for r > 0 such that f(x) /= 0.

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By the previous definition we have that Xss is the domain of definition of the rational map

X ⇢ ProjR(X)G, and we call the morphism Xss → X//G ∶= ProjR(X)G the GIT quotient of

this action of G.

The following theorem will show that Xss → X//G is a good quotient, but from we know

that the presence of non-closed orbits in the semistable locus will prevent the good quotient

from being a geometric quotient. This motivates the definition of stability.

Definition 25. A point x ∈ X is stable if dimGx = 0 (the stabilizer is finite) and there is a

G-invatiant homogeneous polynomial f ∈ R(X)G+ such that x ∈ Xf = {x ∈ X ∣f(x) /= 0} and the

action of G on Xf is closed.

First, we’d like to prove that the set of stable points is open in X. We need the following:

Lemma 4. The dimension of the stabilizer subgroup viewed as a function X → N is upper-

semicontinuous, that is, for every n the set {x ∈X ∶ dimGx ≥ n} is closed in X.

Proof. Consider the graph of the action Γ = (pX , σ) ∶ G ×X →X ×X and the fiber product

P X

G ×X X ×X

ϕ

∆

Γ

where ∆ is the diagonal morphism. Then the k-points of P consist of pairs (g, x) such that

g ∈ Gx. The function on P which sends p = (g, x) ∈ P to the dimension of Pϕ(p) ∶= ϕ−1(ϕ(p)) is

upper-semicontinuous (see Grothendieck’s EGA [3] IV 13.1.3) i.e. for all n we have that {p ∈ P ∶

dimϕ(p) ≥ n} is closed in P . By restricting to the closed subscheme X ≅ {(e, x)∣x ∈ X} ⊂ P we

can conclude that the set {x ∈X ∣dimGx ≥ n}is closed in X for all n.

Lemma 5. The stable and semistable sets Xs and Xss are open in X.

Proof. The set of semistable points is open by definition, since it is the complement of the

nullcone, which is closed. To prove that the set of stable points is open consider Xc ∶= ∩Xfwhere the union is taken over f ∈ R(X)G+ such that the action of G on Xf is closed. Xc is open

in X, so we need to prove that Xs is open in Xc. By the previous lemma we have that the

function x ↦ dimGx is upper-semicontinuous, so the set of points in X with zero-dimensional

stabilizer is open in X, so Xs is open since it is the intersection of two open sets in X (Xc and

the set of points with finite stabilizer).

Now we come to the main theorem of this section, which will prove an important tool to

have in moduli theory.

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Theorem 3. For a linear action of a reductive group G on a closed subscheme X ⊂ Pn, let

ϕ ∶Xss → Y ∶=X//G the GIT quotient. Then:

(i) ϕ ∶ Xss → Y ∶= X//G is a good quotient for the G-action on the open subset Xss of

semistable points in X. Moreover, Y is a projective scheme.

(ii) There is an open subscheme Y s ⊂ Y such that ϕ−1(Y s) = Xs and that the GIT quotient

restricts to a geometric quotient ϕ ∶Xs → Y s.

(iii) A k-point x ∈ X(k) is stable if and only if x is semistable, its orbit G ⋅ x is closed in Xss,

and its stabilizer Gx has dimension zero.

For proof see [15] Theorem 3.14.

Example 10. Let us look at an example. Consider the linear action of G = Gm on X = Pn given

by

t ⋅ [x0 ∶ ... ∶ xn] = [t−1x0 ∶ tx1 ∶ ... ∶ txn]

We have that R(X) = k[x0, ..., xn] graded by degree. We claim that the functions x0x1, ..., x0xngenerate the ring of invariants. Let f ∈ R(X), then

f = ∑m=(m0,...,mn)

a(m)xm00 ⋯xmnn

and for f ∈ G

t ⋅ f = ∑m=(m0,...,mn)

a(m)tm0−m1−⋯−mnxm00 ⋯xmnn

So we see that f is G-invariant if and only if a(m) = 0 for all m such that m /= ∑ni=1mi. If

m = ∑ni=1mi we can write

xm00 xm11 ⋯x

mnn = (x0x1)m1⋯(x0xn)mn

so if f is G-invariant we have f ∈ k[x0x1, ..., x0xn]. Hence

R(X)G = k[x0x1, ..., x0xn] = k[y0, ...yn−1]

Taking Proj we get the GIT quotient X//G = Pn−1. We can write down the rational morphism

ϕ in this case:

ϕ ∶ Pn =X ⇢X//G = Pn−1

Clearly the nullcone is given by

N = {[x0 ∶ ... ∶ xn] ∈ Pn ∶ x0 = 0 or (x1, ..., xn) = 0}

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Therefore we have our semistable points

Xss = ∪ni=1Xx0xi = {[x0 ∶ ... ∶ xn] ∈ Pn ∶ x0 /= 0